ACCURATE, HIGH SPEED PREDICTIVE MODELING OF PASSIVE DEVICES A Thesis Presented to The Academic Faculty by Ravi Poddar In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy In Electrical and Computer Engineering Georgia Institute of Technology January 1998 Copyright ©1998 by Ravi Poddar ACCURATE, HIGH SPEED PREDICTIVE MODELING OF PASSIVE DEVICES Approved: ____________________________________ Martin A. Brooke ____________________________________ Phillip E. Allen ____________________________________ Joy Laskar Date Approved__________________ ii DEDICATION To My Parents and Grandparents iii ACKNOWLEDGEMENT I would like to thank my thesis advisor, Dr. Martin A. Brooke, for his constant support, flexibility, flow of ideas, and patience during this research program. He has allowed me to explore new ideas, but has constantly helped me stay focused on the main objectives of the research work. His strong affiliations with other research groups, including the microwave applications, optoelectronics and MEMS groups, have allowed me to interact with and gain insight from student members and faculty from those areas. I would especially like to thank Dr. Joy Laskar of the microwave applications group and his graduate students for their assistance in the use of high frequency test equipment, which has been fundamental to the verification of this work. I would also like to thank the many students of the various groups and the Microelectronics Research Center staff for assistance in fabrication of devices, and general help and advice. Last but not least, I would like to thank my family for their constant support and encouragement during this research program. iv TABLE OF CONTENTS DEDICATION..............................................................................................................iii ACKNOWLEDGEMENT............................................................................................iv TABLE OF CONTENTS .............................................................................................. v LIST OF FIGURES....................................................................................................xiii SUMMARY ............................................................................................................ xxviii CHAPTER I Introduction............................................................................................. 1 1.1. Thesis Organization ......................................................................................... 4 CHAPTER II Background............................................................................................ 6 2.1. Introduction ..................................................................................................... 6 2.2. Analytical Models............................................................................................ 7 2.3. Measurement Based Models ............................................................................ 9 2.4. Numerical Full-Wave Methods ...................................................................... 10 2.5. Discussion ..................................................................................................... 12 2.6. Summary ....................................................................................................... 14 CHAPTER III Passive Device Modeling Methodology.............................................. 15 3.1. Introduction ................................................................................................... 15 3.2. Design and Modeling Flowchart .................................................................... 17 3.3. Building Blocks ............................................................................................. 20 v 3.4. Build Block Definition................................................................................... 25 3.5. Test Structures ............................................................................................... 28 3.6. Test Structure Equivalent Circuit Extraction .................................................. 30 3.6.1. Case Study: RLC Filter Frequency Resolution and Extraction Example......................................................................................36 3.6.2. Case Study: Partial Element Equivalent Circuit (PEEC) Extraction and Sensitivity Analysis ..............................................................38 3.6.3. Case Study: 4 Segment RLC Circuit Extraction and Sensitivity Analysis......................................................................................40 3.7. Library Based Implementation ....................................................................... 45 3.8. Summary ....................................................................................................... 46 CHAPTER IV Modeling of Resistors ......................................................................... 47 4.1. Introduction ................................................................................................... 47 4.2. Modeling Procedure....................................................................................... 49 4.3. Detailed Resistor Modeling Procedure ........................................................... 50 4.4. Processing and Measurement ......................................................................... 55 4.5. Modeling and Parameter Extraction ............................................................... 57 4.5.1. Sensitivity Analysis ............................................................................58 4.5.2. Model Extraction................................................................................66 4.6. Results........................................................................................................... 71 4.7. Summary ....................................................................................................... 79 CHAPTER V Modeling of Interdigital Capacitors.................................................... 80 5.1. Introduction ................................................................................................... 80 vi 5.2. Modeling Procedure....................................................................................... 81 5.3. Detailed Modeling Procedure......................................................................... 82 5.4. Processing and Measurement ......................................................................... 88 5.5. Modeling and Parameter Extraction ............................................................... 90 5.5.1. Sensitivity Analysis ............................................................................91 5.5.2. Model Extraction................................................................................97 5.6. Results......................................................................................................... 102 5.7. Conclusion................................................................................................... 107 CHAPTER VI Modeling of Planar Spiral Inductors............................................... 108 6.1. Introduction ................................................................................................. 108 6.2. Modeling Procedure..................................................................................... 109 6.3. Detailed Modeling Procedure....................................................................... 111 6.4. Method-of-Moments Simulation .................................................................. 116 6.5. Modeling and Parameter Extraction ............................................................. 116 6.5.1. Sensitivity Analysis .......................................................................... 117 6.5.2. Model Extraction.............................................................................. 128 6.6. Results......................................................................................................... 134 6.7. Conclusion................................................................................................... 143 CHAPTER VII Modeling of Fully 3-Dimensional Passive Device........................... 144 7.1. Introduction ................................................................................................. 144 7.2. Modeling Procedure..................................................................................... 147 7.3. Detailed LTCC Structure Modeling Procedure............................................. 148 7.3.1. Solenoid Inductor and Gridded Plate Capacitor Building Blocks ...... 151 vii 7.4. Solenoid Inductor and Gridded Plate Capacitor Test Structures.................... 154 7.5. Structure Fabrication and Measurement ....................................................... 157 7.6. Modeling and Parameter Extraction ............................................................. 164 7.6.1. Sensitivity Analysis .......................................................................... 165 7.6.2. Model Extraction.............................................................................. 178 7.7. Results......................................................................................................... 185 7.8. Summary ..................................................................................................... 191 CHAPTER VIII Conclusions and Recommendations ............................................. 192 8.1. Summary of Research and General Conclusions........................................... 192 8.2. Discussion ................................................................................................... 195 8.2.1. Test Structure Design ....................................................................... 195 8.2.2. Number of Test Structures................................................................ 196 8.2.3. Simultaneous Optimization............................................................... 196 8.3. Recommendations........................................................................................ 197 8.3.1. Recommendations for Building Blocks............................................. 197 8.3.2. Recommendations for Test Structure Design .................................... 198 8.3.3. Recommendations for Statistical Modeling....................................... 198 8.3.4. Recommendations for Parameter Extraction and Optimization ......... 199 8.3.5. Recommendations for Implementation ............................................. 199 8.4. Final Conclusions ........................................................................................ 199 APPENDIX A Sensitivity Analysis of 4 Segment RLC Circuit ............................... 201 A.1. Introduction ................................................................................................ 201 APPENDIX B Current Flow Visualization Software............................................... 212 viii B.1. Introduction................................................................................................. 212 B.2. Algorithm.................................................................................................... 214 B.2.1. Network Solution Methodology ....................................................... 216 B.2.2. Mathematical Implementation.......................................................... 218 B.3. Visualization Results................................................................................... 222 B.4. Source Code................................................................................................ 228 B.4.1. Fundamental Structure Geometry Input and Matrix Generator ......... 228 B.4.2. Input and Output Point Definition Routine and Solver ..................... 230 B.4.3. Linear Solver Routine ...................................................................... 231 B.4.4. Nodal Elimination Routine............................................................... 231 APPENDIX C HSPICE CIRCUIT OPTIMIZATION............................................. 233 C.1. Introduction................................................................................................. 233 C.2. Input File Parameters .................................................................................. 234 C.2.1. Desired Goal Definition ................................................................... 234 C.2.2. Definition of Circuit Parameters for Optimization............................ 236 C.2.3. Criteria for Successful Optimization ................................................ 237 C.2.4. Optimization Execution ................................................................... 237 C.3. Complete Optimization Control File Example ............................................. 238 C.4. Simultaneous S-parameter Circuit Optimization .......................................... 240 APPENDIX D Circuits and Data for Serpentine Resistor Modeling ...................... 245 D.1. Introduction ................................................................................................ 245 D.2. Test Structure 1........................................................................................... 246 D.2.1. Circuit Optimization Input File ........................................................ 246 ix D.2.2. Measured S-Parameter Data............................................................. 248 D.3. Test Structure 2........................................................................................... 249 D.3.1. Circuit Optimization Input File ........................................................ 249 D.3.2. Measured S-Parameter Data............................................................. 254 D.4. 9-Segment Resistor ..................................................................................... 256 D.4.1. Circuit File ...................................................................................... 256 D.4.2. Measured S-Parameter Data............................................................. 265 APPENDIX E Circuits and Data for Interdigital Capacitor Modeling .................. 271 E.1. Introduction................................................................................................. 271 E.2. Test Structure 1 ........................................................................................... 271 E.2.1. Circuit Optimization Input File ........................................................ 271 E.2.2. Measured S-Parameter Data............................................................. 273 E.3. Test Structure 2 ........................................................................................... 277 E.3.1. Circuit Optimization Input File ........................................................ 277 E.3.2. Measured S-Parameter Data............................................................. 280 E.4. Test Structure 3 ........................................................................................... 282 E.4.1. Circuit Optimization Input File ........................................................ 282 E.4.2. Measured S-Parameter Data............................................................. 285 E.5. 10-Segment Interdigital Capacitor ............................................................... 290 E.5.1. 9-Segment Resistor and 10-Segment Interdigital Capacitor Series Equivalent Circuit ................................................................................ 290 E.5.2. Measured S-Parameter Data for 10-Segment Interdigital Capacitor .. 294 x E.5.3. Voltage Magnitude and Phase of RC Series Circuit Terminated in 50 Ohm Resistor ....................................................................................... 296 APPENDIX F Circuits and Data for Planar Spiral Inductor Modeling ................. 302 F.1. Introduction ................................................................................................. 302 F.2. Test Structure 1 ........................................................................................... 303 F.2.1. Circuit Optimization Input File......................................................... 303 F.2.2. Method-of-Moments S-Parameter Data ............................................ 304 F.3. Test Structure 2 ........................................................................................... 312 F.3.1. Circuit Optimization Input File......................................................... 312 F.3.2. Method-of-Moments S-Parameter Data ............................................ 314 F.4. Test Structure 3 ........................................................................................... 323 F.4.1. Circuit Optimization Input File......................................................... 323 F.4.2. Method-o-Moments S-Parameter Data ............................................. 327 F.5. 4-Turn Spiral Inductor ................................................................................. 335 F.5.1. Circuit File for 4-Turn Spiral Inductor.............................................. 335 F.5.2. Method-of-Moments S-Parameter Data ............................................ 344 APPENDIX G Circuits and Data for Low Temperature Cofired Ceramic (LTCC) Structures Modeling.................................................................................................. 348 G.1. Introduction ................................................................................................ 348 G.2. Test structure 1 ........................................................................................... 349 G.2.1. Circuit Optimization Input File ........................................................ 349 G.2.2. S-Parameter Measured Data............................................................. 351 G.3. Test Structure 2........................................................................................... 352 xi G.3.1. Circuit Optimization Input File ........................................................ 352 G.3.2. Measured S-Parameter Data............................................................. 354 G.4. Test Structure 3........................................................................................... 359 G.4.1. Circuit Optimization Input File ........................................................ 359 G.4.2. Measured S-Parameter Data............................................................. 362 G.5. Test Structure 4........................................................................................... 366 G.5.1. Circuit Optimization Input File ........................................................ 366 G.5.2. Measured S-Parameter Data............................................................. 368 G.6. Solenoid Inductors - 4 Coils, with 6 and 8 Turns per Coil........................... 373 G.6.1. Inductor Equivalent Circuit.............................................................. 373 G.6.2. Measured S-Parameter Data............................................................. 377 G.7. Capacitor Benchmark Structure................................................................... 382 G.7.1. Equivalent Circuit............................................................................ 382 G.7.2. Measured S-Parameter Data............................................................. 384 REFERENCES.......................................................................................................... 390 xii LIST OF FIGURES Figure 3.2-1. Design and Modeling Flowchart...............................................................19 Figure 3.3-1. Uncoupled and coupled PEEC circuits with associated building blocks ....23 Figure 3.3-2. Interdigital capacitor segment end piece and possible second order (nearest neighbor coupling) equivalent circuit.....................................................................24 Figure 3.4-1. Serpentine resistors designed from the same set of building blocks...........27 Figure 3.5-1. Test structures for serpentine resistor modeling ........................................29 Figure 3.6-1. Test structure for initial guess computation...............................................36 Figure 3.6-2. Voltage gain of low pass filter ..................................................................37 Figure 3.6-3. Expanded view of overshoot region in gain response of low pass filter .....37 Figure 3.6-4. Sensitivity plot of Z11 for PEEC circuit.....................................................40 Figure 3.6-5. Circuit for impedance parameter sensitivity analysis.................................41 Figure 3.6-6. Sensitivity analysis of Z11 for circuit of Figure 3.6-5 ................................44 Figure 3.6-7. Sensitivity analysis of Z21 for circuit of Figure 3.6-5................................44 Figure 4.3-1. Current distribution plot of serpentine resistor structure............................52 Figure 4.3-2. Enlarged view of U shaped section of serpentine resistor..........................53 xiii Figure 4.3-3. Current contour plot showing current crowding effects in serpentine resistor ..............................................................................................................................53 Figure 4.3-4. Test structures and primitives for meander resistor modeling....................54 Figure 4.4-1. Photograph of fabricated structures for meander resistor modeling. ..........56 Figure 4.4-2. Photograph of predictively modeled 9 segment resistor ............................57 Figure 4.5-1. S21 and S11 sensitivity with respect to line inductance in the uncoupled square building block in test structure 1. ................................................................59 Figure 4.5-2. S21 and S11 sensitivity with respect to capacitance to ground in the uncoupled square building block for test structure 1...............................................59 Figure 4.5-3. S21 and S11 sensitivity with respect to line resistance in the uncoupled square building block in test structure 1. ................................................................60 Figure 4.5-4. S21 and S11 sensitivity with respect to shunt capacitance in the uncoupled square building block in test structure 1. ................................................................60 Figure 4.5-5. S21 and S11 sensitivity with respect to capacitance to ground in the probe pad building block in test structure 1......................................................................61 Figure 4.5-6. S21 and S11 sensitivity with respect to line inductance in the probe pad building block in test structure 1. ...........................................................................61 Figure 4.5-7. S21 and S11 sensitivity with respect to line resistance in the probe pad building block test structure 1. ...............................................................................62 xiv Figure 4.5-8. S21 and S11 sensitivity with respect to shunt capacitance in the probe pad building block in test structure 1. ...........................................................................62 Figure 4.5-9. S21 and S11 sensitivity with respect to capacitance to ground in the coupled squares building block in test structure 2................................................................63 Figure 4.5-10. S21 and S11 sensitivity with respect to mutual inductance in the coupled squares building block in test structure 2................................................................63 Figure 4.5-11. S21 and S11 sensitivity with respect to coupling capacitance in the coupled squares building block in test structure 2................................................................64 Figure 4.5-12. S21 and S11 sensitivity with respect to capacitance to ground in the Ushaped building block in test structure 2. ...............................................................64 Figure 4.5-13. S21 and S11 sensitivity with respect to line resistance in the U-shaped building block in test structure 2. ...........................................................................65 Figure 4.5-14. S21 and S11 sensitivity with respect to line inductance in the U-shaped building block in test structure 2. ...........................................................................65 Figure 4.5-15. S21 and S11 sensitivity with respect to shunt capacitance in the U-shaped building block in test structure 2. ...........................................................................66 Figure 4.5-16. Building blocks, equivalent circuits and parameter values for serpentine resistor modeling. ..................................................................................................68 Figure 4.5-17. Measured vs. modeled results for test structure 1. (a) S21 real and imaginary response. (b) S11 real and imaginary response. .......................................69 xv Figure 4.5-18. Measured vs. modeled results for test structure 2. (a) S21 real and imaginary response. (b) S11 real and imaginary response........................................70 Figure 4.6-1. Serpentine resistor and associated building blocks. ...................................72 Figure 4.6-2. Measured vs. predicted results for 9 segment resistor. (a) S21 real and imaginary response. (b) S11 real and imaginary response........................................73 Figure 4.6-3. Resistor divider circuit..............................................................................74 Figure 4.6-4. MDS generated vs. predicted results for voltage divider circuit. (a) Voltage magnitude response. (b) Voltage phase response....................................................75 Figure 4.6-5. 6 Segment LC circuit with 9 segment resistor used as termination. ...........76 Figure 4.6-6. MDS generated vs. predicted results for 6 segment LC circuit with resistive termination. (a) S21 real and imaginary response. (b) S11 real and imaginary response.................................................................................................................77 Figure 4.6-7. MDS generated vs. results using ideal 17.88Ω resistor for 6 segment LC circuit. (a) S21 real and imaginary response. (b) S11 real and imaginary response....78 Figure 5.3-1. Interdigital capacitor and associated building blocks.................................84 Figure 5.3-2. Contour and indexed color intensity plots of current distribution in ladder shaped structure.....................................................................................................85 Figure 5.3-3. Contour plot of current in T-shaped section within ladder structure. .........86 Figure 5.3-4. Test structures and building blocks for interdigital capacitor and serpentine resistor modeling. ..................................................................................................87 xvi Figure 5.4-1. Fabricated interdigital capacitor - test structure 3......................................89 Figure 5.4-2. Fabricated interdigital capacitor – 10 segment capacitor predictively modeled.................................................................................................................89 Figure 5.4-3. Fabricated RC structure predictively modeled...........................................90 Figure 5.5-1. S21 and S11 sensitivity of test structure 3 with respect to line to line coupling capacitance (CM)...................................................................................................92 Figure 5.5-2. S21 and S11 sensitivity of test structure 3 with respect to stub to line coupling capacitance (C2) in the shielded stub. ....................................................................93 Figure 5.5-3. S21 and S11 sensitivity of test structure 3 with respect to line to ground capacitance in the IDC fingers. ..............................................................................93 Figure 5.5-4. S21 and S11 sensitivity of test structure 3 with respect to line to ground capacitance in the shielded stub. ............................................................................94 Figure 5.5-5. S21 and S11 sensitivity of test structure 3 with respect to line inductance in finger segments......................................................................................................94 Figure 5.5-6. S21 and S11 sensitivity of test structure 3 with respect to line inductance in shielded stub..........................................................................................................95 Figure 5.5-7. S21 and S11 sensitivity of test structure 3 with respect to line to line mutual inductance between finger segments. .....................................................................95 Figure 5.5-8. S21 and S11 sensitivity of test structure 3 with respect to line resistance in finger segments......................................................................................................96 xvii Figure 5.5-9. S21 and S11 sensitivity of test structure 3 with respect to line resistance in shielded stub..........................................................................................................96 Figure 5.5-10. Building blocks, equivalent circuits, and parameters for IDC and resistor modeling................................................................................................................98 Figure 5.5-11. S-parameter measured and modeled results for test structure 1................99 Figure 5.5-12. S-parameter measured and modeled results for test structure 2.............. 100 Figure 5.5-13. S-parameter measured and modeled results for test structure 3.............. 101 Figure 5.6-1. RLC resonant tank circuit....................................................................... 104 Figure 5.6-2. Measured and predicted results for 10 segment interdigital capacitor...... 105 Figure 5.6-3. Actual (MDS) and predicted resonator voltage magnitude and phase...... 106 Figure 6.3-1. Spiral inductor and associated building blocks........................................ 112 Figure 6.3-2. Indexed color intensity plots of current distribution in spiral inductor..... 113 Figure 6.3-3. Contour plot of X and Y directed current gradients showing current crowding in spiral inductor. ................................................................................. 114 Figure 6.3-4. Test structures and building blocks for spiral inductor modeling............. 115 Figure 6.5-1. S21 and S11 sensitivity of test structure 1 with respect to line-to-ground capacitance in the uncoupled square building block. ............................................ 119 Figure 6.5-2. S21 and S11 sensitivity of test structure 1 with respect to line inductance in the uncoupled square building block. ................................................................... 119 xviii Figure 6.5-3. S21 and S11 sensitivity of test structure 1 with respect to line resistance in the uncoupled square building block.......................................................................... 120 Figure 6.5-4. S21 and S11 sensitivity of test structure 1 with respect to shunt capacitance in the uncoupled square building block. ................................................................... 120 Figure 6.5-5. S21 and S11 sensitivity of test structure 2 with respect to line-to-line coupling capacitance in the coupled squares building block................................................ 121 Figure 6.5-6. S21 and S11 sensitivity of test structure 2 with respect to line-to-ground capacitance in the U building block...................................................................... 121 Figure 6.5-7. S21 and S11 sensitivity of test structure 2 with respect to shunt capacitance in the U building block. ........................................................................................... 122 Figure 6.5-8. S21 and S11 sensitivity of test structure 2 with respect to line-to-line mutual inductance in the coupled squares building block................................................. 122 Figure 6.5-9. S21 and S11 sensitivity of test structure 2 with respect to line-to-ground capacitance in the coupled squares building block................................................ 123 Figure 6.5-10. S21 and S11 sensitivity of test structure 2 with respect to line inductance in the U-shaped building block. ............................................................................... 123 Figure 6.5-11. S21 and S11 sensitivity of test structure 3 with respect to line inductance in the uncoupled squares section of the coupled corner building block. .................... 124 Figure 6.5-12. S21 and S11 sensitivity of test structure 3 with respect to line resistance in the uncoupled squares section of the coupled corner building block. .................... 124 xix Figure 6.5-13. S21 and S11 sensitivity of test structure 3 with respect to shunt capacitance in the uncoupled squares section of the coupled corner building block. ................ 125 Figure 6.5-14. S21 and S11 sensitivity of test structure 3 with respect to line-to-line coupling capacitance in the coupled corner building block................................... 125 Figure 6.5-15. S21 and S11 sensitivity of test structure 3 with respect to line-to-ground capacitance in the coupled corner building block. ................................................ 126 Figure 6.5-16. S21 and S11 sensitivity of test structure 3 with respect to shunt capacitance in the coupled corner building block. ................................................................... 126 Figure 6.5-17. S21 and S11 sensitivity of test structure 3 with respect to line-to-line mutual inductance in the coupled corner building block................................................... 127 Figure 6.5-18. S21 and S11 sensitivity of test structure 3 with respect to line inductance in the coupled corner building block. ....................................................................... 127 Figure 6.5-19. S21 and S11 sensitivity of test structure 3 with respect to line resistance in the coupled corner building block. ....................................................................... 128 Figure 6.5-20. Building blocks, equivalent circuits, and parameters for spiral inductor modeling.............................................................................................................. 130 Figure 6.5-21. S-parameter measured and modeled results for test structure 1.............. 131 Figure 6.5-22. S-parameter measured and modeled results for test structure 2.............. 132 Figure 6.5-23. S-parameter measured and modeled results for test structure 3.............. 133 Figure 6.6-1. 4 turn spiral inductor predictively modeled............................................. 135 xx Figure 6.6-2. Z-parameter circuit configurations for inductor analysis (top) MDS configuration (bottom) circuit predictive model configuration.............................. 137 Figure 6.6-3. Measured and predicted results for Z11(dB) of four turn spiral inductor. 138 Figure 6.6-4. Measured and predicted results for Z11(phase) of four turn spiral inductor. ............................................................................................................................ 138 Figure 6.6-5. LC resonant tank circuit.......................................................................... 139 Figure 6.6-6. Actual (MDS) and predicted LC circuit output voltage magnitude. ......... 140 Figure 6.6-7. Actual (MDS) and predicted LC circuit output voltage phase. ................ 140 Figure 6.6-8. LC circuit with 2 4-turn inductors in parallel. ......................................... 141 Figure 6.6-9. Actual (MDS) and predicted LC circuit output voltage magnitude. ......... 142 Figure 6.6-10. Actual (MDS) and predicted LC circuit output voltage phase................ 142 Figure 7.3-1. Solenoid inductor geometry.................................................................... 150 Figure 7.3-2. Gridded plate capacitor geometry. .......................................................... 151 Figure 7.3-3. Solenoid inductor building blocks........................................................... 153 Figure 7.3-4. Gridded plate capacitor building block. ................................................. 154 Figure 7.4-1. Test structures for solenoid inductor modeling........................................ 156 Figure 7.4-2. Additional test structure for gridded plate capacitor modeling................. 157 Figure 7.5-1. Physical layout of LTCC coupon............................................................ 159 Figure 7.5-2. Photograph of top side of fabricated LTCC coupon. ............................... 160 xxi Figure 7.5-3. Photograph of bottom side of LTCC coupon with last embedded layer partially visible. ................................................................................................... 161 Figure 7.5-4. Photograph of cross section of metal line in a LTCC structure along the line length (photograph courtesy of National Semiconductor Corp.) ........................... 162 Figure 7.5-5. Photograph of cross section of metal line across line width (short) (photograph courtesy of National Semiconductor Corp.)...................................... 163 Figure 7.5-6. Photograph of cross section of 2 via stack (photograph courtesy of National Semiconductor Corp.).......................................................................................... 164 Figure 7.6-1. S11 and S21 sensitivity responses of test structure 1 with respect to capacitance to ground in the interconnect line building block............................... 167 Figure 7.6-2. S11 and S21 sensitivity responses of test structure 1 with respect to line inductance in the interconnect line building block................................................ 167 Figure 7.6-3. S11 and S21 sensitivity responses of test structure 1 with respect to capacitance-to-ground in the probe pad building block......................................... 168 Figure 7.6-4. S11 and S21 sensitivity responses of test structure 1 with respect to line inductance in the probe pad building block. ......................................................... 168 Figure 7.6-5. S11 and S21 sensitivity responses of test structure 2 with respect to capacitance-to-ground of the top conductor in the inductor coil building block. ... 169 Figure 7.6-6. S11 and S21 sensitivity responses of test structure 2 with respect to line inductance of the top conductor in the inductor coil building block. ..................... 169 xxii Figure 7.6-7. S11 and S21 sensitivity responses of test structure 2 with respect to line resistance of the top conductor in the inductor coil building block. ...................... 170 Figure 7.6-8. S11 and S21 sensitivity responses of test structure 2 with respect to line conductance of the top conductor in the inductor coil building block. .................. 170 Figure 7.6-9. S11 and S21 sensitivity responses of test structure 2 with respect to line-toground capacitance of the bottom conductor in the inductor coil building block... 171 Figure 7.6-10. S11 and S21 sensitivity responses of test structure 2 with respect to line inductance of the bottom conductor in the inductor coil building block. ............... 171 Figure 7.6-11. S11 and S21 sensitivity responses of test structure 2 with respect to line resistance of the bottom conductor in the inductor coil building block. ................ 172 Figure 7.6-12. S11 and S21 sensitivity responses of test structure 2 with respect to line conductance to ground of the bottom conductor in the inductor coil building block. ............................................................................................................................ 172 Figure 7.6-13. S11 and S21 sensitivity responses of test structure 2 with respect to via capacitance in the inductor coil building block..................................................... 173 Figure 7.6-14. S11 and S21 sensitivity responses of test structure 2 with respect to via inductance in the inductor coil building block. ..................................................... 173 Figure 7.6-15. S11 and S21 sensitivity responses of test structure 3 with respect to coupling capacitance in the interacting inductor coil building block. .................................. 174 xxiii Figure 7.6-16. S11 and S21 sensitivity responses of test structure 3 with respect to line-toline mutual inductance in the interacting inductor coil building block. ................. 174 Figure 7.6-17. S11 and S21 sensitivity responses of test structure 4 with respect to capacitance-to-ground of the top plate in the gridded capacitor building block..... 175 Figure 7.6-18. S11 and S21 sensitivity responses of test structure 4 with respect to capacitance-to-ground of the bottom plate in the gridded capacitor building block. ............................................................................................................................ 175 Figure 7.6-19. S11 and S21 sensitivity responses of test structure 4 with respect to mutual capacitance between the plates in the gridded capacitor building block. ............... 176 Figure 7.6-20. S11 and S21 sensitivity responses of test structure 4 with respect to mutual inductance between the plates in the gridded capacitor building block. ................ 176 Figure 7.6-21. S11 and S21 sensitivity responses of test structure 4 with respect to line inductance for both plates in the gridded capacitor building block........................ 177 Figure 7.6-22. S11 and S21 sensitivity responses of test structure 4 with respect to line resistance for both plates in the gridded capacitor building block. ........................ 177 Figure 7.6-23. Z-parameter MDS circuit configuration for inductor and capacitor analysis................................................................................................................ 179 Figure 7.6-24. Building blocks, equivalent circuits and parameter values for solenoid inductor and gridded plate capacitor modeling. .................................................... 180 xxiv Figure 7.6-25. Measured vs. modeled results for test structure 1. (a) S21 real and imaginary response. (b) S11 real and imaginary response...................................... 181 Figure 7.6-26. Measured vs. modeled results for test structure 2. (a) Z11 magnitude response. (b) Z11 phase response. ......................................................................... 182 Figure 7.6-27. Measured vs. modeled results for test structure 3. (a) Z11 magnitude response. (b) Z11 phase response. ......................................................................... 183 Figure 7.6-28. Measured vs. modeled results for test structure 4. (a) Z11 magnitude response. (b) Z11 phase response. ......................................................................... 184 Figure 7.7-1. Fabricated solenoid inductors. ................................................................ 185 Figure 7.7-2. Measured and predicted results for Z11 (dB) for 4-coil, 6 turn per coil inductor. .............................................................................................................. 187 Figure 7.7-3. Measured and predicted results for Z11 (phase) for 4-coil, 6 turn per coil inductor. .............................................................................................................. 187 Figure 7.7-4. Measured and predicted results for Z11 (dB) for 4-coil, 8 turn per coil inductor. .............................................................................................................. 188 Figure 7.7-5. Measured and predicted results for Z11(phase) for 4-coil, 8 turn per coil inductor. .............................................................................................................. 188 Figure 7.7-6. Large gridded plate capacitor used to test capacitor building block model validity. ............................................................................................................... 189 Figure 7.7-7. Measured and predicted results for Z11 (dB) for large capacitor. ............. 190 xxv Figure 7.7-8. Measured and predicted results for Z11(phase) for large capacitor. .......... 190 Figure A.1-1 Circuit for impedance parameter sensitivity analysis............................... 202 Figure A.1-2. Z11 and Z21 real and imaginary components for RLC circuit.................. 207 Figure A.1-3. Z11 sensitivity with respect to C and L for RLC circuit, real and imaginary parts. ................................................................................................................... 208 Figure A.1-4. Z11 sensitivity with respect to R and R2 for RLC circuit, real and imaginary parts. ................................................................................................................... 209 Figure A.1-5. Z21 sensitivity with respect to C and L for RLC circuit, real and imaginary parts. ................................................................................................................... 210 Figure A.1-6. Z21 sensitivity with respect to R and R2 for RLC circuit, real and imaginary parts. ................................................................................................................... 211 Figure B.1-1. Possible corner building block and usage in two structures .................... 213 Figure B.2-1. Representative impedance grid. Each box represents and impedance...... 215 Figure B.2-2. Definition of S-shaped region on impedance grid................................... 216 Figure B.2-3. Impedance and corresponding entries in MNA matrix............................ 217 Figure B.2-4. Mapping operation between computed voltage vector and 2D voltage matrix for actual geometrical structure being analyzed......................................... 219 Figure B.2-5. MNA matrix sparsity pattern for serpentine resistor analysis.................. 220 xxvi Figure B.2-6 Contour and indexed image plots of current distribution for two different geometry bends.................................................................................................... 221 Figure B.3-1. Indexed current intensity plot of gridded structure. ................................ 223 Figure B.3-2. Current gradient magnitude contour plot................................................ 224 Figure B.3-3. Contour plots of X and Y directed current gradients showing current crowding effects. ................................................................................................. 225 Figure B.3-4. Current profile plot through axis A-A’. .................................................. 226 Figure B.3-5. Current profile plot through axis B-B’.................................................... 227 Figure B.3-6. Current profile plot through axis C-C’.................................................... 227 xxvii SUMMARY A novel procedure is presented for accurate, high frequency electrical behavior predictive modeling of passive devices with interactions. The developed method is based upon defining structural building blocks and equivalent circuits, associating design rules with them, and characterizing them through the use of test structures. The test structures are designed such that they are comprised only of sensitive combinations of defined building blocks, and they are measured over a wide band of frequencies using network analysis techniques. Building block equivalent circuit models are derived from the measured test structure data by nonlinear optimization methods. The method has been experimentally verified for all different classes of passive devices, including resistors, capacitors, and inductors, in both planar and 3-dimensional configurations. The method has also been verified on circuits using these components, with good results obtained in both cases. xxviii CHAPTER I INTRODUCTION Advances in technology are making possible systems that are faster and more powerful than ever before. Research and development in academia and industry are constantly finding new ways to improve system performance. Most of the advances have been in the area of integrated circuit (IC) technology and fabrication. While on-chip frequencies are already at the 500 MHz level in commercial products, such as in the Alpha microprocessor developed at Digital Equipment Corporation, communication with off-chip devices such as DRAM still occur at a much reduced rate. In fact, frequencies at the board level at even the 100-200 MHz range are difficult to obtain cheaply, mainly due to the parasitic effects of simple interconnect lines on the printed circuit board. While most systems do include a number of integrated circuits, they usually also include a large number of passive devices. In general, the majority of passive devices that are required are kept off chip and outside the package, in order to reduce costs and to reserve area for active structures. In some cases, however, such as in analog and RF chips, vital passive devices may be integrated on chip. This generally is required when accurate passives are required, or when high speed signals must be propagated through these structures, as would be the case for resistors in a resistor utilizing digital to analog converter, or for capacitors and inductors in a high-Q cutoff filter or oscillator. 1 Increasing on-chip transistor count is allowing more functionality to be integrated on to a single integrated circuit. This has led to combining functions of several separate chips into one IC, thereby eliminating the need for other ICs altogether. This integration has lead to considerable reductions in overall board space requirements, with overall active component counts dropping by significant percentages. Passive component board area has actually increased as a percentage of overall board space in recent times, and is now becoming a limiting factor for further reductions in board size. Reducing printed circuit board area would result in much smaller, lighter, and more reliable systems which could potentially impact every component using electronic circuits. The advantages of miniaturization have driven the development of new technologies to remove passives from on top of the printed circuit board, and instead to embed them within a substrate or a package. Processes such as thick and thin film processing have allowed for the deposition of materials at mil and micron level linewidths and spacings within multichip modules, and techniques such as low temperature cofired ceramic (LTCC) processing allow for thick film printing and stacking of passive devices in a multilayer (well over 30 layers are possible), low-cost substrate. LTCC fabrication techniques show tremendous promise for integration of a large number of passive components into a multilayer ceramic substrate, with the possibility of combining it along with an integrated circuit within a standard IC size package. Passive devices manufactured in these technologies take on certain representative shapes. Resistors, for example, are usually designed in straight lines or in a serpentine 2 fashion, while capacitors are usually designed with interdigitated fingers in the planar form, and as a parallel plate device in 3 dimensions. Inductors too have basically two shapes, one is planar spiral structure, while a 3 dimensional implementation involves generating a solenoid with two different layers of metal and many deep vias. In order to utilize any new technology efficiently for design work, good behavior predictions of the various components involved is very important. In integrated circuit design work, for example, good models for transistors are crucial to help obtain fabricated circuits that match designed specifications. In this area itself, transistor models are in continuous development, and based on technology enhancements, have evolved from SPICE level I, II, and III, to the current BSIM family of models. In the same manner, accurate, frequency dependent, wide band models of passive components are very important for successful high speed circuit design. Most practical passive devices have complex geometries, nonuniform current flow, and correspondingly complex field patterns. All passive components suffer from parasitic effects which can affect the electrical behavior of the device at different frequencies. In addition, for small planar and 3 dimensional structures, the structures can become electrically long above some frequency, and start exhibiting transmission line behavior. Coupling effects within the structure itself can also affect performance, and this kind of behavior can easily manifest itself in passive components where many long lines run adjacent to each other, as is the case in serpentine resistors. All of these phenomena, coupled with non-ideal processing effects, make predictive modeling of such structures very difficult. 3 1.1. Thesis Organization In this dissertation, a novel method of modeling of passive structures will presented, with several case studies examined in detail. The first part of this thesis is dedicated to the development and procedures of the method, while the second section shows the application of the method to the predictive modeling of the three main classes of passive devices, namely the resistor, capacitor, and inductor. Procedures and results are shown initially for planar devices. Modeling of fully 3-dimensional devices with interactions is also examined, and good results are shown for devices fabricated in a state-of-the-art multilayer low temperature cofired ceramic substrate process. A brief chapter-by-chapter outline of the thesis is given below. Chapter 2 discusses the background and origin of the problem, and presents some of the major work already performed in this area. As will be seen, most of the work originates from the microwave engineering arena, where planar passive structures have been used for a considerable length of time and at high frequencies for microwave/RF applications. Chapter 3 presents a detailed description of the passive predictive modeling methodology developed under this research program. Chapter 4 shows the application of the method to the modeling of planar serpentine resistor structures. The results are compared against measurements, and are verified up to 5-10 GHz frequency range. Chapter 5 also shows application of the method, in this case to interdigital capacitors, and combined capacitor resistor circuits. Again, results are verified against measurements. Chapter 6 shows application of the method to the modeling of planar spiral inductors. 4 Results are obtained and verified against simulations obtained from a Method of Moments simulator. Chapter 7 presents the application of the method to full 3 dimensional structures manufactured in a thick film low temperature cofired ceramic process. The structures modeled include gridded plate capacitors and solenoid inductors in both series and parallel configurations with interactions. The results obtained are compared against actual measurements with good agreement up to the first resonance. Chapter 8 draws conclusions regarding this work and provides recommendations for further research. Several appendices are also included which document some detailed circuit sensitivity analysis; the development of a current visualization tool; optimization procedures; and the various circuit models and measured data of passive device test and benchmark structures studied under this research work. 5 CHAPTER II BACKGROUND 2.1. Introduction As presented in Chapter 1, accurate modeling of integrated passive components is becoming very important for the successful design and fabrication of compact, high performance systems that utilize such devices. Highly miniaturized passive components have been used extensively in the microwave/RF community and have been fabricated on GaAs, and high speed silicon substrates for use in microwave circuits. Clearly, at microwave frequencies, good frequency dependent models of passive components must be obtained for successful design. As a result of this requirement, much of the work in the area of passive component modeling originates from the microwave engineering community. Modeling of miniature passive components usually falls into three categories; analytical equation based models; measurement based models, and numerical full wave electromagnetic models. There has also been some interesting work published using neural networks for the modeling of passive devices. In this chapter, an overview of the various methods will be presented; details can be found in the various references. 6 2.2. Analytical Models In this section passive component equation models that are based on fundamental principles are discussed. There are very few entirely analytical models that do not require any kind of special numerical computation, such as numerical integration, except for very simple structures. Most of the methods do require some form of numerical analysis, but do not require gridding, and solving large matrix equations, as is the case for the direct full wave methods. Many analytical expressions have also been derived from simulation and curve fitting, and not directly from first principles. It is neither practical nor useful to present all analytical formulas and methods for modeling of passive components, but some representative results are discussed. Expressions for electrically small straight line, circular, and rectangular inductors can be found in [1]. Completely analytical results (without any numerical techniques involved) are given for the inductance of a straight line, taking into account metal thickness, and also attempts to take into account high frequency skin effect losses. Analytical equations are also provided for the single loop, circular and rectangular inductors, but all of these require numerical methods involving numerical integration and the use of elliptical integrals. A generalization of the method described in [1] for the modeling of multilayer spiral inductors is given in [2]. Models for circular spiral inductors are also presented in [3], which is based on some early work on exact evaluation of inductance of circular line segments. This paper also presents an overview of the various methods for inductor 7 modeling that had been attempted earlier, and the reader is referred to its list of references for further information. Another paper based on early work [4] is presented in [5], in which the author presents some modifications to a very early empirically derived formula for square spiral inductor modeling, with good results. Capacitor models have been shown in various papers. Parallel plate capacitor models are usually modeled using microstrip coupled lines and conformal mapping theory [1]. Interdigital capacitors have been modeled within a microwave CAD package using built in models for coupled microstrips, T-junctions, etc. [1] These models themselves are based upon curve fit or table lookup models derived from full wave simulations or measurements of the various pieces [6]. Multilayer interdigital capacitors have also been modeled analytically using complex conformal mapping techniques [7]. In general, there appears to be a myriad of different analytical models available for the modeling of passive devices. They clearly fall into two classes; derived from first principles and derived from curve fitting from simulation or measurement. It is interesting to note that resistive effects are either not modeled at all, or are simply included in the form of a series resistance term, since many microstrip models assume very low loss conductors. This is also probably a major factor contributing to the fact that very little work can be found on modeling of serpentine resistors, except by use of full wave methods. 8 2.3. Measurement Based Models In many cases, models of passive components are only developed after they have been fabricated and measured. This technique has many advantages, since it can be applied to any arbitrary structure, and takes into account processing effects such as nonuniform dielectric thicknesses, which would be very difficult to achieve with any other method. For high frequency applications, network analysis or time domain reflectometry (TDR) measurements are taken to characterize the device under measurement. In network analysis, a scattering parameter response over a wide band of frequencies is obtained, while for TDR a time domain voltage profile is obtained, with peaks and valleys representing capacitive or inductive discontinuities within the structure. Frequency domain S-parameter measurements allow device models to be constructed in several ways. One way is to simply use the S-parameters as a black box model of the device, and use that for design applications. Some simulators allow the use of S-parameter datasets as models (such as Hewlett-Packard MDS), but this is still not widely supported. The other approach is to fit the measured S-parameter data to a circuit model using optimization. Simple circuit representations of the various passive components can be found in [6], and these are often used in the literature. Examples of using S-parameter measurements to model passive structure are given in [8], [9] and [10]. In time domain reflectometry, a very short duration pulse is injected into the structure under test, and an effective reflected impulse response is obtained. The response usually contains peaks, valleys, oscillations, and relaxations, and these phenomena can be 9 related to inductive, capacitive, resistive or some combination discontinuities. The position of the discontinuity within the structure can also be estimated, since an earlier one will manifest itself earlier on the TDR output. Based upon the actual time scale, the relative position of the discontinuity can be determined, since the velocity of propagation is known (or can be easily found). An overview of time domain reflectometry can be found in [11]. Examples of where TDR has been used in passive component modeling can be found in [12] and [13]. 2.4. Numerical Full-Wave Methods Numerical electromagnetic full-wave simulation methods are undoubtedly the most flexible and general of all the modeling methods. These methods essentially apply Maxwell’s equations to an arbitrary geometry structure and compute electric and magnetic field patterns. The methods generally require segmenting the structure under analysis into many small pieces, and solving equations on each piece in order to obtain the response of the whole structure. Simulation time is directly related to the number of grid cells and frequency points, and as a result, simulation times can become very long for complex structures. Problems are especially difficult when there are many discontinuities in a structure, such as a gridded plate, since many grid cells have to be created to model complex behavior at the corners and edges of the discontinuities. There are several different numerical methods that have been used for full wave analysis of structures. These methods include the finite and boundary element methods, 10 the finite-difference time domain method, the method of moments, the transmission line matrix method, the 3-D spectral domain analysis [14], and the mixed potential integral equation method [15]. Some good general overview papers describing the most popular methods are [16], [17], [18] and [19]. One of the more popular methods of simulation is the dyadic Green’s function based method of moments algorithm (MoM), since it eliminates one degree of freedom by assuming infinitely thin conductors, but still allows for multilayer (conductors on different layers) simulations. This falls into the class of 2 ½-dimensional methods, however, simulations with this method can take many hours also, but are radically faster, although more inaccurate than full 3-D methods. There are many articles in the literature showing the use of this method for modeling of passive structures. Some representative ones are [20], [21], [22] and [23]. Another method that has become very popular for passive device modeling is the finite difference time domain (FDTD) method [24]. Unlike the method of moments, this is a full 3-D method, and does take into account conductor heights. Much work can be found where the FDTD method has been applied to the modeling of spiral inductors with air bridges [25], [26], and [27]. Additionally interdigital capacitors have also been modeled using the method [28]. The method has also been used to model an entire library of components, including discontinuities in [29] and [30]. For the modeling of full 3-D devices, such as those that can be manufactured in a multilayer low temperature cofired ceramic (LTCC) process, the finite element method seems to be the simulation method of choice [31] [32]. 11 In addition to the standard methods of modeling, a new neural network based approach has been presented [33]. This method is based upon training a neural network on the S-parameters of various devices, with the network inputs being design parameters such as widths and lengths. The method has shown very good results for an inductor modeling application. 2.5. Discussion All of the above methods have advantages and disadvantages. The analytical methods can be very useful if good results can be obtained for a particular process. Derivations of expressions from measured or simulated structures can be a time consuming and difficult process, and for general structure design many degrees of freedom will be required. For example, for spiral inductor modeling, the designer will at least need to vary the number of turns in an inductor and also the horizontal and vertical dimensions, yielding 3 degrees of freedom. Additionally, in order to model parasitics, even more variables are introduced, and this will tend to complicate the generation of analytical models further. However, if development time is acceptable, and expressions can be developed, then accurate analytical models are very useful and extremely fast. Models developed from measurements are very accurate, since they account for processing effects such as inhomogeneous dielectrics, uneven conductors, and similar effects which are very difficult to model with any other method. The main problem associated with this method is lack of flexibility. The generated models only apply to the 12 devices actual fabricated, with no accurate method of scaling to model other structures. The method can be used to model a large library of components, and a designer must choose components from that library alone to obtain accurate models. If the designer requires a component that does not exist in the library, it must first be fabricated and then modeled. In general, developing a comprehensive library of modeled components could be difficult and time consuming. Full wave analysis is a very useful method for investigating the electrical behavior of an arbitrary structure. The analysis is usually quite accurate if enough time is spent to input the substrate and structure geometries accurately. However, due to the meshing nature of these methods, the more complex the input system, the greater the number of mesh points, and the longer the simulation time. Even for relatively simple structures, such as planar resistors and interdigital capacitors, simulation times can run into hours for accurate high frequency simulations using a method of moments approach. Finite element simulations usually take much longer since an entire 3-dimensional volume must be meshed and solved. The use of these tools for component design is probably not very practical (although it has been attempted [34]), since design by nature is an iterative process, and each iteration could take many hours or even days of simulation time on modern workstations. 13 2.6. Summary In this chapter, an overview of the various methods of passive component design has been presented and classified. A brief discussion has also been presented, treating some of the more important issues regarding each type of method. In the next chapter a novel test structure and building block based modeling methodology that has been developed under this research program will be presented. In the chapters following that, the method will be demonstrated and verified on various types of passive structures, including resistors, capacitors, inductors, and full 3-D structures manufactured in a LTCC process. 14 CHAPTER III PASSIVE DEVICE MODELING METHODOLOGY 3.1. Introduction This chapter describes in detail the development of a new high speed, high frequency modeling methodology for passive devices with interactions [35], [36], and [37]. The method produces circuit models of structures which are constructed from equivalent circuits of building blocks. The building block equivalent circuits are derived from test structures and measurements, and thus automatically take into account effects of processing fluctuations and nonideal material properties. The generated circuit models simulate in a standard circuit solver and occur very quickly, usually on the order of minutes or seconds, thus providing a major speedup over methods that do not utilize lumped elements. The method is applicable to both electrically long and short structures, and is independent of technology or the process in which the structures are fabricated. The building block paradigm of this method as well as the production of circuit models that simulate very quickly make this method very well suited for circuit design applications. 15 Accuracy of the modeling method is solely dependent upon the accuracy of the modeled building blocks. Extremely accurate models that are valid up to very high frequencies can be obtained if long interaction distances and retardation effects are taken into account. The method is also flexible enough that circuit models do not have to be used for the building blocks. If necessary, multiport parameter black box representations can be used, although many circuit simulators do not accept direct multiport parameters as input. The fundamental idea behind the modeling procedure is that most designed passive structures are comprised of several key geometrical building blocks, that is, they can be constructed from several building block cells representing individual parts of the structure. These building blocks can de defined in a number of ways, but careful selection can result in relatively few building blocks being needed. If accurate models for each of the building blocks along with interaction information can be obtained, then any arbitrary structure comprised of those building blocks can be modeled accurately using the individual block models. Building block models are extracted by the use of test structures. Test structures comprising a complete set of the identified building blocks are designed and manufactured in the process of interest, and two port frequency measurements are performed on them by use of a network analyzer. In general, any test structure will be comprised of several different building blocks. Using the measured data, optimization and extraction routines are performed in order to extract passive RLC models for each of the embedded building blocks. These building blocks and their associated models can be 16 used to predict the behavior of other arbitrary geometry structures made in the same process as the test structures, if they are comprised of the modeled building blocks in a specified and correct manner. The method generates equivalent circuits of the devices, and the predictions are obtained simply through circuit simulation utilizing standard SPICE-like software. 3.2. Design and Modeling Flowchart The entire developed modeling methodology can be concisely described in a flow diagram. The diagram is shown in Figure 3.2-1. The details behind the method will be described at length in the remainder of the chapter, but a short description of the complete process will help clarify the process. The first step in the modeling process is to identify what sort of devices are to be modeled in a process, identify building blocks and consequently design rules. Once the building blocks have been defined, the next step is to design and fabricate test structures to characterize them. This is then followed by measurement of the test structures in order to aid building block circuit extraction. The measured data is then used to set up optimizations to extract equivalent circuits of the test structures and building blocks, and is also used to determine initial guesses. Once successful optimizations have been achieved, the building blocks with associated models and design rules are combined in a library. Once a valid library is constructed, a designer can then use it to construct a new passive device. Design rule compliance can be verified through the use of a geometry 17 based design rule checker. If the check fails, then the designer can take one of two routes – he or she can either redesign the device until it is compliant, or can attempt to generate models for the section that is causing errors by defining new building blocks and test structures and going through the characterization procedure. Once the design passes the design rule checker, then accurate models of the device are output which can then be simulated in a circuit simulator. 18 Modeling Identify Building Blocks, Define Geometries Design and Fabricate Test Structures Perform Measurements, Obtain Initial Guess Data Execute Optimizations, Extract Building Block Equivalent Circuits Associate Building Block Equivalent Circuits and Design Rules in Library/Technology File Design Design Passive Device Design Rule Check Fail Pass Accurate Model Figure 3.2-1. Design and Modeling Flowchart 19 3.3. Building Blocks The general idea behind the proposed modeling methodology is that accurately modeling small pieces of a structure will allow us to model the behavior of a larger structure composed of some combination of those pieces. This idea, but with many very small pieces, is the premise behind the classical, well understood method of finite element analysis (FEA) [38]. In FEA, a large, complex problem is broken down into a huge number of simple subproblems by segmenting the structure of interest into many pieces and applying the relevant boundary conditions or external excitations to the appropriate elements. Every piece, or element, is characterized by a relatively simple functional mapping (basis function) on its boundaries, which is continuous within itself and between adjacent elements. As the element sizes become smaller, the effect of the simple basis function reduces, and hence the results become more accurate. As is to be expected, the gain in accuracy with finite elements comes at a price. For most nontrivial structures, a large number of elements are needed, and for a threedimensional problem, the complexity of the problem increases dramatically. Computer memory requirements can very quickly become enormous, even using some of the most advanced FEA packages commercially available, with computation times leading into many hours, if not days, on some of the fastest workstations available today. In addition to huge computational expense, if the problem size exceeds available physical memory, a very large part of computer time is wasted in simply performing memory management 20 tasks, and in many cases, only a small percentage of actual CPU resources is spent on actual solution computation. In the method presented here, there is no need to segment the problem in to a large number of small pieces in order to obtain accurate results. In this method, relatively large pieces compared to finite elements are considered, even for complex current regions such as corners. The reason this is possible is that there is no assumption made about basis functions within the pieces or building blocks, since their individual behavior is derived from measurements of test structures, or if measurements are not available or possible, them from accurate simulations of the test structures utilizing exhaustive FEA, moment methods or similar procedures. Since basis functions are not assumed, the size of the block has no effect on accuracy, and thus it can be chosen to be relatively large. The main goal is to generate a function that represents the behavior of the block, and then utilize that functional description in the analysis of other structures comprised of those blocks in a specified way. The main restriction on the size of the block is our ability to model its behavior well with a relatively simple system. In a strict sense, this is not as much a restriction as it is a preference - generating intricate models to represent complex behavior is acceptable, although it may be difficult. Our objective is to be able to predict the electrical behavior of arbitrary geometry passive devices in a standard circuit simulator. In order to achieve this, each building block is modeled as a SPICE compatible RLC circuit. For relatively simple uncoupled building blocks, such as for modeling a piece of a straight line, simple RLC models based on the partial element equivalent circuit (PEEC) [39], [40] are used for equivalent 21 circuits. In the case of blocks where coupling needs to be taken into account, e.g. coupled material squares (corresponding to coupled adjacent lines or interacting material squares), coupled PEEC models are used, connected by coupling capacitances and mutual inductances (Figure 3.3-1). Although circuit models are being used here, this is not a requirement. Any functional description or data table representing each of the building blocks could be used just as well, however, this would require specialized simulators and tools which may not be readily available. The circuit level modeling approach works well in any standard SPICE compatible circuit simulator. 22 2 1 1 2 R Uncoupled Line Bldg. Block 1 CC CC L R C CC 3 CC 1 R 2 L L LM 4 Coupled Line Bldg. Block CM CC 2 R L C R 3 LM CC L C L R 4 Figure 3.3-1. Uncoupled and coupled PEEC circuits with associated building blocks The PEEC circuit has been shown to be equivalent to Maxwell's Equations for small sections of material of approximately 1/10 wavelength long. The level of coupling also refers to the level of the equivalent circuit, for example, when considering only second level coupling, that is coupling effects from nearest neighbors only, then we have second order equivalent circuits, as shown on the lower part of Figure 3.3-1. For higher order coupling effects to be taken into account, higher order building blocks and equivalent circuits are required. For example, to account for coupling from both the nearest and second nearest neighbor, we would need to include an additional PEEC 23 circuit and include capacitive couplings between all the center nodes of the circuits, and include mutual inductances between all left and right side inductors respectively. For more complex building blocks, however, equivalent circuits are derived on a per case basis. This would be required, for example, in the case of a building block representing one square of material surrounded on three sides by material, as might be found at the ends of segments in interdigital capacitors. A possible circuit representing this structure is shown in Figure 3.3-2. 1 1 2 2 3 3 Figure 3.3-2. Interdigital capacitor segment end piece and possible second order (nearest neighbor coupling) equivalent circuit Clearly, the circuit model for the capacitor end piece is fairly complex, but even so, certain behavior may not be adequately predicted using such a model. In order to obtain good models of complex geometry building blocks, we may have to resort to Sparameter table models to capture all electrical behavior within them. 24 3.4. Build Block Definition In order to effectively use the building block based method, building blocks for the structures of interest must be defined. The first step involved in accomplishing this is a determination of the kinds of geometries that are to be considered for the design and modeling of a particular kind of device. For example, a particular linewidth and spacing for interdigital capacitors should be fixed. The number of variations possible on the physical layout of a particular type of structure are infinite, however, generating building blocks for every conceivable layout is not practical. In order to utilize the proposed modeling methodology efficiently, design rules must be derived to dictate what geometries of structures will be allowable, and once this is determined, building blocks and test structures can be built and characterized. It should be emphasized that even with one linewidth and spacing, a large variety of structures can de designed, since there is no explicit limit on length or number of segments. In order to determine the actual geometries of the building blocks, some form of current analysis should be performed to ensure that building block boundaries are defined along regions of uniform current flow. Although this is not a requirement in the strictest sense (a large number of context sensitive building blocks can be built up, e.g. a corner adjacent to one square, a corner adjacent to another corner, etc.), it helps simplify the modeling procedure if adhered to. In order to do this, a high speed, low frequency, current visualization tool was developed under this research, and details about it are given in Appendix B. 25 In general, the modeling procedure essentially leads to generation of design rules in order to achieve good modeling accuracy. This philosophy is actually adapted widely by analog integrated circuit design houses where only certain geometry transistors are allowable due to the existence of good models for them. These geometry based design rules are easily implementable in most commercially available integrated circuit design packages, and thus these programs can be used for designing passive component in compliance with set design rules. These design rules may seem to be very restrictive at first glance, but even a small set actually can allow for a huge variety of different structures to be designed. For example, consider the case where design rules for serpentine resistors are under development (Figure 3.4-1). For a particular thick film process, minimum linewidth and spacing rules dictate that lines must be a minimum of 5 mils wide with interline spacings of 5 mils. If design rules are established which force 5 mil lines and 5 mils spacings for lines in serpentine resistors designed in this process, with only right angle bends allowed at the ends of lines to connect two adjacent lines together, and with a minimum line length of 35 mils, a wide variety of structures can be designed. All serpentine resistors which comply with the design rules can be as long as desired beyond 35 mils in 5 mil increments, and there are no restrictions on the number of segments. This clearly represents a very large set of resistor values that can be designed and modeled accurately. The allowable geometries can easily be increased by defining new building blocks and test structures, and hence the method is highly expandable. 26 Coupled Line Building Block U Building Block Uncoupled Line Building Block Figure 3.4-1. Serpentine resistors designed from the same set of building blocks 27 3.5. Test Structures Once building blocks have been defined, the next step is to characterize and develop models for them. This is achieved through the use of carefully designed test structures. The test structure set is designed such that it is comprised of all the predefined building blocks. The equivalent circuit of the test structure is constructed by combining the equivalent circuits of each of building blocks of which it is comprised. Once designed and fabricated in the process of interest, high frequency measurements of the test structures are taken which are then used to characterize each of the building blocks. An example of a possible test structure set for the modeling of serpentine resistors is shown in Figure 3.5-1. Although some standard formulations exist for obtaining high frequency scattering parameter data sets for some building blocks, such as coupled lines, and Tjunctions in microstrip configurations, the measurement based method provides some significant advantages. Since measurements of the devices are taken, no assumptions are made regarding material properties, layered dielectrics, or imperfect substrates, since all these effects are taken into account in the measured data. This makes the modeling procedure entirely process independent. It has been experimentally shown to work on highly nonuniform alumina substrates as well as multilayer low temperature cofired ceramic (LTCC) substrates. Additional process related nonlinearities are also taken into account, such as uneven metal deposition or printing, jagged edges, etc. In addition, with very fine lines, metal loss becomes an important factor, and this too can be taken into 28 account by the measurement based method. Test structures are also not limited to 2 dimensions. Structures can easily be defined in 3 dimensions to help characterize and predict the behavior of 3D passives; for example, those that are fabricated in LTCC processes. An important feature of the measurement based method is the ability to collect statistical information on the process of interest. This can be accomplished by fabricating test structures repeatedly on different runs of a process, and extracting building block models for them each time. A range of values for each circuit parameter can be constructed which in turn can generate statistical yield information for the designed passive device. This sort of information would be very important in any kind of volume precision application, for example in high-Q filter designs. Coupled Square Primitive U-Shaped Bend Primitive Figure 3.5-1. Test structures for serpentine resistor modeling 29 One important issue that must be addressed when designing the test structures is that all the required building blocks contribute enough to the overall response to be measurable, that is, the structure must be sensitive to all the building blocks. Since the structure responses are frequency dependent, the various blocks and their corresponding circuit models will have circuit components which will yield different sensitivities to the output parameters at different frequency points. The varying sensitivity of the output parameters with respect to frequency of the individual circuit parameters helps us to extract the circuit parameters for the various blocks. 3.6. Test Structure Equivalent Circuit Extraction In this section an overview of the theory behind extracting equivalent circuits and their corresponding parameters from measured data is presented. As was mentioned above, high frequency measurements using network analysis techniques are performed to obtain two port scattering parameters. In the equations and formulae below, impedance parameters are used, mainly because they are easier to derive analytically. There is no difference in the information obtained from impedance or scattering parameters, however, since they are essentially equivalent due to the existence of known transformations which map one to the other [41]. Due to the extremely high degree of nonlinearity of equations for deembedding the circuit parameters, analytical solutions are seldom possible, optimization of model parameters is usually the only recourse available. The objective of this section is to show 30 that, with enough frequency points, and sufficient sensitivity, accurate equivalent circuit extraction of individual building blocks from multi building block test possible. Vandermonde analysis is presented to aid in clarifying how the extraction method works. Sensitivity analysis is also discussed as a tool to help determine the "relative uniqueness" of the extracted parameters. Finally, in order to clarify some of the issues, several simple examples are presented in detail. For an arbitrary passive circuit network, we can obtain multiport parameters such as impedance parameters which are essentially quotient polynomial functions in frequency, and which have the form Zij ( V, ω ) = a ( V, ω ) b( V, ω ) (3-1) where V is the passive element circuit parameters vector and ω is frequency in radians per second. Expanding the quotient polynomial, we obtain a( V , ω ) a0 + a1ω + a 2 ω 2 +L+ a k ω k Zij ( V, ω ) = = b( V , ω ) b0 + b1ω + b2 ω 2 +L+ bn ω n (3-2) where a k and bk represent combinations of various circuit parameter values. Equation 32 can be represented by a qth order polynomial by performing a moment matching approximation. This yields an expression of the form c0 + c1ω + c2 ω 2 + c3ω 3 + L + cq ω q ≈ Z ij (ω) 31 (3-3) where the ck terms represent moments of the system. Moment matching techniques are widely used in integrated circuit interconnect analysis for network simplification and timing analysis [42]. Details of the moment matching approximation can be found in the literature [43]. For a physical circuit whose circuit parameters are not known, measurements can be performed to obtain impedance parameters at different frequencies. The various ck are combinations of the various circuit component parameters and are constant over frequency. At frequencies where nonlinear phenomena such as skin effect start to take place, this assumption no longer holds. Equation (3-3) implies that if measurements of the system can be obtained at different and a sufficient number of frequency points, then the various ck can be deembedded. Systems of this type are known as Vandermonde systems. The Vandermonde formulation proceeds with a linear system of the following form, with the right hand sides of the equations being the measured impedance parameter. 1 ω 01 1 1 ω1 M M 1 1 ω n q L ω 0 c1 Z ij (ω0 ) q L ω1 c2 Z ij (ω1 ) = O M M M q L ω q cq Z ij (ω q ) (3-4) which can also be written as V Tc = Z (3-5) Where V is the Vandermonde matrix and is nonsingular for distinct ω k . This equation can be solved using well investigated methods that can take into account problems such 32 as ill-conditioning that can easily arise in such systems, especially when large frequency values are used. The fact that we can formulate a non-singular system of equations for a linear network, with each equation generated by a different frequency point, leads to the idea that simply by sampling the system over frequency, we can obtain all the information necessary to deembed the various circuit parameters which comprise that system. Once the polynomial coefficients are obtained, we are faced with the challenge of determining the original circuit parameters from them. This problem is very difficult in general, due to the high degree of nonlinearity that is encountered at this point, since the various polynomial coefficients are comprised of products, sums, or some combination of the different circuit parameters. In most cases, due to the nonlinearity present in these problems, the only practical method for extracting circuit component parameter values from measured impedance or scattering parameters is by a process of nonlinear optimization. There are many optimization techniques available, and some of the more popular ones are the Newton-Raphson algorithm, the steepest descent and other gradient methods, fixed-point routines and hybrid methods, which combine several different methods together. The hybrid methods have gained popularity due to their ability to handle a wide variety of problems with better methods of recovering from incorrect search directions, intelligent error based parameter incrementing, etc. The optimization algorithm chosen in our case was Leavenberg-Marquardt due to its ability to choose between the inverseHessian and steepest descent methods, which allowed the search algorithm to switch 33 depending on whether RMS error at a particular step was relatively large or small [44]. This algorithm is implemented in the Hspice circuit simulator [45], and this tool that has been used extensively for this purpose in this research. The method generally converged in a reasonable amount of time for many different circuit configurations, with good results. Optimization over many frequency points and a wide band is necessary, in order to insure that the output parameters are sensitive to all of the important circuit parameters at some points over the entire frequency range. Sensitivity is very important, since an output parameter which is not sensitive to a particular circuit parameter over a wide frequency band implies that any value (within a reasonable range) of that circuit parameter will not influence the output response, and hence a completely incorrect value can be extracted. It is also possible (and is usually the case), however, that even over the entire frequency range of interest, a particular circuit parameter does not influence the output response, and in this case the lack of sensitivity is probably valid, and is not a result of an incorrect optimization technique. In actual practice, a low sensitivity usually does imply that the parameter does not affect the output responses significantly. Wide band sampling with sufficient frequency resolution is crucial in order to capture all the major reactive effects over a band of interest. In order to deembed a parasitic, it must be observed, and in most instances parasitics are small enough that they only manifest themselves at higher frequencies. A very important factor which must be considered in any optimization is the initial starting vector. This is usually the factor that determines success or failure of a 34 particular optimization run. Initial guesses in this case are obtained from measurement as much as possible and then modified based on geometry of building blocks. Additionally, results obtained from a successful optimization of one test structure can be used as initial guesses for unknown building block circuits of another. First pass initial guesses for the building block circuit parameters are determined based on actual measured scattering parameter data. Formulas found in the literature are used for these guesses [46] [47], and have proved to work well in general. The parameter values yielded by these equations represent overall test structure resistance, capacitance and inductance. In order to apportion them correctly to each building block, appropriate scaling factors must be applied, based on the geometry and area or volume used by the blocks. As an example, consider the building block in Figure 3.6-1. It is constructed of 24 building blocks of type 1 and 1 block of type 2. The area of block 2 is 8 times that of block 1. If through measurements, it was determined that the entire structure possessed a capacitance C, then on a per unit area basis, each block would be assigned an initial guess capacitance value of C/32. Correctly apportioned, each block of type 1 would have a capacitance of C/32 and block 2 would have a capacitance of C/4. Similar arguments can be applied to resistance and inductance initial guesses also. In reality, due to the nonuniform current flow in block 2, the optimized values of all components will be significantly different, probably less than type 1 blocks on a per unit basis. This is due to the fact that current does not occupy the complete area of the block, and is concentrated on the inner edges, especially in the corners. 35 1 2 Figure 3.6-1. Test structure for initial guess computation 3.6.1. Case Study: RLC Filter Frequency Resolution and Extraction Example As a simple illustration, consider the case of a low pass filter with a voltage gain transfer function given by V (ω ) = 1 . 1 + jωRC − ω 2 LC (3-6) A plot of the response for the particular values of L=0.1 µH, and C=1 nF with values of R ranging from 0.05 to 1.6 Ω are shown in Figure 3.6-1. Referring to the figure, if frequency samples are taken only at low frequencies (up to 10 MHz), then the system will be seen to have a constant gain response, and a incorrect equivalent circuit will be extracted. Frequency samples over the entire band of interest must be taken, and with considerably good resolution in order to extract a valid model. The issue of resolution is also illustrated in Figure 3.6-2, which is an expanded view of the overshoot region of Figure 3.6-1. As mentioned, the plot contains 5 curves, each one for a different resistance value. The effect of the different resistance values can only be seen in the overshoot 36 region, with different heights corresponding to different values of resistance, and thus to correctly deembed them, sufficient frequency resolution must be obtained in this area. 20.2178 30 20 10 20. log A w , R , L, C i 20. log A w , 2. R, L, C i 20. log A w , 4. R, L, C i 20. log A w , 8. R, L, C i 20. log A w , 0.25. R, L, C i 0 10 20 30 -34.4047 40 6 1 10 6.58546e+006 7 1 10 8 1 10 w i 9 1 10 7.31412e+008 Figure 3.6-1. Voltage gain of low pass filter 24.7905 25 20 20. log A w , R , L, C i 20. log A w , 2. R , L, C i 20. log A w , 4. R , L, C i 20. log A w , 8. R , L, C i 20. log A w , 0.25. R , L, C i 15 10 5 0 0 7 1 10 1.92194e+007 8 1 10 w i 9 1 10 3.14556e+008 Figure 3.6-2. Expanded view of overshoot region in gain response of low pass filter 37 Several examples of sensitivity analysis for some different circuits are presented. Actual analytical expressions are only shown in very few cases, since the expressions in general become very complex and are difficult to interpret. The output variables of interest are the impedance parameters Z11 and Z21, since the circuits are generally symmetric. Scattering parameters are not used since they are more difficult to calculate, however, well known transformations exist between impedance and scattering parameters, and thus they are essentially equivalent. 3.6.2. Case Study: Partial Element Equivalent Circuit (PEEC) Extraction and Sensitivity Analysis Complete analytical results are shown for the uncoupled PEEC circuit (Figure 3.3-1, top). Although it is unlikely that we will design any test structure solely modeled by only one PEEC circuit, the procedure and results obtained are instructive. Sensitivity analyses of more complex circuits can be carried out by the use of circuit simulators. For the PEEC circuit, the impedance parameters are given by Z11 (ω ) = R + jωL 1 + 2 1 + jωRCS − ω LCS jωC (3-7) 1 jωC (3-8) Z21 (ω ) = Taking derivatives of the impedance parameters with respect to the individual parameters, and normalizing to remove the effects of scaling, result in the following expressions for Z11 38 ∂Z11 1 jωCS ( R + jωL ) R = − 2 ∂R 1 + jωRCS − ω LCS 1 + jωRCS − ω 2 LCS Z11 (ω ) jω jω 2 CS ( R + jωL ) L ∂Z11 = + ∂L 1 + jωRCS − ω 2 LCS 1 + jωRCS − ω 2 LCS Z11 (ω ) −1 C ∂Z11 = 2 jωC Z11 (ω ) ∂C (3-9(a-d)) ∂Z11 − ( R + jωL )( jωR − ω 2 L ) CS = (1 + jωRCS − ω 2 LCS ) 2 Z11 (ω ) ∂CS and for Z21, we obtain ∂Z21 −1 C = = −1 2 ∂C jωC Z21 (ω ) ∂Z21 ∂Z21 ∂Z21 = = = 0. ∂R ∂L ∂CS (3-10(a,b)) The sensitivity results for Z11 are plotted on Figure 3.6-1 for a circuit with circuit parameter values given by R=1 Ω, L = 1 nH, C=0.5 pF, and CS = 50 fF. As can clearly be seen on the plot, the various circuit parameters affect the output response to a different degree over a wide frequency range. Clearly, the parameters which affect the response most significantly are L and C, the series inductance and capacitance to ground, followed by shunt capacitance CS, and lastly by series resistance R. It is important to note that the series resistance does not have a sensitivity of 0, as might be expected at high frequencies, rather it has a peak of approximately 0.35 at the resonance frequency of 7.1GHz. Z21 in this case has a constant normalized sensitivity value of -1, indicating that at all frequencies, Z21 is influenced equally by the one parameter that it consists of; capacitance to ground. 39 30 20 DL w , R, L, C , CS i DC w , R, L, C , CS i 10 DCS w , R, L , C, CS i DR w , R, L, C , CS i 0 10 9 1 10 10 1 10 f i 11 1 10 Figure 3.6-1. Sensitivity plot of Z11 for PEEC circuit The fact that all the circuit parameters in this case yield good sensitivities implies that the test structure is well designed, and the contribution of the building block can be deembedded. Of course, this was a foregone conclusion for this example, since only one building block was used. 3.6.3. Case Study: 4 Segment RLC Circuit Extraction and Sensitivity Analysis A second, more detailed example is also now described, showing the sensitivities of impedance parameters to two different resistance values in the circuit shown in Figure 3.6-1. Due to the complexity of this problem, only partial results are shown here, a full analysis can be found in Appendix A. The circuit is a four segment RLC ladder network, with the resistance value in the last segment being a different value with respect to the rest of the circuit. Analytical results are shown in the appendix for the impedance 40 parameters Z11 and Z21 with their normalized sensitivities with respect to R and R2 as shown in the circuit. As will become evident from the analysis details, it is usually not practical to compute the parameters manually, due to the huge complexity of the problem, even for a very simple circuit such as this. + R V1 L R L C R L C R2 C + L V2 C I1 I2 - - Figure 3.6-1. Circuit for impedance parameter sensitivity analysis. Impedance parameters are defined by the following relationship V1 Z11 V = Z 2 21 Z12 I1 . Z22 I 2 (3-11) In order to extract Z11 and Z21 , we can simply open circuit I 2 and calculate the impedance parameter responses as Z11 = V1 V and Z21 = 2 I1 I1 (3-12) with V2 being the voltage across the capacitor in the last segment of the circuit. Node equations can be written, and the voltages at all the nodes computed. Using this approach, we obtain the expressions for the impedance parameters. With the aid of symbolic computation tools, we obtain analytical results, the details of which can be found in the 41 appendix. Briefly, we obtain complex quotient polynomial expressions for both impedance parameters of the form Θ(ω 0 ) Θ(ω 8 ) Z11 = . and Z21 = Θ(ω 7 ) Θ(ω 7 ) (3-13) Normalized sensitivities of each of the impedance parameters with respect to the circuit resistances are computed with the first finite difference approximation, which is actually a procedure which is very well suited for use within circuit simulators to compute small signal sensitivities. Most commercial simulators do not have the capability of calculating AC sensitivity, but do have the ability of computing a response several times after altering a particular parameter. The classical definition for normalized sensitivity for a function F to a parameter h is given by S hF = ∂F h ⋅ . ∂h F (3-14) Using the first finite difference to approximate the first order derivative of F, we obtain the finite difference form of the expression: S hF = F ( V , h + ∆h ) − F ( V, h ) h ⋅ , ∆h F ( V, h ) ∆h small (3-15) where V represents the vector of unchanging variables of F, h is the parameter in consideration, and ∆h is the increment in h. ∆h must be kept small with respect to h in order for this expression to be accurate. This expression can also be written as 42 S hF = F ( V , h(1 + X )) − F ( V , h ) , XF ( V , h ) X small , (3-16) which is somewhat easier to compute. Using these definitions, the sensitivities of the impedance parameters Z11 and Z21 with respect to both resistances R and R2 were calculated. The values for the various circuit elements used were R=0.2 Ω, R2=0.1 Ω, L=0.1 µH, and C=1 nF. The plots for both sensitivities are shown in Figure 3.6-2 and Figure 3.6-3. The traces labeled DR represent normalized sensitivity with respect to R, and correspondingly, the traces label DR2 represent sensitivity with respect to R2. There are several important issues which surface in these results. One of the most striking factors is that Z11 is far more sensitive to changes in resistance values than Z21 . This is evidenced by the vertical scale on the plots, where Z11 reaches a maximum of almost 1, whereas Z21 does not even reach -0.1. Additionally, even though the value of R2 is quite small, Z11 is fairly sensitive to it at about 150 MHz, with a value of approximately 0.22. By contrast, Z21 is almost entirely insensitive to R2, with values staying below -0.01. These results imply that if one is trying to deembed resistance values for a network from measured two-port parameters, including Z11 in the optimizations is necessary, especially if resistance values are low. A very important factor which has become evident from this example is that the two separate impedance parameters yield completely different information about the network. By using both of them, we are utilizing 2 sets of completely different underlying 43 equations, and this can help considerably in the optimization and parameter deembedding process. 0.968015 1 0.8 0.6 DR w , R, L, C , R2 i 0.4 DR2 w , R, L, C, R2 i 0.2 0 -0.0523815 0.2 7 1 10 1.23134e+007 8 1 10 w i 9 1 10 2.39629e+008 Figure 3.6-2. Sensitivity analysis of Z11 for circuit of Figure 3.6-1 -1.98307e-006 0 0.01 0.02 DR w , R , L, C, R2 i 0.03 DR2 w , R, L, C, R2 i 0.04 0.05 -0.0541372 0.06 7 1 10 1.23134e+007 8 1 10 w i 9 1 10 2.39629e+008 Figure 3.6-3. Sensitivity analysis of Z21 for circuit of Figure 3.6-1 44 3.7. Library Based Implementation Once equivalent circuits for the building blocks have been successfully extracted, they can be inserted into a design library. The components in the library will have associated with them the geometries of the various building blocks along with input/output port connectivity information, the developed circuit models, and applicable design rules. In modern day electrical engineering CAD systems, highly flexible rules can be defined for design rule checking purposes for physical circuit design. This kind of tool can be used for this work to check for design rule compliance of designed structures, that is, to make sure that the constructed designs are constructed only from the modeled building blocks. Once a useful library has been developed, it can be used for new component design purposes. A designer can then consult the library to create a new structure, and once complete, run it through the design rule checker to ensure compliance. If the structure is not design rule compliant, the offending portions will be highlighted. The designer can then either modify his or her design to make it compliant, or if preferred, can define the error producing structure as a building block and go through the modeling procedure to characterize it. Once the design passes design rule checks, accurate models can be obtained which can then be simulated in a SPICE compatible circuit simulator to obtain predictions of behavior. 45 3.8. Summary In this chapter, a detailed description of the modeling methodology was presented. The concept of building blocks was discussed, and a test structure and measurement based characterization procedure was described. Circuit extraction and nonlinear optimization were discussed at length, along with several examples to illustrate some important issues. A method of implementation of the entire procedure within existing EDA frameworks was also presented. In the following chapters, the discussed modeling method will be applied to the predictive modeling of serpentine resistors, interdigital capacitors and spiral inductors. Additionally, the method will also be applied to some fully 3-dimensional passive structures manufactured in a multilayer low temperature cofired ceramic process. 46 CHAPTER IV MODELING OF RESISTORS 4.1. Introduction Resistors are an important component in many electrical systems. They are used in many areas, including circuit termination, filtering, voltage scaling, and in active circuits such as digital/analog converters and operational amplifiers just to name a few. Clearly, in applications where resistors play in integral role in circuit performance, their behavior must be accurately modeled and taken into consideration at design time. For relatively low frequency systems, resistors can be approximated as ideal components, with low error. However, for higher frequencies, even for those available in current CMOS technology, this assumption is no longer valid. The parasitic effects of resistors must be taken into account in order to obtain accurate models at these frequencies. An example of a high frequency passive system which uses resistors is a filter designed to meet the Digital Enhanced Cordless Telecommunications (DECT) standard up to 5.7 GHz. Clearly, in order to design a passive filter up to such a high frequency, the behavior of the passive components which comprise the filter must be accurately modeled up to that frequency. For these structures, coupling and parasitics must be taken 47 into account to model non-ideal behavior; for example, at some frequency, and depending upon their geometry, resistors become capacitive and then start to resonate, at which point they become essentially useless as resistors. Additionally, high frequency spectral content within signals can cause glitches and signal integrity problems due to reflections, and these phenomena must also be modeled. Accurately predicting these failure modes is very important to ensure that designs operate as expected at higher frequencies. Accurate modeling of resistors using non-lumped element methods, as discussed earlier, can be difficult and time consuming. In this chapter, the high frequency modeling of serpentine resistors using the methodology presented in this thesis is described [48]. As will be shown, accurate predictive modeling results of a 9 segment serpentine resistor have been obtained and verified experimentally up to 5-10 GHz frequency range. Additionally, the generated circuit model has been used in several circuit configurations, with good results. The circuits themselves have not been verified experimentally, but accurate predictions of the fabricated circuit have been obtained using the measured parameters as a model for the multisegment serpentine resistor in a microwave circuit simulator; Hewlett Packard Microwave Design System (MDS). The results obtained by using the complete circuit model of the resistor in the various circuit configurations, and simulated with a standard circuit simulator are compared against results obtained from MDS. It will be shown that in general, the described method produces results that agree well with MDS predictions, up to 10-20 GHz in various circuit configurations. 48 4.2. Modeling Procedure The procedure for modeling the resistors proceeded in the method described in Chapter 3. A brief description of the various steps involved is now presented. 1. The first step involved a determination of what geometry structures were to be considered and allowed in order to set up a practical set of building blocks and test structures to be measured and characterized. 2. This step required entering the geometry of a target structure into custom current flow visualization software (Appendix B) in order to determine the nature of low frequency current distribution through the device. Since the current visualization software could not model coupling behavior, the generated current distribution plots were only approximate for high frequency behavior, but were useful for helping determine building block geometries. Building blocks were to be cut along cross sections of uniform current distribution only. 3. Once the various building blocks had been determined, the next step was to design test structures the help model the various building blocks accurately. Additionally, a sensitivity analysis needed to be performed on the test structure equivalent circuits to ensure that the various parameters could be accurately deembedded. 4. At this point, test structures are fabricated and tested. High frequency network analysis and DC resistance measurements are taken. 49 5. Test data is used to form optimization input files for the test structures. Initial guesses are made based on the measured results for each structure. Once optimization for one structure is complete, the results are used for the remaining optimizations. 6. Circuit models of the building blocks are obtained. 4.3. Detailed Resistor Modeling Procedure The first step involved in the resistor modeling procedure was a determination of what types of resistor geometries were to be modeled. Since the theoretical number of possible layouts for a resistor (or any passive structure) is infinite, a restricted set had to be defined in order to determine a sufficiently small set of primitive blocks that would require characterization. Although at first glance, this type of restriction would seem harsh, it is not impractical. Looking at most designs with integrated passive components, most resistors are laid out in one of two ways; straight lines or serpentine structures, with the former the layout of choice for high frequency applications. In this case, attention was focused on the serpentine case for several reasons. First, serpentine resistors are more efficient in substrate area when compared to straight lines for the same resistance value, and if modeled correctly and efficiently, may have larger application in the high frequency arena. Secondly, the serpentine structure also presents a more difficult modeling problem due to higher levels of parasitics, such as coupling effects between the 50 segments of the structure, which could considerably affect the overall system response [49]. Resistor modeling with equal linewidths and spacing were considered. In the case presented here, 25 µm linewidth and spacing resistors were modeled. The serpentine geometry dictated that there were three main fundamental building blocks that required characterization: a square building block with connections on opposite sides, a U-shaped section connecting two parallel segments of the resistor together, and a coupled block segment to characterize line to line coupling behavior on a per square basis. Due to the fact that testing of these structures was required, one more building block was added - the probe pad. The actual sizes of these building blocks could only be determined through the use of the current flow visualization tool. Coupling was only considered with respect to nearest neighbors, but higher order coupling could be taken into account, however, this would require more complex test structures (but possibly not more in number) and a more complex extraction procedure. For the 5-10 GHz range, the higher order coupling would not result in an appreciable increase in modeling accuracy for these structures. The current visualization software was used to predict current flow through a representative serpentine resistor. Plots of current distribution and an enlarged view of the U shaped corner are shown in Figure 4.3-1 and Figure 4.3-2 and a contour plot showing the current crowding effect, is shown in Figure 4.3-3. Referring to the diagrams, the cutoff points for each of the primitives were at the areas where the current contours stopped changing rapidly, thus indicating constant current flow between the boundaries of the building blocks. Using this approach, the pad primitive was taken to be the large 51 pad square plus one adjacent line square. The material square and coupled material square were taken as one unit of material square each, and the U shaped primitive was represented by 3 squares on each of the horizontal and vertical axes of the U. Figure 4.3-1. Current distribution plot of serpentine resistor structure. 52 Figure 4.3-2. Enlarged view of U shaped section of serpentine resistor. Figure 4.3-3. Current contour plot showing current crowding effects in serpentine resistor 53 In order to model the four stated building blocks, two test structures were built (Figure 4.3-4). For clarity, the ground lines and pads are not shown in the figure, but the two test structures are designed for compatibility with a ground-signal-ground coplanar probe system. The first test structure is simply a line with probe pads on its ends; the purpose of this structure is to help characterize basic uncoupled material parameters, including self resistance, inductance, and capacitance. The second test structure is a 3segment meander resistor; this structure allows passive characterization of the U-shaped corner segments as well as line to line mutual inductance and coupling capacitance. The structures were characterized using D.C. measurements to determine resistances and network analysis techniques up to 20GHz so that parasitics would be observable in the Sparameter response. Pad Primitive Coupled Square Primitive Material Square Primitive U-Shaped Bend Primitive Figure 4.3-4. Test structures and primitives for meander resistor modeling. 54 It is of interest to note that a structure with first order coupling is actually a 4-port structure, whereas the test structures themselves are only 2-port devices, and thus only standard and repeatable 2-port measurements are necessary. A 4-port device is considerably more difficult to measure in practice than a 2-port, since many different excitation and loading iterations are required. To consider second and higher order coupling requires 6 (or more) port devices, at which point accurate measurement may be prohibitively difficult. Our method of simply measuring two port structures and extracting all required multiport information is a significant advantage over attempting to measure coupling between physically disconnected devices. 4.4. Processing and Measurement The test structure design was deposited on a 96 % alumina substrate which had a surface roughness of approximately +/- 1.5 µm. All processing was done at the Georgia Tech Microelectronics Research Center by MiRC cleanroom staff and students. The processing details can be found in Appendix C. A photograph of the fabricated structures is shown in Figure 4.4-1. The test structures were measured using network analysis techniques, a DC curve tracer, and a high precision multimeter. For the high frequency measurements, a HP 8510C network analyzer was used in conjunction with a Cascade Microtech probe station and ground-signal-ground configuration probes. Calibration was accomplished using a supplied substrate and utilization of the line-reflect-match (LRM) calibration method. 55 Data was gathered for each of the test structures at over 200 frequency points between 500MHz and 20GHz and stored with the aid of computer data acquisition software and equipment. DC I-V measurements of the test structures were also made in order to determine component resistances. At DC, parasitic capacitance and inductance have no effect on the response and the measured resistance value, once properly apportioned, can be used directly in the models of the building blocks. Figure 4.4-1. Photograph of fabricated structures for meander resistor modeling. 56 Figure 4.4-2. Photograph of predictively modeled 9 segment resistor 4.5. Modeling and Parameter Extraction At this stage, the objective is to generate circuit models for each of the defined building blocks. The circuit topologies for the uncoupled and coupled building blocks are shown in Figure 4.5-1. As discussed in Chapter 3, the fundamental circuit is based on the partial element equivalent circuit (PEEC) [39] which has been used for interconnect analysis [40] and general three dimensional high frequency structure simulation [50]. Coupling behavior is represented by the coupling capacitance between the middle nodes of the two PEEC circuits, as well as by mutual inductances between the left upper and left lower branch inductors in the model, and likewise, for the right hand side. These circuits represent models for the building blocks only; the test structure and resistor 57 circuits are comprised of many of the building block circuits connected in accordance with the structure geometry. 4.5.1. Sensitivity Analysis In order to determine whether individual building block circuit components could be deembedded from the designed test structures, a sensitivity analysis was performed. The sensitivity analysis was performed on the test structure equivalent circuits with respect to each building block circuit parameter that was desired to extracted. The results of the sensitivity analysis showed exactly how the S-parameters varied when one circuit parameter was differentially modified. Normalized plots of the various sensitivities are shown. In general, a non-zero non-flat response shows that the output is affected by the parameter over frequency, and thus should be extractable. Test structure 1 sensitivity plots are shown in Figure 4.5-1 to Figure 4.5-8. As can be seen from the plots, all parameters affect the output in at least one of the real or imaginary parts of the S-parameters, except for some of the pad parameters and shunt capacitances. Clearly though, the response of the structure is dominated by inductive effects. Test structure 1 sensitivity plots are shown in Figure 4.5-9 - Figure 4.5-15. Here also, the output parameters are quite sensitive to all parameters for both the coupled line and U-shaped bend building blocks, except for shunt and coupling capacitances. Additionally, it is interesting to note that mutual inductance starts becoming increasingly important at higher frequencies. 58 1.5 Normalized Sensitivity 1 0.5 0 -0.5 L S11 (R) L S11(I) L S21(R) L S21(I) -1 1.00E+08 1.00E+09 1.00E+10 1.00E+11 Frequency (Hz) Figure 4.5-1. S21 and S11 sensitivity with respect to line inductance in the uncoupled square building block in test structure 1. 0.4 Normalized Sensitivity 0.2 0 -0.2 -0.4 -0.6 -0.8 1.00E+08 C S11 (R) C S11(I) C S21(R) C S21(I) 1.00E+09 1.00E+10 1.00E+11 Frequency (Hz) Figure 4.5-2. S21 and S11 sensitivity with respect to capacitance to ground in the uncoupled square building block for test structure 1. 59 1 R S11 (R) R S11(I) 0.8 R S21(R) Normalized Sensitivity R S21(I) 0.6 0.4 0.2 0 -0.2 1.00E+08 1.00E+09 1.00E+10 1.00E+11 Frequency (Hz) Figure 4.5-3. S21 and S11 sensitivity with respect to line resistance in the uncoupled square building block in test structure 1. 0.008 0.007 Normalized Sensitivity 0.006 0.005 0.004 0.003 0.002 0.001 CC S11 (R) 0 -0.001 CC S11(I) CC S21(R) CC S21(I) -0.002 1.00E+08 1.00E+09 1.00E+10 1.00E+11 Frequency (Hz) Figure 4.5-4. S21 and S11 sensitivity with respect to shunt capacitance in the uncoupled square building block in test structure 1. 60 0.04 Normalized Sensitivity 0.02 0 -0.02 -0.04 CPAD S11 (R) -0.06 CPAD S11(I) CPAD S21(R) CPAD S21(I) -0.08 1.00E+08 1.00E+09 1.00E+10 1.00E+11 Frequency (Hz) Figure 4.5-5. S21 and S11 sensitivity with respect to capacitance to ground in the probe pad building block in test structure 1. 0.2 Normalized Sensitivity 0.15 0.1 0.05 0 LPAD S11 (R) -0.05 LPAD S11(I) LPAD S21(R) LPAD S21(I) -0.1 1.00E+08 1.00E+09 1.00E+10 1.00E+11 Frequency (Hz) Figure 4.5-6. S21 and S11 sensitivity with respect to line inductance in the probe pad building block in test structure 1. 61 0.1 0.08 Normalized Sensitivity 0.06 0.04 0.02 0 -0.02 RPAD S11 (R) RPAD S11(I) -0.04 RPAD S21(R) RPAD S21(I) -0.06 1.00E+08 1.00E+09 1.00E+10 1.00E+11 Frequency (Hz) Figure 4.5-7. S21 and S11 sensitivity with respect to line resistance in the probe pad building block test structure 1. 0.006 0.005 Normalized Sensitivity 0.004 0.003 0.002 0.001 0 CCPAD S11 (R) CCPAD S11(I) -0.001 CCPAD S21(R) CCPAD S21(I) -0.002 1.00E+08 1.00E+09 1.00E+10 1.00E+11 Frequency (Hz) Figure 4.5-8. S21 and S11 sensitivity with respect to shunt capacitance in the probe pad building block in test structure 1. 62 0.3 0.2 Normalized Sensitivity 0.1 0 -0.1 -0.2 -0.3 -0.4 C S11 (R) C S11(I) -0.5 C S21(R) -0.6 1.00E+08 C S21(I) 1.00E+09 1.00E+10 1.00E+11 Frequency (Hz) Figure 4.5-9. S21 and S11 sensitivity with respect to capacitance to ground in the coupled squares building block in test structure 2. 0.2 0.15 Normalized Sensitivity 0.1 0.05 0 -0.05 -0.1 -0.15 LM S11 (R) LM S11(I) -0.2 LM S21(R) LM S21(I) -0.25 1.00E+08 1.00E+09 1.00E+10 1.00E+11 Frequency (Hz) Figure 4.5-10. S21 and S11 sensitivity with respect to mutual inductance in the coupled squares building block in test structure 2. 63 0.01 0.008 Normalized Sensitivity 0.006 0.004 0.002 0 -0.002 -0.004 -0.006 CM S11 (R) CM S11(I) -0.008 CM S21(R) CM S21(I) -0.01 1.00E+08 1.00E+09 1.00E+10 1.00E+11 Frequency (Hz) Figure 4.5-11. S21 and S11 sensitivity with respect to coupling capacitance in the coupled squares building block in test structure 2. 0.1 Normalized Sensitivity 0.05 0 -0.05 -0.1 C2 S11 (R) C2 S11(I) C2 S21(R) C2 S21(I) -0.15 1.00E+08 1.00E+09 1.00E+10 1.00E+11 Frequency (Hz) Figure 4.5-12. S21 and S11 sensitivity with respect to capacitance to ground in the Ushaped building block in test structure 2. 64 0.22 0.17 Normalized Sensitivity 0.12 0.07 0.02 -0.03 -0.08 -0.13 1.00E+08 R2 S11 (R) R2 S11(I) R2 S21(R) R2 S21(I) 1.00E+09 1.00E+10 1.00E+11 Frequency (Hz) Figure 4.5-13. S21 and S11 sensitivity with respect to line resistance in the U-shaped building block in test structure 2. 0.3 Normalized Sensitivity 0.2 0.1 0 -0.1 -0.2 L2 S11 (R) L2 S11(I) L2 S21(R) -0.3 1.00E+08 L2 S21(I) 1.00E+09 1.00E+10 1.00E+11 Frequency (Hz) Figure 4.5-14. S21 and S11 sensitivity with respect to line inductance in the U-shaped building block in test structure 2. 65 0.006 0.005 Normalized Sensitivity 0.004 0.003 0.002 0.001 0 -0.001 CC S11 (R) CC S11(I) -0.002 CC S21(R) CC S21(I) -0.003 1.00E+08 1.00E+09 1.00E+10 1.00E+11 Frequency (Hz) Figure 4.5-15. S21 and S11 sensitivity with respect to shunt capacitance in the Ushaped building block in test structure 2. 4.5.2. Model Extraction The extraction of the circuit model parameters was achieved in several steps. Due to the highly nonlinear nature of the generated system equations with respect to circuit parameter values, a procedure of hierarchical optimization with respect to measured Sparameter and DC resistance data was chosen. All optimizations and simulations were done using the Hspice circuit simulator on Sun Sparcstation 20 series workstations. The starting point or initial guesses of the circuit parameters were crucial for correct optimization results, and in order to achieve this, an initial optimization was done assuming that each test structure was comprised of just one building block, utilized repetitively across the length of the structure on a per square basis. The initial guess for 66 these circuit parameters were derived from the measured S-parameters in a test structure, and then dividing by the number of blocks used in order to extract the valid R,L,C, and CC values for the circuit model. This method was very effective for obtaining a good starting point for the optimizations of the test structure circuits. The first test structure optimized was structure 1 shown in Figure 4.3-4. The goal was to extract the parameters of the contact pad and the uncoupled material square. The initial guesses were inserted, and the circuit was optimized with respect to measurements up to 10GHz. Once the optimization was complete, the computed models were taken and used as valid model parameters for their respective building blocks for test structure 2, shown in Figure 4.3-4. The remaining parameters to be computed for this structure were the line to line coupling parameters (mutual inductance and coupling capacitance), and the parameters for the U-shaped corner. Additionally, line to ground capacitance had to be recomputed for the material square primitive in the presence of adjacent lines. Optimizations were done on measurements performed up to 10GHz. Both optimizations completed with very low residual sum of squares error, indicating accurate results. The modeling results for test structure 1 and 2 are shown in Figure 4.5-2 and Figure 4.5-3 respectively. The various circuit models and parameters for the different building blocks are shown in Figure 4.5-1. 67 CC CC 2 1 1 R L L C R=0.08 Ohm L= 1.2E-11 H C = 1.4e-15 F CC = 1.8e-15 F Probe Pad Bldg. Block 1 CC R L LM 4 2 CC 1 3 Coupled Line Bldg. Block 2 1 C L C LM L R L 2 CC CC R R R=0.4 Ohm L= 3.7E-11 H C = 5.3e-15 F CC=2.7e-15 F U-Shaped Bend Bldg. Block 1 4 L C 2 Uncoupled Line Bldg. Block R=0.09 Ohm L= 1E-11 H C = 1.1e-15 F CM=0.4e-15 F CC = 1.4e-15 F LM = 0.20 CC CC R 2 R CC 1 1 3 L CM CC R 2 R L L R 2 C R=0.08 Ohm L= 1E-11 H C = 2.7e-15 F CC = 1.2e-15 F Figure 4.5-1. Building blocks, equivalent circuits and parameter values for serpentine resistor modeling. 68 1.00E+00 8.00E-01 6.00E-01 S21(R) Measured S21(R) Modeled S21(I) Measured S21(I) Modeled Real S21 4.00E-01 2.00E-01 0.00E+00 Imaginary -2.00E-01 -4.00E-01 -6.00E-01 1.00E+08 1.00E+09 1.00E+10 1.00E+11 Frequency (Hertz) 2.50E-01 2.00E-01 S11(R) Measured S11(R) Modeled S11(I) Measured S11(I) Modeled S11 1.50E-01 Imaginary 1.00E-01 Real 5.00E-02 0.00E+00 1.00E+08 1.00E+09 1.00E+10 1.00E+11 Frequency (Hertz) Figure 4.5-2. Measured vs. modeled results for test structure 1. (a) S21 real and imaginary response. (b) S11 real and imaginary response. 69 1 0.8 0.6 S21 0.4 S21(R) Measured S21(R) Modeled S21(I) Measured S21(I) Modeled Real 0.2 0 Imaginary -0.2 -0.4 -0.6 1.00E+08 1.00E+09 1.00E+10 1.00E+11 Frequency (Hertz) 0.25 0.2 S11(R) Measured S11(R) Modeled S11(I) Measured S11(I) Modeled S11 0.15 Imaginary 0.1 Real 0.05 0 1.00E+08 1.00E+09 1.00E+10 1.00E+11 Frequency (Hertz) Figure 4.5-3. Measured vs. modeled results for test structure 2. (a) S21 real and imaginary response. (b) S11 real and imaginary response. 70 4.6. Results The computed fundamental building block models were used to predict the behavior of a 9 segment meander resistor constructed using a combination of those blocks. The resistor was then used in several simple circuits to assess the accuracy of the model in common applications. An equivalent circuit of the resistor was constructed by replacing each building block in the structure with its equivalent extracted circuit (Figure 4.6-1). Since only first level coupling was taken into account, each material square in each segment of the resistor was coupled to its nearest neighbor by a pair of mutual inductances and a coupling capacitance. As inferred from the circuit description, the resulting circuit of the 25µm linewidth and 300µm length per segment 9 segment resistor was a complex, highly interconnected system, consisting of approximately 700 nodes. The longest path length of the resistor was approximately 0.35 wavelengths long at 10GHz. In spite of the large circuit size, AC small signal analysis proceeded very quickly, with the entire circuit simulation completed in under 2 minutes. The predicted Sparameters were compared to measured values of the same structure; the results are shown in Figure 4.6-2. Both real and imaginary parts of S11 and S21 were well predicted up to 5GHz. In comparison, the same structure was designed and simulated in a method of moments solver with a 3GHz meshing frequency. The structure required 72 min. to complete, while consuming approximately 50MB of system memory and utilizing 2 processors in a multiprocessing Sun workstation. Thus, for this example, a speedup factor 71 of approximately 35 was obtained. For more complex structures, simulation time of the method of moments solver would increase dramatically, whereas using our approach, simulation time would increase only with the number of elements in the equivalent circuit. Coupled Line Building Block U Building Block Uncoupled Line Building Block Figure 4.6-1. Serpentine resistor and associated building blocks. 72 1 0.8 0.6 S21 0.4 S21(R) Measured S21(R) Predicted (This Paper) S21(I) Measured S21(I) Predicted (This Paper) Real 0.2 0 -0.2 Imaginary -0.4 -0.6 -0.8 1.00E+08 1.00E+09 1.00E+10 1.00E+11 Frequency (Hertz) 0.7 S11(R) Measured 0.6 S11(R) Predicted (This Paper) S11(I) Measured 0.5 S11(I) Predicted (This Paper) S11 0.4 Real 0.3 0.2 0.1 Imaginary 0 -0.1 1.00E+08 1.00E+09 1.00E+10 1.00E+11 Frequency (Hertz) Figure 4.6-2. Measured vs. predicted results for 9 segment resistor. (a) S21 real and imaginary response. (b) S11 real and imaginary response. 73 In addition to confirming an S-parameter match between predicted and measured values, two of these resistors connected in a voltage divider configuration was also considered. Since an actual divider structure was not constructed, the target response was generated by using the measured S-parameters of the resistor as a model, and constructing an equivalent circuit in a RF simulator that was able to utilize the measured data directly (Hewlett Packard MDS). The circuit is shown in Figure 4.6-3. The divider circuit generated with our modeling approach was created using standard SPICE netlist techniques. The circuit model for the 9-segment resistor was enclosed within a subcircuit, and then two subcircuits were used to construct the divider. The simulated response of both MDS and the circuit simulator are shown in Figure 4.6-4. The predicted resistor model circuit response models the divider behavior extremely well, matching the results generated by MDS up to approximately 10 GHz. In addition, the unusual voltage divider peaking behavior of the MDS response beyond 10 GHz was mimicked by our circuit, with the voltage peak frequency predicted slightly earlier in this case. X + X Vout - X=9 Segment Resistor Figure 4.6-3. Resistor divider circuit. 74 Voltage Magnitude (dB) 15 Predicted (This Paper) MDS Result 10 5 0 -5 -10 1.00E+07 1.00E+08 1.00E+09 1.00E+10 1.00E+11 Frequency (Hertz) 20 Voltage Phase (degrees) 0 -20 Predicted (This Paper) MDS Result -40 -60 -80 -100 -120 -140 -160 -180 1.00E+07 1.00E+08 1.00E+09 1.00E+10 1.00E+11 Frequency (Hertz) Figure 4.6-4. MDS generated vs. predicted results for voltage divider circuit. (a) Voltage magnitude response. (b) Voltage phase response. 75 The resistor model was also tested in a 6 segment LC circuit, with the resistor used as a termination. The inductance and capacitance were chosen such that a characteristic impedance per segment of 50Ω was obtained. The circuit is shown in Figure 4.6-5. The circuit was again simulated in both MDS (using measured parameters) and in the circuit simulator using the constructed model, but this time, a two port Sparameter simulation was done. The results of the simulation are shown in Figure 4.6-6. Both S11 and S21, real and imaginary parts are predicted well up to 20GHz. For comparison purposes, the circuit performance using an ideal resistor is shown also in Figure 4.6-7. From these plots it is evident that the ideal resistor model does not predict high frequency behavior well. Both circuits simulated here clearly illustrate the importance of modeling resistive passive components along with their associated parasitics in order to obtain accurate simulation results at high frequencies. L Port 1 L C X=9 Segment Resistor L C L C L C L C C X Port 2 L=1nH C = 0.4pF Figure 4.6-5. 6 Segment LC circuit with 9 segment resistor used as termination. 76 1 0.8 0.6 S21(R) MDS S21(R) - This Paper S21(I) MDS S21(I) - This Paper 0.4 S21 0.2 Real 0 -0.2 -0.4 Imaginary -0.6 -0.8 -1 1.00E+08 1.00E+09 1.00E+10 Frequency (Hertz) 1 0.8 0.6 S11(R) MDS S11(R) - This Paper S11(I) MDS S11(I) - This Paper S11 0.4 0.2 0 -0.2 Imaginary -0.4 -0.6 -0.8 1.00E+08 Real 1.00E+09 1.00E+10 Frequency (Hertz) Figure 4.6-6. MDS generated vs. predicted results for 6 segment LC circuit with resistive termination. (a) S21 real and imaginary response. (b) S11 real and imaginary response. 77 1.00E+00 S21(R) MDS S21(R) - Ideal R S21(I) MDS S21(I) - Ideal R 8.00E-01 6.00E-01 4.00E-01 Real S21 2.00E-01 0.00E+00 -2.00E-01 -4.00E-01 Imaginary -6.00E-01 -8.00E-01 -1.00E+00 1.00E+08 1.00E+09 1.00E+10 Frequency (Hertz) 1.00E+00 8.00E-01 6.00E-01 S11(R) MDS S11(R) - Ideal R S11(I) MDS S11(I) - Ideal R S11 4.00E-01 2.00E-01 0.00E+00 Imaginary -2.00E-01 -4.00E-01 -6.00E-01 -8.00E-01 1.00E+08 Real 1.00E+09 Frequency (Hertz) 1.00E+10 Figure 4.6-7. MDS generated vs. results using ideal 17.88Ω Ω resistor for 6 segment LC circuit. (a) S21 real and imaginary response. (b) S11 real and imaginary response. 78 4.7. Summary In this chapter, accurate modeling of serpentine resistors using the modeling methodology described earlier in this thesis has been achieved and verified experimentally. The entire procedure has been described in detail, including building block and test structure development, equivalent circuit extraction, and model verification, with results presented at each stage. The models have shown accuracy up to ~10 GHz in both direct resistor models as well as within circuits, with simulation speeds far greater than that of conventional electromagnetic or RF simulators. This procedure creates highly flexible circuit level models of the resistors, which are extremely useful since they can be incorporated into the circuit design stage to investigate how they will affect circuit performance, and which cannot easily be obtained from method of moments or other conventional non-lumped element simulation and modeling methods. Additionally, the method is very well suited for circuit design applications, since resistor designs can be changed incrementally, and behavior predictions can be obtained very quickly. In the next chapter, the application of the method to interdigital capacitor modeling will be discussed. 79 CHAPTER V MODELING OF INTERDIGITAL CAPACITORS 5.1. Introduction Interdigital capacitors (IDCs) play an important role in integrated electrical systems. They are used in a wide variety of circuits, including resonators, oscillators, and filters, just to name a few. IDCs are used to perform functions including DC blocking, frequency filtering and impedance transformation. They are cheap to manufacture, since they are planar devices, unlike the parallel plate or metal-insulator-metal (MIM) variety. In high frequency systems which use these devices, accurate models of them must be obtained in order to model their behavior at high frequencies. As in the case with the resistors, IDCs suffer from many parasitic effects which can cause them to resonate or behave unexpectedly at high frequencies, and capturing these effects is of paramount importance in order to accurately model systems which use them. In this chapter the application of the developed modeling methodology is applied to the modeling of interdigital capacitors. As will be shown, accurate predictive modeling results of a 10 segment interdigital capacitor have been obtained and verified experimentally up to 5-10 GHz frequency range. Additionally the behavior of a 80 fabricated series resistor-capacitor circuit is predicted and verified by constructing a resonant tank circuit in a microwave circuit simulator (Hewlett Packard Microwave Design System (MDS)) using an ideal inductance and the measured RC data against an actual model constructed using the presented method, with good results. 5.2. Modeling Procedure Modeling of interdigital capacitors proceeded in the method described in Chapter 3. A brief description of the various steps involved is now presented. 1. The first step involved a determination of what geometry structures were to be considered and allowed in order to set up a practical set of building blocks and test structures to be measured and characterized. 2. This step required entering the geometry of a target structure into the custom current flow visualization software in order to determine the nature of the current distribution through the device. Building blocks were to be cut along cross sections of uniform current distribution only. In this case, the current visualization software was unable to handle physically disjoint structures, so geometrically equivalent joint structures were used. 3. Once the various building blocks had been determined, the next step was to design test structures the help model the various building blocks accurately. Additionally, a 81 sensitivity analysis needed to be performed on the test structure equivalent circuits to ensure that the various parameters could be accurately deembedded. 4. At this point, test structures are fabricated and tested. High frequency network analysis and DC resistance measurements are taken. 5. Test data is used to form optimization input files for the test structures. Initial guesses are made based on the measured results for each structure. Once optimization for one structure is complete, the results are used for the remaining optimizations. 6. Circuit models of the building blocks are obtained. 5.3. Detailed Modeling Procedure The first step involved in the interdigital capacitor and resistor modeling procedure was a determination of which geometry structures were to be modeled. Since the theoretical number of possible layouts for these devices is infinite, a restricted set had to be defined in order to determine a sufficiently small set of primitive blocks that would require characterization. Although at first glance, this type of restriction would seem harsh, it is not impractical. Even with only one linewidth and interline spacing allowed, a huge array of devices can be designed with large line lengths and many segments. The described procedure is equally applicable to electrically long and short structures, since the only limiting factor in this method is the accuracy of the building block, and not on how many are used. 82 For the devices discussed in this paper, equal linewidths and spacings of 30 um were considered. The basic structure of the capacitor and resistor lead to the identification of 5 fundamental building blocks that required characterization (Figure 5.3-1). The resistor blocks were the same as those discussed in chapter 5. The first building block was simply a 30um x 30um square of material connected on two opposite sides by additional material. The second building block was two interacting, but physically disconnected squares of material to account for codirectional and contradirectional coupling between segments. The third and fourth blocks were a U-shaped piece of material, used in serpentine resistors to connect adjacent line segments together, and a shielded stub piece to model the end of a capacitor finger surrounded on three sides by the conductor of the opposite terminal. The fifth block was simply a probe pad with a short 1 square stub, necessitated by the fact that all the test structures that would be required to model these blocks needed to be physically tested by probing. 83 Material Square Building Block Pad Building Block First Order Coupled Building Block Shielded Stub Building Block Figure 5.3-1. Interdigital capacitor and associated building blocks. A representative ladder structure was input in to the current visualization software and analyzed. The output current density and contour plots are shown in Figure 5.3-2 and Figure 5.3-3. According to the structure geometry, it was assumed that the majority of nonuniform current flow would occur in the T-shaped region connecting the long vertical section and the horizontal segments together, and this could be approximated with a ladder shaped structure. Although the T-section did represent a region of nonuniform current flow, due to a lack of coupling information between the T and the opposite terminal conductor segment, the T section itself was not used as a building block. Instead, a hybrid building block was constructed which comprised 2 adjacent ½ T-sections and 84 the stub of the opposite conductor finger, and was named the shielded stub building block (Figure 5.5-1). Examination of the output plots from the current visualization software lead to the determination of the geometries of the various building blocks. The single square and the coupled square building blocks were only one square in width as expected. The U shaped block was a total of 7 squares in length with the two horizontal sections of the U extending for a length of 3 squares each in order for the current flow distribution to be constant across the boundaries of the building block. For the same underlying reason, the shielded stub primitive was determined to be 9 squares long. The probe pad primitive was the same size as previously mentioned due to the simple current flow through the feed line interface and into the actual device. Figure 5.3-2. Contour and indexed color intensity plots of current distribution in ladder shaped structure. 85 Figure 5.3-3. Contour plot of current in T-shaped section within ladder structure. Three test structures were built in order to model the stated building blocks, (Figure 5.3-4). The test structures are designed for compatibility with a ground-signalground coplanar probe system, but for clarity, the ground lines and pads are not shown in the figure. The first test structure is simply a line with probe pads on its ends; the purpose of this structure is to help characterize basic uncoupled material parameters, including self resistance, inductance, and capacitance. The second test structure is a 3-segment meander resistor; this structure allows passive characterization of the U-shaped corner segments as well as line to line mutual inductance and coupling capacitance. The third structure is a simple interdigital capacitor. The purpose of this structure is to help characterize the shielded stub primitive and also refine coupling capacitances. As might 86 be expected, mutual inductances have almost no effect on this structure until the device is conducting at high frequencies. Test Structure 1 Material Square Primitive Pad Primitive Test Structure 2 Coupled Square Primitive U-Shaped Bend Primitive Test Structure 3 Shielded Stub Primitive Figure 5.3-4. Test structures and building blocks for interdigital capacitor and serpentine resistor modeling. 87 5.4. Processing and Measurement The test structure resistor material was Ti/Au deposited on a 96% alumina substrate. An electron beam evaporation system was used to deposit 0.04µm of titanium followed by a 0.2 µm layer of gold. The thin layer of titanium was used to improve adhesion of the gold to the substrate. Following deposition, the resistors were defined using standard photolithography and etch back. The photoresist was hard baked for five minutes at 125°C in order to stabilize it before etching. The gold was etched in a heated KCN solution for 1 minute followed by a buffered oxide etch to remove the titanium. Due to the surface roughness of the substrate - approximately +/- 1.5 µm, the edges of the resistor were jagged, but the lines were continuous. A photograph of several fabricated structures are shown in Figure 5.4-1, Figure 5.4-2, and Figure 5.3-1. The test structures were measured using a network analyzer, and a high precision multimeter. For the high frequency measurements, a HP 8510C network analyzer was used in conjunction with a Cascade Microtech probe station and ground-signal-ground configuration coplanar probes. Calibration was accomplished using a supplied impedance standard substrate and utilization of the line-reflect-match (LRM) calibration method. Data was gathered for each of the test structures at over 200 frequency points between 45MHz and 20GHz and stored with the aid of computer data acquisition software and equipment. 88 Figure 5.4-1. Fabricated interdigital capacitor - test structure 3. Figure 5.4-2. Fabricated interdigital capacitor – 10 segment capacitor predictively modeled. 89 Figure 5.4-3. Fabricated RC structure predictively modeled. 5.5. Modeling and Parameter Extraction Following measurement of test structures, the next step is to extract circuit models for all the building blocks from which the test structures are comprised. As with the resistor case, the fundamental circuit used is the partial element equivalent circuit (PEEC) which has been used extensively for interconnect and arbitrary shaped conductor high frequency analysis. The PEEC circuit takes into account couplings, but does not take into account retardation effects. Depending on actual building block geometries, the PEEC circuit model is modified as needed. For example, for modeling of coupling behavior, coupling capacitances and mutual inductances are included between parallel segments, 90 and for complex geometries, such as the shielded stub, additional elements and ports are added as well. The building blocks, extracted circuits and parameter values are shown in Figure 5.5-1. 5.5.1. Sensitivity Analysis A sensitivity analysis was performed on the test structures with respect to the individual building block circuit components to determine their relative importance and the level of influence on the test structure S-parameter output responses. The sensitivity responses for test structures 1 and 2 are not shown here since they are almost identical to the sensitivities obtained in the serpentine resistor modeling case, since the same test structures were used there (although fabricated on a different run). The reader is referred to chapter 5 for an investigation of these results. Results for test structure 3 are presented here. A significant difference in sensitivity to different circuit parameters is shown here when compared to the other two cases, mainly due to the fact that the two terminals of the structure are physically disconnected. In particular, there is very low sensitivity to inductance, especially at low frequencies, in both self and mutual inductances. This is easily explained by the fact that current flow is practically zero at low frequencies, but begins to increase at higher frequencies as the capacitor begins to conduct. Clearly, for accurate inductance extractions, results from the fully connected structures must be used, due to their high degree of sensitivity of S-parameters to the various inductances. 91 From the sensitivity response plots, it is evident that for interdigital capacitors, the most critical parameters are coupling capacitance, the capacitance between the shield and the opposite conductor in the shielded stub primitive, followed by line to ground capacitances. The next tier of importance goes to line and mutual inductances which start becoming important at higher frequencies. Line resistance have a relatively small effect in the frequency range of interest, since the IDC test structure is essentially an open circuit, and thus exhibits very high impedance. 1.4 1.2 Normalized Sensitivity 1 0.8 0.6 0.4 0.2 0 -0.2 DCM S11(R) DCM S11(I) -0.4 DCM S21(R) DCM S21(I) -0.6 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 5.5-1. S21 and S11 sensitivity of test structure 3 with respect to line to line coupling capacitance (CM). 92 0.35 0.3 Normalized Sensitivity 0.25 0.2 0.15 0.1 0.05 0 -0.05 DCC2 S11(R) DCC2 S11(I) -0.1 -0.15 1.00E+08 DCC2 S21(R) DCC2 S21(I) 1.00E+09 1.00E+10 Frequency (Hz) Figure 5.5-2. S21 and S11 sensitivity of test structure 3 with respect to stub to line coupling capacitance (C2) in the shielded stub. 0.4 0.3 Normalized Sensitivity 0.2 0.1 0 -0.1 DCSQ S11(R) -0.2 DCSQ S11(I) DCSQ S21(R) DCSQ S21(I) -0.3 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 5.5-3. S21 and S11 sensitivity of test structure 3 with respect to line to ground capacitance in the IDC fingers. 93 0.15 Normalized Sensitivity 0.1 0.05 0 -0.05 DCC S11(R) -0.1 DCC S11(I) DCC S21(R) DCC S21(I) -0.15 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 5.5-4. S21 and S11 sensitivity of test structure 3 with respect to line to ground capacitance in the shielded stub. 0.14 0.12 0.1 Normalized Sensitivity 0.08 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 DLSQ S11(R) DLSQ S11(I) DLSQ S21(R) DLSQ S21(I) -0.08 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 5.5-5. S21 and S11 sensitivity of test structure 3 with respect to line inductance in finger segments. 94 0.05 0.04 Normalized Sensitivity 0.03 0.02 0.01 0 -0.01 DLC S11(R) DLC S11(I) -0.02 DLC S21(R) DLC S21(I) -0.03 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 5.5-6. S21 and S11 sensitivity of test structure 3 with respect to line inductance in shielded stub. 0.012 0.01 Normalized Sensitivity 0.008 0.006 0.004 0.002 0 -0.002 DLM S11(R) -0.004 DLM S11(I) DLM S21(R) DLM S21(I) -0.006 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 5.5-7. S21 and S11 sensitivity of test structure 3 with respect to line to line mutual inductance between finger segments. 95 0.015 Normalized Sensitivity 0.01 0.005 0 -0.005 DRSQ S11(R) DRSQ S11(I) DRSQ S21(R) DRSQ S21(I) -0.01 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 5.5-8. S21 and S11 sensitivity of test structure 3 with respect to line resistance in finger segments. 0.012 0.01 Normalized Sensitivity 0.008 0.006 0.004 0.002 0 -0.002 -0.004 DRC S11(R) DRC S11(I) -0.006 DRC S21(R) DRC S21(I) -0.008 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 5.5-9. S21 and S11 sensitivity of test structure 3 with respect to line resistance in shielded stub. 96 5.5.2. Model Extraction Circuit model parameter extraction proceeded in several steps. Due to the highly nonlinear nature of the problem, a procedure of hierarchical optimization with respect to measured S-parameter and DC data was chosen. Initial guesses for the various parameters were derived from actual measurement data. All optimizations and simulations were done using the Hspice circuit simulator on Sun SPARCstation 20 series workstations. Details of the extraction method can be found in chapter 3. Test structure 1 was initially optimized in order to extract the parameters of the contact pad and the uncoupled material square. The initial guesses were inserted, and the circuit was optimized with respect to measurements up to 10GHz. Once the optimization was complete, the computed models were taken and used as valid model parameters for their respective building blocks for test structure 2. The parameters to be computed for this structure were the first order line to line coupling parameters (mutual inductance and coupling capacitance), and the parameters for the U-shaped corner. Additionally, line to ground capacitance needed to be recomputed for the material square building block in the presence of adjacent lines. Lastly, the third test structure was optimized to extract the value of the shielded stub primitive and to refine the values of the coupling capacitances. All optimizations completed with very low residual sum of squares error, indicating accurate results. The modeling results for test structures 1, 2 and 3 are shown in Figure 5.5-2, Figure 5.5-3, and Figure 5.5-4 respectively. The extracted circuit models and parameters are shown in Figure 5.5-1. 97 CC CC 1 2 2 1 R L Uncoupled Line Bldg. Block L R R=0.07 Ohm L= 1E-11 H C = 2.9e-15 F CC = 1.2e-15 F C CC 1 3 2 4 CC 1 R L CM CC R R LM CC L L R=0.07 Ohm L= 1E-11 H C = 1.1e-15 F CM=1.9e-15 F CC = 1.2e-15 F LM = 0.55 4 R C 2 1 C2 2 3 1 R L L R R=0.6 Ohm L= 1E-10 H C = 7.1e-15 F C2=3.3e-15 F C 3 Shielded Stub Bldg. Block CC CC 1 2 1 2 R L L R C R=0.08 Ohm L= 1.2E-11 H C = 3.6e-15 F CC = 1.5e-15 F Probe Pad Bldg. Block CC CC 1 2 1 R L L C 2 3 C LM Coupled Line Bldg. Block 2 L U-Shaped Bend Bldg. Block R R=0.3 Ohm L= 3.7E-11 H C = 5.3e-15 F CC=2.7e-15 F Figure 5.5-1. Building blocks, equivalent circuits, and parameters for IDC and resistor modeling. 98 5.00E-01 4.50E-01 4.00E-01 3.50E-01 Meas. S11(R) Modeled S11(R) Meas. S11(I) Modeled S11(I) S11 3.00E-01 2.50E-01 2.00E-01 1.50E-01 1.00E-01 5.00E-02 0.00E+00 1.00E+08 1.00E+09 1.00E+10 1.00E+11 Frequency (Hertz) 1.00E+00 8.00E-01 6.00E-01 4.00E-01 S21 2.00E-01 Meas. S21(R) Modeled S21(R) Meas. S21(I) Modeled S21(I) 0.00E+00 -2.00E-01 -4.00E-01 -6.00E-01 -8.00E-01 -1.00E+00 1.00E+08 1.00E+09 1.00E+10 1.00E+11 Frequency (Hertz) Figure 5.5-2. S-parameter measured and modeled results for test structure 1. 99 3.50E-01 3.00E-01 2.50E-01 Meas. S11(R) Modeled S11(R) Meas. S11(I) Modeled S11(I) S11 2.00E-01 1.50E-01 1.00E-01 5.00E-02 0.00E+00 1.00E+08 1.00E+09 1.00E+10 Frequency (Hertz) 1.00E+00 8.00E-01 6.00E-01 S21 4.00E-01 Meas. S21(R) Modeled S21(R) Meas. S21(I) Modeled S21(I) 2.00E-01 0.00E+00 -2.00E-01 -4.00E-01 -6.00E-01 1.00E+08 1.00E+09 1.00E+10 Frequency (Hertz) Figure 5.5-3. S-parameter measured and modeled results for test structure 2. 100 1.00E+00 8.00E-01 6.00E-01 4.00E-01 Meas. S11(R) Modeled S11(R) Meas. S11(I) Modeled S11(I) S11 2.00E-01 0.00E+00 -2.00E-01 -4.00E-01 -6.00E-01 -8.00E-01 -1.00E+00 1.00E+08 1.00E+09 1.00E+10 Frequency (Hertz) 5.00E-01 4.00E-01 S21 3.00E-01 Meas. S21(R) Modeled S21(R) Meas. S21(I) Modeled S21(I) 2.00E-01 1.00E-01 0.00E+00 -1.00E-01 1.00E+08 1.00E+09 1.00E+10 Frequency (Hertz) Figure 5.5-4. S-parameter measured and modeled results for test structure 3. 101 5.6. Results The extracted building blocks were used to predict the behavior of a ten segment interdigital capacitor, with each segment having a line length of 600 um. In addition, a series resistor capacitor structure was fabricated, and the behavior of a RLC resonator circuit was predicted. The RLC resonator circuit itself was not fabricated, but its behavior was simulated by use of the HP MDS simulator which is able to take measured Sparameter datasets and use them directly as models of structures. Both the capacitor and RC circuit were modeled in Hspice by constructing them out of the building block pieces. The performance of the capacitor was compared against measured data, and the performance of the RLC resonant circuit was compared against the output generated from MDS. The equivalent circuit for the interdigital capacitor was constructed by replacing the different geometrical sections with the applicable building block circuit models. Each square of material in the finger segment was modeled with coupled building blocks, which were cascaded in order to generate the required line length. Since only first order coupling was considered, coupling only to the nearest neighbor pieces of material was considered. The ends of each finger were modeled by replacing the stub and surrounding material by the shielded stub primitive, and the probe pads were modeled with their associated circuits. The various building blocks and their locations in the capacitor are shown above in Figure 5.3-1. Since most of the building blocks occurred in a regular and repetitive manner, many subcircuit calls could be used to simplify the overall circuit 102 construction. The finished circuit was quite complex, with over 1000 nodes, but AC analysis completed in under a minute. The predicted and measured results are shown in Figure 5.6-2. As can be seen, the building block based prediction agrees very well with actual measurements, in both the S11 and S21 responses, up to approximately 10 GHz. The next structure considered was the resistor-capacitor series circuit. A 9segment serpentine resistor with each segment being 600 µm long was placed in series with the 10 finger interdigital capacitor (Figure 5.4-3). An important point to note here is that the longest line segment in this system is over 5.6 mm in length, which is electrically over 3/5 wavelength long at 10 GHz. The equivalent circuit for this system was developed using the same procedure as above, except that the parallel coupled lines in the resistor were connected together by the U shaped building block circuit. Once the circuit for the series RC circuit was obtained, it was used to predict the voltage magnitude and phase response of a RLC resonant circuit terminated in a 50 Ω impedance (Figure 5.6-1). 103 L = 10nH Vout R C 50 Ohm Series RC Figure 5.6-1. RLC resonant tank circuit. The RLC resonator is a good demonstration circuit since both capacitance and resistance need to be modeled correctly in order to predict the output resonance point and the shape of the curve. As mentioned earlier, the actual response of the RLC resonator was generated artificially from the HP MDS circuit simulator, in which the RC series combination was modeled using the measured S-parameter data directly. The voltage magnitude and phase data was then compared to results obtained from simulations of the building block based equivalent circuit. The results are shown in Figure 5.6-3. As can be seen, we have extremely good agreement in both magnitude and phase responses up to approximately 10 GHz. The point of resonance is predicted well, with only a slight divergence developing beyond that point. 104 1.00E+00 8.00E-01 6.00E-01 4.00E-01 Meas. S11(R) Predicted S11(R) Meas. S11(I) Predicted S11(I) S11 2.00E-01 0.00E+00 -2.00E-01 -4.00E-01 -6.00E-01 -8.00E-01 -1.00E+00 1.00E+08 1.00E+09 1.00E+10 Frequency (Hertz) 5.00E-01 4.00E-01 3.00E-01 Meas. S21(R) Predicted S21(R) Meas. S21(I) Predicted S21(I) S21 2.00E-01 1.00E-01 0.00E+00 -1.00E-01 -2.00E-01 1.00E+08 1.00E+09 1.00E+10 Frequency (Hertz) Figure 5.6-2. Measured and predicted results for 10 segment interdigital capacitor. 105 0 Voltage Magnitude(dB) -5 -10 -15 -20 V(dB) - MDS V(dB) - Predicted -25 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) 60 40 Voltage Phase (degrees) 20 0 -20 -40 -60 -80 -100 -120 1.00E+08 V(Phase) - MDS V(Phase) - Predicted 1.00E+09 1.00E+10 Frequency (Hz) Figure 5.6-3. Actual (MDS) and predicted resonator voltage magnitude and phase. 106 5.7. Conclusion In this chapter, the modeling method presented in this thesis has been applied to the modeling of interdigital capacitors and also to a series resistor capacitor circuit. A 10 segment interdigital capacitor and a electrically long series serpentine resistor capacitor structure have been modeled predictively with accurate results up to 10 GHz, only using data gathered from 3 test structures. In addition, the resonance of a RLC tank circuit has been predicted well using the developed models. Complete circuit models of all the structures have been developed, and fast simulation speeds on the order of a few minutes have been obtained. In the next chapter, the modeling method will be applied to the modeling of a spiral inductor. 107 CHAPTER VI MODELING OF PLANAR SPIRAL INDUCTORS 6.1. Introduction Planar spiral inductors are used extensively in modern integrated circuits, in both silicon and gallium arsenide technologies. They are particularly common in microwave integrated circuits where they usually are an integral component within the system. These spiral inductors are usually important enough in these type of circuits that it is not uncommon that they occupy 50% or more of overall integrated circuit die area. Inductors are used in key circuit building blocks such as oscillators, matching circuits and filters. Inductors are usually designed to have current flowing in a spiral pattern to generate mutual inductance between currents traveling in the same direction within a structure. For standard components, this has meant that the inductor usually consists of a core with a solenoid around it. These kind of structures have not been very amenable to miniaturization in the surface mount arena, and as a result they have tended to remain quite large, and considerably more so than their resistor and capacitor counterparts. Miniaturization and integration of inductors is very attractive, whenever system size reductions and board space conservation are important. 108 Successful design of systems using spiral inductors requires that accurate models of them exist, particularly at high frequencies. It is particularly important to account for losses (both substrate and conductor) to correctly predict the quality (Q) factor, which is very important for designs. Accurate models of spiral inductors are quite difficult to obtain, and they usually do not predict Q factors well. In this chapter the developed modeling methodology will be applied to the predictive modeling of spiral inductors. Accurate results as compared to a method-of-moments simulator will be shown for a 4 turn spiral inductor for both Q-factor and Z-parameter responses up to and exceeding the first self-resonance. In addition, the model validity will be verified in several different LC resonant circuits, with good results. 6.2. Modeling Procedure Modeling of spiral inductors proceeded using the same flow as was discussed in Chapter 3. The various step involved are now briefly described. 1. The first step involved a determination of what geometry structures were to be considered and allowed in order to set up a practical set of building blocks and test structures to be measured and characterized. In this case, considerably larger structures were considered than in the previous resistor and capacitor examples. 2. In this phase, the actual geometries of the various building blocks are determined. This is accomplished by entering the geometry of a target structure into the custom current flow visualization software and examining the output current distribution. 109 Building block boundaries are cut across sections of approximately uniform current distribution only. 3. Once the various building blocks have been determined, the next step is to design test structures the help model the various building blocks accurately. Additionally, a sensitivity analysis is performed on the test structure equivalent circuits to ensure that the various parameters can be accurately deembedded. 4. At this point, test structures are fabricated and tested. High frequency network analysis and DC resistance measurements are taken. In this case, test structures and the devices were not actually fabricated, due to the unavailability of a two layer process. All structures were simulated in a method of moments (MoM) simulator to approximate actual fabricated behavior. 5. MoM simulation data is used to form optimization input files for the test structures. Initial guesses are made based on the measured results for each structure. Once optimization for one structure is complete, the results are used for the remaining optimizations. 6. Circuit models of the building blocks are obtained. The models, with associated design rules are combined in a library which can then be used for device and circuit design applications. 110 6.3. Detailed Modeling Procedure The first step involved in the spiral inductor modeling procedure was a determination of which geometry structures were to be modeled. As in the earlier cases, a restricted set was defined in order to simplify the problem. The set size, though small, was still adequate to help model a wide range of different structures. For modeling inductors, much larger block sizes were used than was the case for either the serpentine resistors or interdigital capacitors. For the devices discussed in this chapter, equal linewidths and spacings of 10 mils (250 µm) were considered. The basic structure of the inductors lead to the identification of 3 fundamental building blocks that required characterization (Figure 6.3-1). The first building block was simply a 10 mil x 10 mil square of material connected on two opposite sides by additional material. The second building block was two interacting, but physically disconnected squares of material to account for capacitive and inductive coupling between segments. The third and fourth blocks were a U-shaped piece of material, used in the second test structure to connect adjacent line segments together, and a coupled corner piece to model corner effects and the coupling between two of them. The U-shaped piece itself is not needed for spiral inductor modeling, but it is required in the second test structure (shown below) that will be used. A pad was not used in this case since the structures were physically tested. The addition of a probe pad would not add any more test structures, however. 111 Coupled Corner Building Block Uncoupled Square Building Block Coupled Square Building Block Figure 6.3-1. Spiral inductor and associated building blocks. A representative spiral inductor structure was input in to the current visualization software and analyzed. The output current density and contour plots are shown in Figure 6.3-2 and Figure 6.3-3. According to the structure geometry, it was assumed that the majority of nonuniform current flow would occur in the corner regions, and the presence of both X and Y directed gradients in those regions confirmed the assumption. Examination of the output plots from the current visualization software lead to the determination of the geometries of the various building blocks. The single square and the coupled square building blocks were only one square in width as expected due to the fact that their was no spatially differential current flow across their boundaries. The L shaped corner sections were taken to be 3 squares long (the corner square plus one square on 112 either side of the corner connection points) due to the fact that the current flow became uniform again about 1 ½ squares away from the corners. The U-shaped bend was taken to be 7 squares in length as in both the resistor and capacitor cases. As mentioned earlier, the U-shaped bend is not used directly in spiral inductor modeling, but is required in coupling capacitance and mutual inductance extraction in the second test structure, as will be described. Figure 6.3-2. Indexed color intensity plots of current distribution in spiral inductor. 113 Figure 6.3-3. Contour plot of X and Y directed current gradients showing current crowding in spiral inductor. Three test structures were built in order to model the stated building blocks, (Figure 6.3-4). The first test structure is simply a material line; the purpose of this structure is to help characterize basic uncoupled material parameters, including self resistance, inductance, and capacitance. The second test structure is a 3-segment meander resistor; this structure requires characterization of the U-shaped section even though it is not used in the spiral inductor, but its main purpose is to characterize line to line mutual inductance and coupling capacitance. The third structure is a coupled line with a coupled corner bend. The purpose of this structure is to help characterize the coupled corner building block. This structure also uses the U-shaped building block and the coupled line building blocks. 114 Uncoupled Material Square Test Structure 1 Test Structure 2 Coupled Material Square U shaped Bend Test Structure 3 Coupled Corner Primitive Figure 6.3-4. Test structures and building blocks for spiral inductor modeling. 115 6.4. Method-of-Moments Simulation The test structures and benchmark spiral inductors were not actually fabricated for this research, due to the inaccessibility of a two layer high frequency process at that time. Instead, all structures were modeled in a 2 ½-D method of moments (MoM) simulator (Hewlett-Packard Momentum) in order to generate results that mimicked actual fabricated structure behavior. All simulations were run on four hyperSPARC processor Sun SPARCstation 20 series computers, equipped with 512 MB RAM and 4GB disk space. The MoM input substrate was 20 mil thick alumina of dielectric constant 9.6 with a ground plane present on the underside of the substrate. The simulations themselves were set up with a 3 GHz meshing frequency for S-parameter simulations from 100 MHz to 20 GHz. The conductor material was 10 mils wide, with a resistance of 0.1 Ω/square. Simulations for all the test structures completed within one hour each. A benchmark 4 turn spiral inductor was simulated for model verification purposes, but in this case, the simulation required over two hours complete. 6.5. Modeling and Parameter Extraction Once the simulations of the test structures had completed, the next step was to extract circuit models for all the building blocks from which the test structures were comprised. As with the resistor and capacitor cases, the fundamental circuit used for long 116 lengths of line was the unretarded partial element equivalent circuit (PEEC). Although the structures in this case were quite long electrically, and in spite of the fact that retardation was not modeled directly in the PEEC circuit, accurate results were still obtained. Depending upon actual building block geometries, the PEEC circuit model is modified as needed. For example, for modeling of coupling behavior, coupling capacitances and mutual inductances are included between parallel PEEC segments. The coupled corner building blocks are also modeled using coupled PEEC circuits for the actual corner sections but also including coupled material square building blocks on one edge, with the actual coupling component absorbed into the coupling portion of the corner coupling circuit parameters. The shown circuit model is duplicated on both sides of the diagonal cut line. The building blocks, extracted circuits and parameter values are shown in Figure 6.5-1. 6.5.1. Sensitivity Analysis A sensitivity analysis was performed on the test structures with respect to the individual building block circuit components to determine their relative importance and the level of influence on the test structure S-parameter output responses. Although the geometries of test structures 1 and 2 are similar to both those of the resistor and capacitor, they are much larger in actual dimensions. A square of material in this case was taken to be 10 x 10 mils (250 um x 250 um) in size, about a factor of eight larger than in the previous cases. All the test structures exhibit considerably different S-parameter 117 sensitivity responses to the individual circuit components than in previous cases, mainly because of much larger sizes. Sensitivities were computed using the finite difference method discussed in Chapter 3. Sensitivity plots for test structure 1 are shown in Figure 6.5-1 to Figure 6.5-4, for test structure 2 in Figure 6.5-5 to Figure 6.5-10, and for test structure 3 in Figure 6.5-11 to Figure 6.5-19. As can be seen in the plots, all parameters are capable of influencing the S-parameter response considerably, especially at higher frequencies, including resistance and shunt capacitance components. Due to the relatively high sensitivity responses of all the components, they could all be deembedded with repeatability from the circuit optimization procedure. As compared to the sensitivity plots generated from the resistor and capacitor structures, in this case all the parameter sensitivities were much higher because of the overall larger size of the structures. 118 10 8 Normalized Sensitivity 6 4 2 0 -2 -4 S11 (R) -6 -8 S11 (I) S21 (R) S21 (I) -10 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.5-1. S21 and S11 sensitivity of test structure 1 with respect to line-to-ground capacitance in the uncoupled square building block. 10 8 Normalized Sensitivity 6 4 2 0 -2 -4 S11 (R) -6 -8 -10 1.00E+08 S11 (I) S21 (R) S21 (I) 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.5-2. S21 and S11 sensitivity of test structure 1 with respect to line inductance in the uncoupled square building block. 119 3 2.5 Normalized Sensitivity 2 1.5 1 0.5 0 -0.5 -1 S11 (R) S11 (I) -1.5 S21 (R) S21 (I) -2 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.5-3. S21 and S11 sensitivity of test structure 1 with respect to line resistance in the uncoupled square building block. 0.4 0.3 Normalized Sensitivity 0.2 0.1 0 -0.1 -0.2 -0.3 S11 (R) S11 (I) S21 (R) S21 (I) -0.4 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.5-4. S21 and S11 sensitivity of test structure 1 with respect to shunt capacitance in the uncoupled square building block. 120 10 8 Normalized Sensitivity 6 4 2 0 -2 -4 -6 S11 (R) -8 S11 (I) S21 (R) S21 (I) -10 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.5-5. S21 and S11 sensitivity of test structure 2 with respect to line-to-line coupling capacitance in the coupled squares building block. 10 8 Normalized Sensitivity 6 4 2 0 -2 -4 -6 S11 (R) -8 S11 (I) S21 (R) S21 (I) -10 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.5-6. S21 and S11 sensitivity of test structure 2 with respect to line-to-ground capacitance in the U building block. 121 3 Normalized Sensitivity 2 1 0 -1 S11 (R) -2 S11 (I) S21 (R) S21 (I) -3 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.5-7. S21 and S11 sensitivity of test structure 2 with respect to shunt capacitance in the U building block. 10 8 Normalized Sensitivity 6 4 2 0 -2 -4 -6 -8 S11 (R) S11 (I) S21 (R) S21 (I) -10 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.5-8. S21 and S11 sensitivity of test structure 2 with respect to line-to-line mutual inductance in the coupled squares building block. 122 10 8 Normalized Sensitivity 6 4 2 0 -2 -4 S11 (R) -6 -8 S11 (I) S21 (R) S21 (I) -10 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.5-9. S21 and S11 sensitivity of test structure 2 with respect to line-to-ground capacitance in the coupled squares building block. 10 8 Normalized Sensitivity 6 4 2 0 -2 -4 S11 (R) -6 S11 (I) S21 (R) -8 -10 1.00E+08 S21 (I) 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.5-10. S21 and S11 sensitivity of test structure 2 with respect to line inductance in the U-shaped building block. 123 10 8 Normalized Sensitivity 6 4 2 0 -2 -4 S11 (R) -6 -8 S11 (I) S21 (R) S21 (I) -10 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.5-11. S21 and S11 sensitivity of test structure 3 with respect to line inductance in the uncoupled squares section of the coupled corner building block. 5 4 Normalized Sensitivity 3 2 1 0 -1 -2 -3 S11 (R) S11 (I) -4 S21 (R) S21 (I) -5 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.5-12. S21 and S11 sensitivity of test structure 3 with respect to line resistance in the uncoupled squares section of the coupled corner building block. 124 0.5 0.4 Normalized Sensitivity 0.3 0.2 0.1 0 -0.1 -0.2 S11 (R) -0.3 S11 (I) S21 (R) -0.4 S21 (I) -0.5 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.5-13. S21 and S11 sensitivity of test structure 3 with respect to shunt capacitance in the uncoupled squares section of the coupled corner building block. 1 0.8 Normalized Sensitivity 0.6 0.4 0.2 0 -0.2 -0.4 S11 (R) -0.6 S11 (I) S21 (R) -0.8 -1 1.00E+08 S21 (I) 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.5-14. S21 and S11 sensitivity of test structure 3 with respect to line-to-line coupling capacitance in the coupled corner building block. 125 5 4 Normalized Sensitivity 3 2 1 0 -1 -2 -3 S11 (R) S11 (I) S21 (R) -4 S21 (I) -5 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.5-15. S21 and S11 sensitivity of test structure 3 with respect to line-to-ground capacitance in the coupled corner building block. 0.5 0.4 Normalized Sensitivity 0.3 0.2 0.1 0 -0.1 -0.2 S11 (R) -0.3 -0.4 S11 (I) S21 (R) S21 (I) -0.5 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.5-16. S21 and S11 sensitivity of test structure 3 with respect to shunt capacitance in the coupled corner building block. 126 5 4 Normalized Sensitivity 3 2 1 0 -1 -2 S11 (R) -3 S11 (I) S21 (R) -4 S21 (I) -5 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.5-17. S21 and S11 sensitivity of test structure 3 with respect to line-to-line mutual inductance in the coupled corner building block. 10 8 Normalized Sensitivity 6 4 2 0 -2 -4 S11 (R) -6 -8 -10 1.00E+08 S11 (I) S21 (R) S21 (I) 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.5-18. S21 and S11 sensitivity of test structure 3 with respect to line inductance in the coupled corner building block. 127 5 4 Normalized Sensitivity 3 2 1 0 -1 -2 S11 (R) -3 -4 S11 (I) S21 (R) S21 (I) -5 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.5-19. S21 and S11 sensitivity of test structure 3 with respect to line resistance in the coupled corner building block. 6.5.2. Model Extraction Circuit model parameter extraction proceeded in several steps. Due to the highly nonlinear nature of the problem, a procedure of hierarchical optimization with respect to MoM generated S-parameter data was chosen. Initial guesses for the various parameters were derived from actual measurement data. All optimizations and simulations were done using the Hspice circuit simulator on Sun SPARCstation 20 series workstations. Details of the extraction method can be found in Chapter 3. Test structure 1 was initially optimized in order to extract the parameters of the uncoupled material square. The initial guesses were inserted, and the circuit was optimized with respect to measurements up to 10 GHz. Once the optimization was 128 complete, the computed models were taken and used as valid model parameters for their respective building blocks for test structure 2. The parameters to be computed for this structure were the first order line to line coupling parameters (mutual inductance and coupling capacitance), and the parameters for the U-shaped corner. Additionally, line to ground capacitance was recomputed for the material square building block in the presence of adjacent lines which provided a shielding effect. The third test structure was optimized in order to extract the model for the coupled corner building block. All optimizations completed with low residual sum of squares error, indicating accurate results. The modeling results for test structures 1, 2 and 3 are shown in Figure 6.5-2, Figure 6.5-3, and Figure 6.5-4 respectively. The extracted circuit models and parameters are shown in Figure 6.5-1. 129 2 1 R Uncoupled Line Bldg. Block L R CC 1 R CC 2 L C 3 R R=0.06 Ohm L= 5.7E-11 H C = 2.1e-14 F CC=3.5E-14 F CM=5.0e-15 F LM = 0.32 LM CM 4 Coupled Line Bldg. Block CC L LM 2 R=0.06 Ohm L= 7.1E-11 H C = 3.1e-14 F CC=3.5E-14 F 2 L C 3 1 CC CC 1 CC R L L C R 4 1 2 CC CC 1 R L L R C U-Shaped Bend Bldg. Block Eq. circuit on each side of diagonal cut line 1C 1 1 2C CC U R CC L LM 2 L C CC 2 1 CC L 3 2 U R C L R 2C CC CC 1 R 2 L L R 1C R LM CM 4 Coupled Corner Bldg. Block R=0.45 Ohm L= 2.9E-11 H C = 1.8E-13 F CC=5.5E-14 2 R=0.37 Ohm L= 2.0E-10 H C = 2.0E-14 F CM=4.0E-15 F CC=1.0E-14 LM = 0.20 R=0.1 Ohm L= 1.1E-10 H C = 3.7E-14 F CC=1.0E-15 F C Figure 6.5-1. Building blocks, equivalent circuits, and parameters for spiral inductor modeling. 130 3.50E-01 3.00E-01 2.50E-01 2.00E-01 1.50E-01 1.00E-01 5.00E-02 0.00E+00 -5.00E-02 MoM S11(R) -1.00E-01 -1.50E-01 Modeled S11(R) MoM S11(I) Modeled S11(I) -2.00E-01 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) 1.00E+00 8.00E-01 6.00E-01 4.00E-01 2.00E-01 0.00E+00 -2.00E-01 -4.00E-01 -6.00E-01 MoM S21(R) Modeled S21(R) -8.00E-01 MoM S21(I) Modeled S21(I) -1.00E+00 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.5-2. S-parameter measured and modeled results for test structure 1. 131 5.00E-01 4.00E-01 3.00E-01 2.00E-01 1.00E-01 0.00E+00 -1.00E-01 -2.00E-01 -3.00E-01 MoM S11(R) Modeled S11(R) MoM S11(I) -4.00E-01 Modeled S11(I) -5.00E-01 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) 1.00E+00 8.00E-01 6.00E-01 4.00E-01 2.00E-01 0.00E+00 -2.00E-01 -4.00E-01 MoM S21(R) -6.00E-01 -8.00E-01 Modeled S21(R) MoM S21(I) Modeled S21(I) -1.00E+00 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.5-3. S-parameter measured and modeled results for test structure 2. 132 1.00E+00 8.00E-01 6.00E-01 4.00E-01 2.00E-01 0.00E+00 -2.00E-01 -4.00E-01 -6.00E-01 MoM S11(R) Modeled S11(R) MoM S11(I) -8.00E-01 Modeled S11(I) -1.00E+00 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) 1.00E+00 8.00E-01 6.00E-01 4.00E-01 2.00E-01 0.00E+00 -2.00E-01 -4.00E-01 MoM S21(R) -6.00E-01 Modeled S21(R) MoM S21(I) -8.00E-01 Modeled S21(I) -1.00E+00 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.5-4. S-parameter measured and modeled results for test structure 3. 133 6.6. Results The extracted building blocks were used to predict the electrical behavior of a four turn rectangular spiral inductor (Figure 6.6-1). The inductor was designed with the longest outside segment length being 230 mils, yielding an overall size of over 2000 mils, which is over one wavelengths long at 2 GHz. As before, all the building block models were extracted from the test structures only.. In addition, the operation of the inductor in actual circuits was tested. Since no structures were physically fabricated, all comparisons were made to the method-of-moments simulator results. Additionally, the circuits were not actually constructed, but were simulated within the Microwave Design System environment using the S-parameter data as a model. 134 170 mil 230 mil Figure 6.6-1. 4 turn spiral inductor predictively modeled. It is of interest to note that none of the test structures were designed to be inductors, and that none of them had current in parallel branches flowing in the same direction as is usually found in inductive components. The equivalent circuit for the spiral inductor was constructed by replacing the different geometrical sections with the applicable building block circuit models. Each ‘parallel square slice’ of material in parallel line segments was modeled with coupled building blocks, which were cascaded in order to generate the required line length. The corners were modeled using the couple corner building blocks. The various building blocks and their locations within the inductor are shown above in Figure 6.3-1. Since most of the building blocks occurred in a regular and repetitive manner, many subcircuit calls could be used to simplify the overall 135 circuit construction. The finished circuit consisted of the 4 turn inductor consisted of over 2000 elements and 600 nodes. For the spiral inductor modeling case, an impedance parameter analysis was done, to investigate input impedance and phase characteristics of the inductor. Z-parameter data was obtained from measured S-parameter data using the MDS software and the circuit configuration shown in Figure 6.6-2 (top), a similar circuit was used to generate the Zparameter data from the developed model Figure 6.6-2 (bottom). The predicted and actual (MoM generated) Z-parameter results are shown in Figure 6.6-3 and Figure 6.6-4. As can be seen, the building block based prediction agrees very well with actual measurements, in both the Z11(dB) and phase responses, up to the first self-resonance, and actually quite well beyond that also. At these higher frequencies beyond the first self-resonance, general behavior patterns are predicted quite well, although offset slightly in frequency. For most practical applications, the only useful range of any passive component will be well below the first self-resonance, since beyond that point the component exhibits characteristics of its reactive counterpart due to a phase inversion. 136 1 Port 1 Inductor S-Parm. Dataset Port 2 2 Port 2 Port 1 Figure 6.6-2. Z-parameter circuit configurations for inductor analysis (top) MDS configuration (bottom) circuit predictive model configuration. 137 70 60 Z11(dB) 50 Z11(db) Modeled Z11(db) MDS 40 30 20 10 0 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.6-3. Measured and predicted results for Z11(dB) of four turn spiral inductor. 100 80 60 40 Z11(phase) 20 Z11(phase) Modeled Z11(phase) MDS 0 -20 -40 -60 -80 -100 -120 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.6-4. Measured and predicted results for Z11(phase) of four turn spiral inductor. 138 To test the validity of the developed inductor circuit model, the model was used in several different tank circuit configurations. All of the following circuit comparisons have been made with respect to circuits simulated within the MDS simulator using Sparameter data directly as a model for the inductor. In the first circuit, a 1 pF capacitor was placed in parallel with the inductor, with a 50 Ω terminating resistor and an AC small signal excitation (Figure 6.6-5). The voltage magnitude and phase data was then compared to results obtained from circuit simulations of the building block based equivalent circuit. The results are shown in Figure 6.6-6 and Figure 6.6-7. As can be seen, we have extremely good agreement in both magnitude and phase responses up to approximately 10 GHz. The point of resonance is predicted well, as can be more clearly seen in the phase plot since the lossy metal used (0.1 Ω/square) does not show a well pronounced notch in the voltage response. C=1pF Vout 50 Ohm 4-turn inductor Figure 6.6-5. LC resonant tank circuit. 139 5.00E+00 Voltage Magnitude (dB) 0.00E+00 -5.00E+00 -1.00E+01 -1.50E+01 -2.00E+01 -2.50E+01 Modeled Vout(dB) MDS Vout(dB) -3.00E+01 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.6-6. Actual (MDS) and predicted LC circuit output voltage magnitude. 150 Voltage Phase (degrees) 100 50 0 -50 -100 Modeled Vout(phase) MDS Vout(phase) -150 -200 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.6-7. Actual (MDS) and predicted LC circuit output voltage phase. 140 A second, more ambitious LC resonant circuit was also constructed, but this time with two of the inductors in parallel with each other. All “manufactured circuit” results were generated by MDS as before, and all predictions using the circuit model were obtained from a circuit simulator. The constructed circuit is shown in Figure 6.6-8. Actual and predicted circuit responses are shown in Figure 6.6-9 and Figure 6.6-10. As can be seen from the plots, both output voltage magnitude and phase, as well as the selfresonance frequency, are predicted well. Additionally, behavior beyond the first resonance is also tracked, except for some deviations in magnitude and offsets in frequency. As mentioned earlier, components are almost never used in circuits beyond their self-resonant frequency, and thus accurate predictions up to that point are essential, but beyond that have limited use. C=5pF Vout 50 Ohm 2 4-turn inductors Figure 6.6-8. LC circuit with 2 4-turn inductors in parallel. 141 5.00E+00 Voltage Magnitude (dB) 0.00E+00 -5.00E+00 -1.00E+01 -1.50E+01 -2.00E+01 -2.50E+01 Modeled Vout(dB) MDS Vout(dB) -3.00E+01 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.6-9. Actual (MDS) and predicted LC circuit output voltage magnitude. 1.50E+02 Voltage Phase (degrees) 1.00E+02 5.00E+01 0.00E+00 -5.00E+01 -1.00E+02 -1.50E+02 Modeled Vout(phase) MDS Vout(phase) -2.00E+02 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 6.6-10. Actual (MDS) and predicted LC circuit output voltage phase. 142 6.7. Conclusion In this chapter, the modeling method presented in this thesis has been applied to the modeling of spiral inductors and to several LC resonant circuits. A 4 turn spiral inductor has been modeled accurately using building blocks derived from test structures which are of entirely different geometries than the inductor itself, except for the presence of the common building blocks. Accurate results for the electrically long inductor have been obtained up to the first resonance, but with good behavior beyond that also. The inductor model was tested in two different LC resonant circuits, with good predictions of the resonant frequency in both cases. Complete circuit models for the inductors were developed which simulated within a circuit simulator within 1 minute. 143 CHAPTER VII MODELING OF FULLY 3-DIMENSIONAL PASSIVE DEVICE 7.1. Introduction The latest advances in integrated passive manufacturing include the ability to fabricate multilayer passive structures. Technologies such as low temperature cofired ceramic (LTCC) are becoming mainstream, and offer potentially huge savings in overall printed circuit board area. LTCC processes have the advantage of supporting a large number of layers of ceramic tape (well over 30), each capable of accepting a conductor layer. In addition, stacked via technology has been developed, allowing for connectivity between layers. As can be envisioned, a large number of passive components could possibly be moved away from the printed circuit board and integrated into a LTCC substrate within a package, thereby yielding very compact circuit boards. LTCC technology clearly shows considerable potential as an enabling technology for the next generation of highly compact systems. Successful design of LTCC structures requires that accurate models of the various LTCC components exist or can be easily obtained. For high frequency designs, most LTCC structures are electrically long, and due to their full 3-dimensional geometries, 144 have very complex field patterns. Standard modeling methods for microstrip or stripline based structures do not apply for these components, and usually the full wave 2 ½-D or 3-D solution methodologies must be used, such as the method of moments, spectral domain, finite element and finite difference time domain methods. These methods, especially the finite element and finite difference time domain method, are very accurate although very computationally expensive, and for complex 3-D structures, analysis can take many hours, and even days utilizing state-of-the-art computers and software. Due to this drawback, this type of analysis is not well suited for the iterative nature of passive component design, and is probably one of the principle factors which has contributed to the slow progress of heavily integrated LTCC substrates. In this chapter, the modeling methodology developed under this research will be applied to structures manufactured in a LTCC process. Accurate results will be shown for the predictive modeling of full 3-D solenoid style multilayer spiral inductors with interactions. In addition, modeling results will be shown for gridded plate capacitors. In all cases, results have been accurate up to the first self-resonance, beyond which the structures have limited use. A 2 ½-D or 3-D simulation of the LTCC structures was not done, primarily due to a lack of detailed information about the process and the difficulty involved in setting up an accurate run. In order to correctly define the input structure in order to obtain as much accuracy as possible, detailed information regarding the process characteristics must be taken into account. This includes setting up a multi-dielectric system, with varying dielectric thicknesses based upon the presence of metal on a layer or not. This is a result 145 of the characteristic “humping” which occurs in regions where metal is printed coincidentally on several different layers. Additionally, complex via geometries must be taken into account, including catch pads and bulging effects between layers of tape. It should be stressed that simply obtaining all the correct geometries and dielectric thicknesses would require a considerable amount of test structure design, fabrication, cross-sectioning, and measurement, and would probably require a significant amount of time to complete. Also, entering all the required geometries into a field solver would be a painstaking and difficult task. Once everything is entered, and because of the resulting non-planar input definition, it is very probable that a very large number of mesh points will be required to solve the system accurately, which in turn could take a long time to solve. Using the building block based modeling method, circuits have been developed for these structures that are comprised of relatively few components and simulate in a circuit solver in approximately one minute. The results presented in this chapter show the true potential of the developed modeling method. Properly utilized, the building block based modeling method can be an enabling technology for component design in multilayer passive component fabrication processes. 146 7.2. Modeling Procedure The modeling procedure involved in this case was somewhat different than for the planar devices discussed in earlier chapters. A brief outline of the steps involved is described below. 1. The first step involved a determination of what geometry structures were to be considered and allowed in order to set up a practical set of building blocks and test structures to be measured and characterized. For the 3-D inductors, this meant that the widths and separations of the top and bottom conductors that comprised the solenoid as well as the spacings in between parallel solenoids had to be fixed. For the gridded capacitor, this implied that the geometry of each grid square was fixed. 2. The custom current visualization software was not designed to analyze 3-D structures, and hence was not used for differential current flow analysis. Instead, building block sizes were determined based on symmetry and repetition within the structure. As an example, for the solenoid inductor, one building block was defined to be one turn. 3. Test structures for modeling the building blocks were designed, along with target structures against which the models were verified. Additionally, a small signal frequency dependent sensitivity analysis was performed on the test structure equivalent circuits to ensure that the building block circuit parameters could be successfully deembedded. 4. The structures are physically designed (12-layer LTCC process) and fabricated. LTCC process access was granted by National Semiconductor Corp. High frequency 147 S-parameter measurements of the devices are taken by on-wafer ground-signalground probing. 5. The measured data is used to create circuit optimization input files for the test structure equivalent circuits. Initial guesses are made based on the measured results for each structure. Once optimization for one structure is complete, the results are used for the remaining optimizations. 6. Circuit models of the building blocks are obtained. 7.3. Detailed LTCC Structure Modeling Procedure The first step involved in the LTCC modeling procedure was a determination of what types of structures and geometries were to be modeled. As mentioned above, two structures were considered for this process – solenoid spiral inductors and gridded plate capacitors. Solenoid inductors were chosen since inductive components are very useful in high frequency RF designs, and most designs currently employ only planar inductors. However, solenoid inductors may be preferable over planar inductors due to more confined field patterns, and possibly smaller area. The 3-D nature of solenoid inductors requires that modeling be achieved using full wave 3-D solvers, which can result in impractically long run times for design purposes. While full 3-D analysis is extremely useful for obtaining detailed information about the structure under analysis, including field patterns, current density plots, etc., it is not well suited for a design process which is 148 usually iterative in nature. The building block-based modeling method could potentially produce vast improvements in modeling and simulation time for these devices. Gridded plate capacitors were also chosen to be modeled in the LTCC process. Large area metal deposition is difficult to achieve in LTCC technology, and as a result a maximum metal area design rule restriction is usually enforced. LTCC technology is very well suited for fabrication of large valued two-layer metal-insulator-metal (MIM) capacitors for use in applications such as power supply decoupling and filtering. In order to make large capacitors which adhere to the maximum metal area requirements requires that they be manufactured with gridded instead of solid conductor plates. Solid plate capacitors could be used, but then their sizes would be restricted. Modeling of gridded plate capacitors are very important to ensure that they function as intended at high frequencies. The modeling method developed under this work would prove very useful and efficient in the modeling of these gridded structures, as compared to other nonlumped element techniques. The solenoid inductors were designed to have an upper and a lower conductor on different layers, with connections between them made by stacked vias. Both conductors were separated by 6 layers of ceramic tape (to reduced capacitive coupling between the conductors), and were connected by stacked vias (6 layers deep with catch pads on each layer) such that a solenoidal pattern of current flow through the structure was obtained. A diagram showing the general inductor geometry is shown in Figure 7.3-1. The metal conductors were designed to be 10 mils wide, with a spacing between adjacent coils of 30 mils. The bottom conductor was laid out at angle to facilitate connections between the via 149 stacks connecting the layers. Interactions between inductors were also modeled, and a spacing of 10 mils between parallel solenoids was specified, with via stacks from one solenoid being directly opposite via stacks of the parallel solenoid. Top Conductor Input Via Stack Output Bottom Conductor Figure 7.3-1. Solenoid inductor geometry. The gridded plate capacitors were also specified with a metal width of 10 mils, with a “hole” of 40 x 40 mils between the metal lines. The structure was two-layer, with a separation distance of only one layer of ceramic tape in order to maximize capacitance. It was specified that the upper and lower conductors were completely coincident, so that the metal lines of the top conductor completely overlapped the metal lines on the lower conductor. A representative structure is shown in Figure 4.3-1. Ground planes for all devices were specified to be on the lowest layer of the LTCC structure. All connections to the devices were made using a ground-signal-ground probe pad pattern on the top later, 150 with connections made to the devices using stacked vias and interconnect. All interconnect to and between structures was drawn on a single layer. Figure 7.3-2. Gridded plate capacitor geometry. 7.3.1. Solenoid Inductor and Gridded Plate Capacitor Building Blocks Current visualization of the solenoid inductors was not possible, since the custom software was unable to handle full 3-D structures. Instead, in this case, a symmetry based approach was taken. It was unlikely that the current flow at any point in the structure was constant, since the current was constantly changing direction, however, the current flow pattern would be the same from coil to coil. Because of this, each coil with its associated vias was taken to be a single building block for modeling the solenoid coil. Each segment 151 of the coil (the top and bottom conductors and the vias) were modeled with different circuit models. Parallel coil building blocks were also modeled as two parallel, single coils, connected with coupling capacitances and mutual inductances. For a fixed coil geometry, these were the only building blocks defined for solenoid inductor modeling – a rather aggressive but not unpractical set. In addition, the probe pad and interconnect squares were also defined as building blocks. The various building block geometries are shown in Figure 7.3-1. Gridded plate capacitor modeling only required three building blocks. Two of them; the probe pad and the interconnect material square building blocks were already included in the solenoid inductor building block set. The only additional block that was required was the gridded capacitor square block, consisting of surrounding metal lines and one grid “hole”. The resulting building block is shown in Figure 7.3-2. 152 Top Metal Via Coil 1 on this axis Bott o m Metal Coil 2 on this axis Co il B uilding Block includes top conductor, 2 vias and bottom conductor Co upled Co ils B uild ing Block for m odeling interactions betwe en pa rallel so len oid s T o p m e tal layer G S G p r o b e pad Un coupled Interconnect Square Build ing B lock Via to interconnect layer Interconnect Layer Vias to ground Direction of current flow through building block Ground G S G Pro be P a d Build ing B lock Figure 7.3-1. Solenoid inductor building blocks. 153 Gridded Plate Capacitor Building Block Figure 7.3-2. Gridded plate capacitor building block. 7.4. Solenoid Inductor and Gridded Plate Capacitor Test Structures There were a total of five building blocks defined for the modeling of solenoid inductors and gridded plate capacitors. The building blocks for the inductor were a single coil; 2 coupled coils; the probe pad and the interconnect square. The capacitor building blocks included the gridded capacitor square building block, as well as the probe pad and interconnect square blocks which were also used for the inductor. The fact that two of the building blocks were shared allowed for designing and manufacturing only one set of test structures for both the inductor and capacitor. This building block definition lead to the design of 4 test structures. The first test structure was simply a line consisting of interconnect material, with the probe pads on its ends. Modeling of this test structure would allow for 154 characterization of the probe pad building block and the interconnect square building block. The models generated by this block could be shared between both the inductor and capacitor. The 2nd test structure was a single inductor coil with probe pads. This structure allowed for the modeling of the uncoupled inductor coil building block. The next test structure was a serially connected 3-segment parallel coil inductor. This structure helped characterize coupling between parallel coils. The last test structure was a simple gridded plate capacitor, which would allow for the characterization of the gridded capacitor square building block. It is noteworthy that this last test structure was the only additional structure required to model these complex gridded plate capacitors. The various test structures are shown in Figure 7.4-1 and Figure 7.4-2. 155 Interconnect Line Bldg. Block Probe Pad Bldg. Block Test Structure 1 Test Structure 2 Test Structure 3 Inductor Coil Bldg. Block Coupled Inductor Coils Bldg. Block Figure 7.4-1. Test structures for solenoid inductor modeling. 156 Test Structure 4 G ridded Capacitor Plate Bldg. Block Figure 7.4-2. Additional test structure for gridded plate capacitor modeling. 7.5. Structure Fabrication and Measurement The test structure coupon was physically design within the Cadence Virtuoso design environment. A custom technology file for a 12-layer process was developed, and a process design rule compliant test structure coupon was produced. The design was fabricated at the National Semiconductor Corp. LTCC fabrication facility through the RF/Wireless design group. The size of the completed coupon was approximately 2.25” x 2.25”. Each layer of ceramic tape was specified to be 3.6 mils thick with a dielectric 157 constant of 7.8. The metal lines were drawn to be 10 mils wide, and the vias were a diameter of 5.6 mils. The designed mask of the LTCC coupon is shown in Figure 7.5-1 with a photograph of the top side of the fabricated coupon in Figure 7.5-2 and the bottom side showing the last embedded layer in Figure 7.5-3. Manufactured characteristics of lines and vias are also shown in Figure 7.5-4, Figure 7.5-5, and Figure 7.5-6. It is clear from these photographs that lines and vias are not very uniform in this process. The test structures were measured using network analysis techniques. Since very low loss metal was used in the manufacturing process, DC resistance measurements were unreliable and were not used. For the high frequency measurements, a HP 8510C network analyzer was used in conjunction with a Cascade Microtech probe station and ground-signal-ground configuration probes. Calibration was accomplished using a supplied substrate and utilization of the line-reflect-match (LRM) calibration method. Data was gathered for each of the test structures at over 200 frequency points between 45MHz and 5GHz and stored with the aid of computer data acquisition software and equipment. Data points beyond 5 GHz were not taken since most of the devices were already in resonance before that point. 158 Figure 7.5-1. Physical layout of LTCC coupon 159 Figure 7.5-2. Photograph of top side of fabricated LTCC coupon. 160 Figure 7.5-3. Photograph of bottom side of LTCC coupon with last embedded layer partially visible. 161 Figure 7.5-4. Photograph of cross section of metal line in a LTCC structure along the line length (photograph courtesy of National Semiconductor Corp.) 162 Figure 7.5-5. Photograph of cross section of metal line across line width (short) (photograph courtesy of National Semiconductor Corp.) 163 Figure 7.5-6. Photograph of cross section of 2 via stack (photograph courtesy of National Semiconductor Corp.) 7.6. Modeling and Parameter Extraction As in previous cases, the fundamental circuit model used for modeling segments of building blocks was the partial element equivalent circuit (PEEC) with modifications made as necessary. However, in this case shunt capacitances were excluded from the model, since they apparently had very little effect on any of the output responses of the 164 previously discussed passive devices, as can be seen in their small-valued sensitivity responses to these parameters. The physical structure of the probe pad was quite complicated (Figure 7.3-1), but the section which was populated with parallel long stacked vias belonged to the ground plane, and hence it was anticipated that it did not contribute significantly to the overall Sparameter responses. A simple PEEC circuit was used in this case, and as can be seen later, was able to model the pad behavior adequately. The interconnect material square also was modeled with a simple PEEC circuit, and this too gave good results. The inductor coil was modeled with two separate PEEC circuit models; one each for the upper and lower conductors of the inductor, and two LC circuits; one each for the via stacks connecting the two conductors. Due to the geometry of the inductor coils, it was assumed that the majority of coupling would occur between adjacent via posts, and in order to model this, mutual inductances and coupling capacitances were added between the via stacks. This coupling mechanism was modeled accurately, and correctly helped model the behavior of parallel inductor coils as will be shown later. The gridded capacitor plate building block was modeled with four sets of coupled PEEC circuits which represented the metal conductors surrounding the cavity in the capacitor building block. 7.6.1. Sensitivity Analysis In order to determine whether individual building block circuit components could be deembedded from the designed test structures, a sensitivity analysis was performed. 165 The sensitivity analysis was performed on the test structure equivalent circuits with respect to each building block circuit parameter that was desired to extracted. The results of the sensitivity analysis showed exactly how the S-parameters varied when one circuit parameter was differentially modified. Normalized plots of the various sensitivities are shown. In general, a non-zero non-flat response shows that the output is affected by the parameter over frequency, and thus should be extractable. Test structure 1 sensitivity responses are shown in Figure 7.6-1 - Figure 7.6-4. In this set of plots the sensitivities with respect to line resistances were close to zero and are not shown, but apart for that, all the capacitive and inductive parameters show that they affect the output response substantially, particularly at higher frequencies. Test structure 2 responses, shown in Figure 7.6-5 - Figure 7.6-14, show similar results for the reactive components. It is interesting to note, however, that conductance to ground of the top conductor does influence the output response considerably. Line resistances have a measurable effect for these structures also. Test structure 3 data is shown in Figure 7.6-15 and Figure 7.6-16, and large sensitivity responses are obtained for both the coupling capacitance and mutual inductance. Finally, test structure 4 sensitivity plots are shown in Figure 7.6-17 - Figure 7.6-22. As might be expected, the largest sensitivity response occurs for capacitance between the conductors, but the plots also show considerable sensitivity responses for the capacitances to ground of both upper and lower conductors, as well as to inductances, particularly at higher frequencies when the device starts to conduct. 166 0.5 0 Normalized Sensitivity -0.5 -1 -1.5 -2 S11(R) -2.5 S11(I) S21(R) S21(I) -3 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.6-1. S11 and S21 sensitivity responses of test structure 1 with respect to capacitance to ground in the interconnect line building block. 2 1.5 1 Normalized Sensitivity 0.5 0 -0.5 -1 -1.5 -2 -2.5 S11(R) S11(I) -3 -3.5 1.00E+08 S21(R) S21(I) 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.6-2. S11 and S21 sensitivity responses of test structure 1 with respect to line inductance in the interconnect line building block. 167 0.5 0 Normalized Sensitivity -0.5 -1 -1.5 -2 -2.5 S11(R) -3 S11(I) S21(R) S21(I) -3.5 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.6-3. S11 and S21 sensitivity responses of test structure 1 with respect to capacitance-to-ground in the probe pad building block. 1.5 1 Normalized Sensitivity 0.5 0 -0.5 -1 -1.5 S11(R) -2 S11(I) S21(R) -2.5 1.00E+08 S21(I) 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.6-4. S11 and S21 sensitivity responses of test structure 1 with respect to line inductance in the probe pad building block. 168 5 4 Normalized Sensitivity 3 2 1 0 -1 -2 -3 S11(R) S11(I) -4 S21(R) S21(I) -5 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.6-5. S11 and S21 sensitivity responses of test structure 2 with respect to capacitance-to-ground of the top conductor in the inductor coil building block. 5 4 Normalized Sensitivity 3 2 1 0 -1 -2 -3 S11(R) S11(I) -4 S21(R) S21(I) -5 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.6-6. S11 and S21 sensitivity responses of test structure 2 with respect to line inductance of the top conductor in the inductor coil building block. 169 0.5 0.4 Normalized Sensitivity 0.3 0.2 0.1 0 -0.1 S11(R) -0.2 S11(I) S21(R) S21(I) -0.3 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.6-7. S11 and S21 sensitivity responses of test structure 2 with respect to line resistance of the top conductor in the inductor coil building block. 5 4 Normalized Sensitivity 3 2 1 0 -1 -2 -3 S11(R) S11(I) -4 S21(R) S21(I) -5 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.6-8. S11 and S21 sensitivity responses of test structure 2 with respect to line conductance of the top conductor in the inductor coil building block. 170 5 4 Normalized Sensitivity 3 2 1 0 -1 -2 -3 S11(R) S11(I) -4 -5 1.00E+08 S21(R) S21(I) 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.6-9. S11 and S21 sensitivity responses of test structure 2 with respect to lineto-ground capacitance of the bottom conductor in the inductor coil building block. 5 4 Normalized Sensitivity 3 2 1 0 -1 -2 -3 S11(R) -4 S11(I) S21(R) S21(I) -5 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.6-10. S11 and S21 sensitivity responses of test structure 2 with respect to line inductance of the bottom conductor in the inductor coil building block. 171 0.5 0.4 Normalized Sensitivity 0.3 0.2 0.1 0 -0.1 S11(R) -0.2 S11(I) S21(R) S21(I) -0.3 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.6-11. S11 and S21 sensitivity responses of test structure 2 with respect to line resistance of the bottom conductor in the inductor coil building block. 0.02 0.01 Normalized Sensitivity 0 -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 S11(R) -0.07 S11(I) S21(R) S21(I) -0.08 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.6-12. S11 and S21 sensitivity responses of test structure 2 with respect to line conductance to ground of the bottom conductor in the inductor coil building block. 172 5 4 Normalized Sensitivity 3 2 1 0 -1 -2 -3 S11(R) S11(I) -4 S21(R) S21(I) -5 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.6-13. S11 and S21 sensitivity responses of test structure 2 with respect to via capacitance in the inductor coil building block. 5 4 Normalized Sensitivity 3 2 1 0 -1 -2 -3 S11(R) -4 S11(I) S21(R) S21(I) -5 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.6-14. S11 and S21 sensitivity responses of test structure 2 with respect to via inductance in the inductor coil building block. 173 5 4 Normalized Sensitivity 3 2 1 0 -1 -2 -3 S11(R) S11(I) -4 S21(R) S21(I) -5 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.6-15. S11 and S21 sensitivity responses of test structure 3 with respect to coupling capacitance in the interacting inductor coil building block. 5 4 Normalized Sensitivity 3 2 1 0 -1 -2 -3 S11(R) S11(I) -4 S21(R) S21(I) -5 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.6-16. S11 and S21 sensitivity responses of test structure 3 with respect to line-to-line mutual inductance in the interacting inductor coil building block. 174 5 4 Normalized Sensitivity 3 2 1 0 -1 -2 -3 S11(R) S11(I) -4 S21(R) S21(I) -5 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.6-17. S11 and S21 sensitivity responses of test structure 4 with respect to capacitance-to-ground of the top plate in the gridded capacitor building block. 5 4 Normalized Sensitivity 3 2 1 0 -1 -2 S11(R) -3 S11(I) S21(R) -4 -5 1.00E+08 S21(I) 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.6-18. S11 and S21 sensitivity responses of test structure 4 with respect to capacitance-to-ground of the bottom plate in the gridded capacitor building block. 175 5 4 Normalized Sensitivity 3 2 1 0 -1 -2 S11(R) -3 -4 S11(I) S21(R) S21(I) -5 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.6-19. S11 and S21 sensitivity responses of test structure 4 with respect to mutual capacitance between the plates in the gridded capacitor building block. 5 4 Normalized Sensitivity 3 2 1 0 -1 -2 -3 -4 S11(R) S11(I) S21(R) S21(I) -5 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.6-20. S11 and S21 sensitivity responses of test structure 4 with respect to mutual inductance between the plates in the gridded capacitor building block. 176 5 4 Normalized Sensitivity 3 2 1 0 -1 -2 -3 S11(R) S11(I) -4 S21(R) -5 S21(I) 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.6-21. S11 and S21 sensitivity responses of test structure 4 with respect to line inductance for both plates in the gridded capacitor building block. 1.2 1 0.8 Normalized Sensitivity 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 S11(R) S11(I) -0.8 -1 1.00E+08 S21(R) S21(I) 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.6-22. S11 and S21 sensitivity responses of test structure 4 with respect to line resistance for both plates in the gridded capacitor building block. 177 7.6.2. Model Extraction The building block equivalent circuits were extracted from the test structure circuits by a process of nonlinear optimization with respect to measured S-parameters. Initial guesses for the blocks were computed directly form the measured data as described in detail in Chapter 3. All optimizations were accomplished utilizing the Star-Hspice circuit simulator on Sun SPARCstation computers. The test structures referred to in this section are shown in Figure 7.4-1 and Figure 7.4-2. The first test structure optimized was structure 1. This test structure allowed for characterization of the probe pad and the interconnect square building blocks. Optimization was performed over the range of measured data; 45 MHz to 5 GHz. Data beyond 5 GHz was not used due to the fact that all the structures were well beyond their self-resonant frequencies by that point. The next structures optimized were test structures 2 and 3, which allowed the behavior characterization of the single inductor coil, and also the coupled inductor coils building blocks. The models for the pad and interconnect generated by test structure 1 were used here also. The last test structure optimized was test structure 4, and from this device the behavior of the gridded plate capacitor square building block was deembedded. All test structures optimized accurately, and the results of the optimizations are shown in Figure 4.5-2 - Figure 7.6-6. Impedance parameter plots are shown for the inductor and capacitor test structures (structures 2-4), since their behavior can be more easily understood in terms of input impedance and phase. The measured S-parameter data was converted to Z-parameter data using the Microwave Design System (MDS) software, and the circuit setup shown in Figure 7.6-1. 178 1 Port 1 Inductor S-Parm. Dataset Port 2 2 Figure 7.6-1. Z-parameter MDS circuit configuration for inductor and capacitor analysis. 179 1 1 R L L R L R 2 R = 1E-3 L = 1.8E-10 C = 3.1E-13 2 R = 1E-3 L = 8.3E-11 C = 2.8E-14 C 2 G round P r o b e P ad Building Block 1 1 R 2 L C Interconnec t S q u a r e Building Block Lvia Lv ia 1 2 1 T op Metal 2 Ri Via Li Ci Li Ri Lvia = 4.9E-10 Cvia = 8.6E-13 Inductor Coil Building Block Li2 Ri2 Cvia Ci2 Rg2 Bottom C on du ctor Via Stac k Ri = 1E-2 Li = 7.4E-11 Ci = 1.0E=13 Li2 Cvia Rg1 To p C on ductor Bottom Metal Ri2 Via Stack Ri2 = 1.7E-2 Li2 = 4.4E-10 Ci2 = 1.8E-13 Lvia Lv ia 1 1 2 2 Ri Li Ci Li Ri Rg1 Li2 Li2 Ri2 Cvia Ci2 Lm To p C on ductor Coi l 1 on this axi s Ri2 Cvia Rg2 Bo ttom C on du ct or Via Stack Cm 3 4 Lvia Lvia 3 Coi l 2 on this axis 4 Ri Li Ci Li Ri 1(a,b) Li2 Ri2 Cvia Ci2 Rg2 Bottom C on du ctor C o up led Via St ack Lm = 0.4 Cm = 1.4E=13 2(a,b) Li2 Cvia Rg1 To p C on ductor C o u p l e d I n d u c t or Coil Building Block Ri2 Via Stack All other parameters from inductor coil building block. Additional via couplings inserted as needed. 1a 2a 4a 3a 1a 1b 2b 4b 3b 1b Coupled PEEC Circuits A 3(a,b) A D 4(a,b) B G r i d d e d C a p a c i tor Plate Building Block D R LM C L CM L C1 LM B C R L L C2 R R R = 0.47E-1 L = 9.1E-10 C1 = 1.3E-14 C2 = 1.0E-14 CM = 0.89E-13 LM=0.9 Figure 7.6-2. Building blocks, equivalent circuits and parameter values for solenoid inductor and gridded plate capacitor modeling. 180 1.00E+00 8.00E-01 6.00E-01 4.00E-01 S21 2.00E-01 0.00E+00 -2.00E-01 -4.00E-01 -6.00E-01 S21(R) Meas. S21(R) Modeled -8.00E-01 S21(I) Meas. S21(I) Modeled -1.00E+00 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) 4.00E-01 S11(R) Meas. S11(R) Modeled S11(I) Meas. 3.00E-01 S11(I) Modeled S11 2.00E-01 1.00E-01 0.00E+00 -1.00E-01 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.6-3. Measured vs. modeled results for test structure 1. (a) S21 real and imaginary response. (b) S11 real and imaginary response. 181 80 Z11(dB) - MDS Z11(dB) - Modeled 70 60 Z11(dB) 50 40 30 20 10 0 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) 100 80 60 40 Z11(Phase) 20 0 -20 -40 -60 Z11(Phase) - MDS -80 Z11(Phase) - Modeled -100 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.6-4. Measured vs. modeled results for test structure 2. (a) Z11 magnitude response. (b) Z11 phase response. 182 70 60 Z11(dB) 50 40 30 20 10 Z11(dB) - MDS Z11(dB) - Modeled 0 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) 100 80 60 40 Z11(Phase) 20 0 -20 -40 -60 -80 Z11(Phase) - MDS Z11(Phase) - Modeled -100 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.6-5. Measured vs. modeled results for test structure 3. (a) Z11 magnitude response. (b) Z11 phase response. 183 60 Z11(dB) - MDS Z11(dB) - Modeled 50 Z11(dB) 40 30 20 10 0 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) 100 Z11(Phase) - MDS 80 Z11(Phase) - Modeled 60 40 Z11(Phase) 20 0 -20 -40 -60 -80 -100 -120 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.6-6. Measured vs. modeled results for test structure 4. (a) Z11 magnitude response. (b) Z11 phase response. 184 7.7. Results Once the building block equivalent circuits had been deembedded, they were used to predict the behavior of several different inductors with interactions between them (Figure 7.7-1), and also the behavior of a large gridded plate capacitor. All of these structures would be very difficult and time-consuming to model with a fully 3-D nonlumped element simulation and modeling method due to their complex geometries and 3dimensional nature. The structure equivalent circuits were generated by replacing each geometrical building block with its associated circuit. Small signal simulation times usually was within 1 minute using Sun SPARCstation computers. Figure 7.7-1. Fabricated solenoid inductors. 185 Figure 7.7-1 shows 5 different inductors that were fabricated. The top right and lower right inductors represent test structures 2 and 3 respectively. The modeling method was tested on the two inductors on the left hand side; the four coil inductor with six turns per coil, and the four coil inductor with 8 turns per coil. These structures were electrically long, with both being greater than 1 wavelength long at 1 GHz. The predicted electrical behavior of the input impedance, in magnitude and phase, for the 4 coil, 6 turn per coil inductor are shown in Figure 7.7-2 and Figure 7.7-3. As can be seen in the plots, the modeling method shows very good agreement with the actual values in both magnitude and phase responses, up to the first self-resonant frequency. Z11 magnitude and phase results are also shown for the 4 coil, 8 turn per coil series connected inductor in Figure 7.7-4 and Figure 7.7-5. Again, for this test case we have good agreement in both response, with the self resonant frequency being predicted quite well. As has been mentioned, most passive devices are usually only useful well before they become self-resonant, since after that point they reverse their phase characteristics and start to behave like their reactive counterparts. 186 60 50 Z11(dB) 40 30 20 10 0 1.00E+08 Z11(dB) - Actual Z11(dB) - Predicted 1.00E+09 Frequency (Hz) Figure 7.7-2. Measured and predicted results for Z11 (dB) for 4-coil, 6 turn per coil inductor. 100 80 60 Z11(Phase) 40 20 0 -20 -40 -60 -80 -100 1.00E+08 Z11(Phase) - Actual Z11(Phase) - Predicted 1.00E+09 Frequency (Hz) Figure 7.7-3. Measured and predicted results for Z11 (phase) for 4-coil, 6 turn per coil inductor. 187 70 60 Z11(dB) 50 40 30 20 10 Z11(dB) - Actual Z11(dB) - Predicted 0 1.00E+08 1.00E+09 Frequency (Hz) Figure 7.7-4. Measured and predicted results for Z11 (dB) for 4-coil, 8 turn per coil inductor. 100 80 60 Z11(Phase) 40 20 0 -20 -40 -60 -80 -100 1.00E+08 Z11(Phase) Actual Z11(Phase) Predicted 1.00E+09 Frequency (Hz) Figure 7.7-5. Measured and predicted results for Z11(phase) for 4-coil, 8 turn per coil inductor. 188 In addition to testing building block model validity on the solenoid inductors, the gridded plate capacitor models were also tested on a large gridded parallel plate capacitor. An illustration of the capacitor is shown in Figure 7.7-6. Each plate of the capacitor had outside dimensions of 400 x 250 mil, and was thus quite large electrically. Only two building blocks were required for modeling this device; the probe pad and the gridded capacitor plate building block. The electrical prediction results for the Z11 parameter are shown in Figure 7.7-7 and Figure 7.7-8. As with the inductor cases discussed above, both magnitude and phase responses agree well with actual measurements, with the self- resonant frequency being predicted accurately. Figure 7.7-6. Large gridded plate capacitor used to test capacitor building block model validity. 189 70 Z11(dB) - Actual Z11(dB) - Predicted 60 Z11(dB) 50 40 30 20 10 0 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.7-7. Measured and predicted results for Z11 (dB) for large capacitor. 100 80 60 Z11(Phase) Actual Z11(Phase) Predicted Z11(Phase) 40 20 0 -20 -40 -60 -80 -100 1.00E+08 1.00E+09 1.00E+10 Frequency (Hz) Figure 7.7-8. Measured and predicted results for Z11(phase) for large capacitor. 190 7.8. Summary In this chapter, the test structure and building block predictive modeling method developed under this research work was applied to the modeling of complex full 3-D passive structures, and accurate results were obtained and experimentally verified. For all benchmark structures, accurate predictions of electrical behavior in both magnitude and phase, up to their respective self-resonant frequencies were obtained. Since complete circuit element models for the structures were constructed, small signal analysis occurred at very high speed in a standard circuit simulator. For the structures discussed in this paper, simulations completed in under 1 minute for all cases. This particular application demonstrates the potential of the building block and test structure modeling method. Models of full 3-D structures manufactured in an inhomogeneous environment with varying conductor and dielectric thicknesses, as is usually the case for a LTCC process, are usually obtained from very complex full 3-D simulation methods or less accurate 2 ½-D solvers. These simulations can take an impractically long time to complete, and are not well suited for circuit design applications, which are usually iterative in nature. The modeling method developed under this research could prove to be of significant use in simulation and modeling in integrated passive component technologies such as LTCC. 191 CHAPTER VIII CONCLUSIONS AND RECOMMENDATIONS 8.1. Summary of Research and General Conclusions A novel methodology for the modeling of 2 and 3 dimensional integrated passive devices with interactions has been presented in this dissertation. The method is based upon defining geometrical building blocks, and modeling them by the use of test structures, measurement, and nonlinear optimization. The method yields equivalent circuit (although table lookup models can also be used) building blocks that can be used to model any structure designed using a combination of those blocks. Since measurements of test structures are performed, manufacturing process effects are taken into account in the models. The method is general, since any type of structure can be modeled using the same technique, whether it be a resistor, capacitor, or inductor. It is also versatile, in that the number of structures that can be accurately designed and modeled with a given building block set is very large. Additionally, the method is expandable, since new building blocks can be defined, characterized and added to the modeled library on an as-needed basis. The building blocks are modeled by equivalent circuits, although direct multiport 192 parameters can be used, and as a result, large circuits of passive structures are developed. Small signal simulations of these circuits usually occur within a few minutes, which is considerably faster than most general numerical full wave methods. Implementation and ease of use are often overlooked issues in modeling research. This method is easily implementable in a modern circuit design CAD framework, since characterized building blocks can easily be inserted in a library, and block geometries, circuit models, and design rules can all be associated together. The use of this method then does not require any special software beyond what is found in most circuit design houses, and only requires minimal training to use correctly. The method has been tested and experimentally verified on a number of different structures, including serpentine resistors, interdigital capacitors, planar spiral inductors, and full 3-D solenoid inductors and gridded parallel plate capacitors. The only structure not experimentally verified was the planar spiral inductor, due to difficulty in gaining access to multilayer fabrication facilities at that time. Benchmark structures comprised of modeled building blocks were designed to test the validity of the modeling method. In all cases, good predictions of electrical behavior were obtained. Several circuits were also built with the help of Hewlett-Packard Microwave Design System (MDS) to test model validity in actual circuits. Good results were obtained here also. One circuit design that was used quite often was the RLC resonant tank circuit. This circuit was very useful for determining model accuracy, since the position of the resonance was determined by component capacitance and inductance, and the actual shape of the resonance was determined by component resistance. In general, good matches were obtained using the 193 developed circuit model when compared to using the measured data directly as a model of the device. While the results for the planar structures are accurate, the true potential of the modeling method can be seen in the modeling results for the completely 3-dimensional structures designed and fabricated in the low temperature cofired ceramic (LTCC) process. Modeling of fully 3-D structures can usually only be accomplished by the use of numerical full wave methods, such as the finite element method which is actually used quite often for LTCC modeling work. As discussed in Chapter 2, numerical methods usually require a structure to be meshed into small segments based upon current flow or geometrical considerations. Equations are formulated locally for each segment, which are then combined to obtain a solution for the entire structure. For complex geometries, the number of mesh points increase, which directly leads to longer run times. The two structures modeled in the LTCC process - the multilayer solenoid inductor, and the gridded plate parallel plate capacitor, both had many regions of complex geometry and rapidly changing current flow. Without taking into account processing effects such as varying dielectric thicknesses, and complex via geometries that often occur in a LTCC process, models of such structures would more than likely generate very large matrices using a numerical method, and would take extremely long to solve. The developed method, on the other hand, has produced circuit models of various structures which all simulate in under a minute in a standard circuit simulator, and produce accurate results. This developed method could very well prove to be an important technique for circuit design in multilayer processes such as LTCC. 194 8.2. Discussion Several important issues which should be considered in order to obtain successful optimizations and building block models are now discussed. Many of the issues presented here are the result of experience gained under this research, and will help guide the reader in obtaining successful results themselves. 8.2.1. Test Structure Design When designing test structures, it is always a good idea to incorporate a simple straight line test structure, as has been done in all cases in this thesis. Apart for modeling an uncoupled square, it allows for an investigation of basic material properties. A good estimate of basic material parameters is crucial for optimization starting points, especially when attempting a multi-building block optimization. Modeling of coupling is extremely important. In order to model coupling behavior, it would be a good idea to have two distinctly different types of test structures; physically connected (such as a serpentine resistor) and physically disconnected (such as an interdigital capacitor). In a physically connected structure there is significant current flow through the device. Since mutual inductance is affected by current flow, it’s effect will be observable and will be easy to extract. For a physically disconnected device, the major signal transmission mechanism is capacitive coupling, and since the output response is highly dependent upon it, it will be easy to deembed. If only one test structure is used for deembedding both coupling components, one will affect the output response 195 much more than the other, and thus the weaker one may be more difficult to extract. Using both test structures essentially eliminates the lack of sensitivity issue. The coupling information from both of these structures can be combined to develop a single coupled building block. 8.2.2. Number of Test Structures In theory, and as has been the case for all the examples presented in this thesis, only a minimum number of test structures have been used. It may be advisable, however, to design and use more test structures rather than just the minimum required. The additional test structures should be comprised of the same building block set, but used in different configurations, just so that the extracted models can be verified in more cases before finally being entered into a library. 8.2.3. Simultaneous Optimization Once initial modeling of building blocks is complete, a few large test structures can be optimized together to fine tune the extracted building blocks. This can be achieved by the use of simultaneous optimization, where the equivalent circuits for each of the building blocks are forced to be the same for each structure, and all structures are simultaneously optimized with respect to their individual measured results. Simultaneous optimization is considerably more complex and time consuming than single structure optimization, but it has the advantage of being far more likely to uniquely deembed building block circuit models. The reason for this is simple – if the 196 same model is tested in several different environments (test structures) and must work for all of them, then it is more likely to be generally correct. 8.3. Recommendations The modeling methodology has shown good results for both planar and 3-D structures. Further work in validating the method for both kinds of structures is definitely warranted. Various recommendations for the further development of the modeling method are given and listed separately for clarity, and to help properly direct research initiatives in this area. 8.3.1. Recommendations for Building Blocks New building blocks can be defined to take into account higher order couplings. It will very interesting to investigate higher order mutual inductance effects, since mutual inductance decreases much more slowly than coupling capacitance with distance. Additionally, a comprehensive library of modeled building blocks for various substrates and processes can be gradually built up, ultimately developing a library of components which can be used for actual circuit and system design work. Building block circuit models can be improved so that they take into account retardation. This is especially important when attempting to model structures that are quite large. As a starting point, the rPEEC (retarded partial element equivalent circuit) 197 circuit might be used, but then modified later as needed. Development of a method to model building blocks directly with S-parameters will also be very useful. 8.3.2. Recommendations for Test Structure Design In order to ensure accurate building block models are developed, a more comprehensive test structure set can be constructed and used in the building block model extraction stage, as outlined above. Since accurate building block models are crucial to modeling success, significant effort needs to be expended to ensure the models are as accurate as possible. A large number of structures should also be built to test model validity. 8.3.3. Recommendations for Statistical Modeling One advantage that this method definitely has over others is that statistical models for the building blocks can be developed. This can be achieved by fabricating test structures on various runs of a process and extracting building blocks each time. Statistical models for each of the building blocks can be developed over time, and these can be used to intelligently predict fabricated passive component yield. A probability density function approach to achieving intelligent yield estimations based upon fabricated test structures can be found in [51]. Statistical modeling and accurate yield prediction are very important for reducing production costs. 198 8.3.4. Recommendations for Parameter Extraction and Optimization The use of simultaneous optimization techniques can help ensure model validity. This should be used in the future, at least after initial model extraction, to refine all the extracted building block models. Development of methods to help with optimization initial guesses is also worthy of attention, since in many cases successful optimization convergence and the number of iterations required is highly dependent upon a good starting point. This is especially true when trying to optimize many parameters at the same time. 8.3.5. Recommendations for Implementation The developed modeling method is highly amenable to implementation within an existing EDA framework. A successful implementation can not only aid research efforts, but can also help the method gain wider acceptance and industrial use. It is a well known fact that many good ideas never get implemented, simply because it is too difficult or cumbersome to do so. 8.4. Final Conclusions This research program has allowed us to develop a novel, accurate, and practical, modeling method for predicting the high frequency behavior of small geometry passive devices. This thesis has described the modeling method in detail, and it has been demonstrated on a variety of two and three-dimensional devices. Good results have been 199 obtained for all structures, but in the author’s opinion, the results for the LTCC structures are particularly impressive. Miniaturized and integrated passive structures will undoubtedly find increasing use in modern and future compact, lightweight, and high performance devices. I believe the predictive modeling method, once properly developed and matured, can be used to great advantage for the successful design and manufacture of such systems. 200 APPENDIX A SENSITIVITY ANALYSIS OF 4 SEGMENT RLC CIRCUIT A.1. Introduction In this appendix, a detailed sensitivity analysis of simple RLC circuit is presented. The circuit is a 4 segment RLC ladder network, with the resistance value in the last segment being a different value with respect to the rest of the circuit. Analytical results are shown for the impedance parameters Z11 and Z21 with their normalized sensitivities with respect to the various circuit parameters shown in the circuit. Actual equations for the various sensitivities are not presented, but the manner in when they are computed is, along with all associated plots. The circuit under analysis is shown in Figure A.1-1. As can be seen, the value of the resistance R2 in the last RLC segment is different from all other resistances in the circuit. This analysis will show the relative importance of the Z11 and Z21 parameters to the circuit parameters. 201 + R L V1 R C L R L C R2 C + L V2 C I1 I2 - - Figure A.1-1 Circuit for impedance parameter sensitivity analysis. Circuit impedance parameters are defined by V1 Z11 V = Z 2 21 Z12 I1 Z22 I 2 (A-1) and are computed by using standard nodal equation formulations. With the use of software symbolic mathematical tools, the following relationships were derived for Z11 and Z21 . Z11 = k 0 + k1ω + k2 ω 2 + k3ω 3 + k4 ω 4 + k5ω 5 + k 6ω 6 + k7 ω 7 + k8ω 8 l1ω + l2ω 2 + l3ω 3 + l4ω 4 + l5ω 5 + l6ω 6 + l7ω 7 + l8ω 8 where the k i and li are given by the following expressions 202 (A-2) k0 = 1 k1 = j (9CR + CR2 ) k 2 = 10CL − 9 R 2 C 2 − 6 RR2 C 2 k3 = − j (2C 3 R3 + 6C 2 LR2 + 24C 2 LR + 5C 3 R2 R ) k 4 = 11R 2 LC 3 + 10 RR2 LC 3 + 15L2 C 2 + C 4 R3 R2 . (A-3) k5 = j(3C 4 R 2 LR2 + 5C 3 L2 R2 + 16C 3 L2 R + C 4 R 3 L ) k 6 = −(3C 4 RL2 R2 + 7 L3C 3 + 3C 4 R 2 L2 ) k 7 = − j( C 4 L3 R2 + 3C 4 RL3 ) k8 = C 4 L4 The denominator terms are given by l1 = j 4C l2 = jC( CR2 + 7CR ) l3 = − jC( 2 R 2 C 2 − 4 RR2 C 2 − 10CL ) l4 = C( C 3 R2 R 2 + 4C 2 LR2 + 8C 2 LR ) . (A-4) l5 = jC( R 2 LC 3 + 6 L2 C 2 + 2 RR2 LC 3 ) l6 = − C(2C 3 L2 R + C 3 L2 R2 ) l7 = − jC 4 L3 Z21 has a simpler representation, in that the numerator of the expression simply has a 1 in it, whereas the denominator is the same as for Z11 . The expression for Z21 is given by 203 Z21 = 1 . l1ω + l2 ω + l3ω + l4ω + l5ω 5 + l6ω 6 + l7 ω 7 + l8 ω 8 2 3 4 (A-5) It is strikingly clear from the polynomial coefficients of these expressions that the terms are extremely nonlinear, and that solving for them would be a very difficult if not impossible task. It is for this reason that during the parameter extraction process nonlinear optimization is utilized, with a large number of frequency points over a wide frequency band. The sensitivity analysis is conducted over a frequency band of interest and with high resolution in order to determine which parameters affect the output responses, and to what degree. This analysis helps determine whether the circuit (which originated from a test structure) is adequate to deembed the parameters of interest. A sensitivity value of 0 implies that the circuit parameter does not influence the output at all over the band of interest, whereas large values imply a large influence. Parameters which affect the output values considerably will be more easy to deembed than those of lower sensitivity. Taking into account a wide range of frequencies also helps in establishing uniqueness, since the different parameters affect the output response differently over a frequency band, and in order to minimize error over the entire band, a unique value is likely to be extracted. It is possible, however, that the sensitivity responses of several different parameters for one of the impedance parameters track each exactly over a band. In this case, unique parameter extraction from that particular output parameter will not be possible, and additional equations must be obtained - for example, from the other impedance parameter. 204 Normalized circuit sensitivities are computed over frequency and with respect to each of the circuit parameters for the circuit shown above. Clearly, deriving symbolic results for the sensitivities would be a complex and pointless task, due to the very complicated expressions that would be generated. For this reason, numerical normalized sensitivities are computed. The equation for sensitivity of F with respect to parameter h this is given by S hF = F ( V , h + ∆h ) − F ( V, h ) h ⋅ , ∆h F ( V, h ) ∆h small (A-6) where V represents the vector of unchanging variables of F, h is the parameter in consideration, and ∆h is the increment in h. ∆h must be kept small with respect to h in order for this expression to be accurate. With some manipulation, this then results in impedance parameter sensitivity equations of the form DR(ω, R, L, C, R2 ) = Z11(ω ,105 . R, L, C, R2 ) − Z11(ω , R, L, C, R2) Z11(ω , R, L, C, R2)0.05 (A-7) for normalized sensitivity of Z11 with respect to R over frequency. Similarly formed equations can be obtained for all the circuit parameters and other impedance parameters. This form of the equation is excellent for implementation on computers. Several plots are now presented for the impedance parameters and sensitivities for the circuit being discussed. Some representative circuit values are chose with R=0.2 Ω, R2=0.1 Ω, L=0.1 µH, and C=1 nF for these computations. Direct impedance parameter 205 plots are shown in Figure A.1-2, with Z11 sensitivities in Figure A.1-3 and Figure A.1-4, and Z21 sensitivities in Figure A.1-5 and Figure A.1-6. Several interesting issues arise from these results. Z11 and Z21 are much more sensitive to changes in C and L as opposed to R and R2, as evidenced by the vertical scale on the plots. However, sensitivity with respect to the resistances is not 0, although it is fairly small. However, actual optimization results show that even this small sensitivity is adequate to deembed both R and R2. Another point of interest is that the sensitivities of C and L to Z11 are almost identical, and track each other over the entire frequency band, in both the real and imaginary parts of the response. This implies that dembedding C and L uniquely will be difficult, even though Z11 is highly sensitivity value to both the parameters. In looking at the sensitivity of Z21 with respect to C and L, it is clear that they are not identical, and do not track each other over the band. Utilization of this fact is what enables us to extract the C and L values. This illustrates the importance of utilizing at least two different parameters and not just one in order to obtain successful optimizations. In all of the research work completed under this program, optimizations were performed using two two-port parameter values. 206 10 5 Re Z11 w , R, L, C, R2 i 0 Re Z21 w , R, L, C, R2 i 5 10 6 1 10 7 1 10 8 1 10 9 1 10 8 1 10 9 1 10 w i 100 50 Im Z11 w , R, L, C, R2 i 0 Im Z21 w , R, L, C, R2 i 50 100 6 1 10 7 1 10 w i Figure A.1-2. Z11 and Z21 real and imaginary components for RLC circuit 207 20 10 0 Re DL w , R, L, C, R2 i Re DC w , R, L, C, R2 i 10 20 30 40 50 7 1 10 8 1 10 w i 9 1 10 8 1 10 w i 9 1 10 30 20 Im DL w , R, L, C, R2 i 10 Im DC w , R, L, C, R2 i 0 10 7 1 10 Figure A.1-3. Z11 sensitivity with respect to C and L for RLC circuit, real and imaginary parts. 208 0.3 0.25 0.2 Re DR w , R, L, C, R2 i Re DR2 w , R, L, C, R2 i 0.15 0.1 0.05 0 0.05 7 1 10 8 1 10 w i 9 1 10 8 1 10 w i 9 1 10 0.4 0.2 Im DR w , R, L, C, R2 i 0 Im DR2 w , R, L, C, R2 i 0.2 0.4 7 1 10 Figure A.1-4. Z11 sensitivity with respect to R and R2 for RLC circuit, real and imaginary parts. 209 40 20 Re DL w , R, L, C, R2 i 0 Re DC w , R, L, C, R2 i 20 40 7 1 10 8 1 10 w i 9 1 10 8 1 10 w i 9 1 10 30 20 Im DL w , R, L, C, R2 i 10 Im DC w , R, L, C, R2 i 0 10 7 1 10 Figure A.1-5. Z21 sensitivity with respect to C and L for RLC circuit, real and imaginary parts. 210 0 0.01 Re DR w , R, L, C, R2 i 0.02 Re DR2 w , R, L, C, R2 i 0.03 0.04 7 1 10 8 1 10 w i 9 1 10 8 1 10 w i 9 1 10 0.15 0.1 0.05 Im DR w , R, L, C, R2 i 0 Im DR2 w , R, L, C, R2 i 0.05 0.1 0.15 7 1 10 Figure A.1-6. Z21 sensitivity with respect to R and R2 for RLC circuit, real and imaginary parts. 211 APPENDIX B CURRENT FLOW VISUALIZATION SOFTWARE B.1. Introduction A key issue in the proposed modeling methodology is the determination of the sizes and shapes of the various building blocks. Since the method is based on connecting various blocks together, it is desirable to verify for simple structures in a fairly tightly constrained design rule set, that the addition of a piece of material does not affect the current flow through the building block in question. This implies that simple building blocks should be designed such that they have constant input and output impedance in a structure regardless of which blocks are attached to them; that is, the building blocks are context insensitive. This is not a requirement in a less constrained design rule environment in which a larger number of possible geometry structures are allowed. In this case we could have separate, context sensitive models for a material square, for example, for which in one instance it is connected to another material square, and in the other it is connected to a corner piece. In this article, building blocks of the first type will be discussed. 212 For illustrative purposes, consider the two structures shown in Figure B.1-1. In this example, we will try to determine if the shaded piece is a valid corner building block structure. Simply by visual inspection, it is difficult to ascertain that the current flow through the blocks is the same in both instances, that is, the current distribution across the boundaries of the building block does not change for either case. A well known example of this situation is the 2/3rd rule for corners in which a corner is assumed to have 2/3rd the resistance value of a square in a straight piece of material due to the nonuniform current flow through that piece. Possible Corner Building Block Figure B.1-1. Possible corner building block and usage in two structures 213 B.2. Algorithm For this research, a software program was written to help visualize the current flow through arbitrary geometry 2 dimensional planar passive devices. The main program is essentially a two dimensional circuit solver and a voltage and current visualization tool. The voltage and current distributions and their corresponding 2 dimensional gradients can be viewed graphically as indexed colormaps or contour plots. The software takes as input a description of the structure to be solved, as well as the input and output positions, which can be entered using a mouse on a graphical user interface window. Next, a 2 dimensional circuit is constructed with a user controlled accuracy by specifying a grid size. An example schematic of a 6x6 impedance grid is shown in Figure B.2-1. Each grid point represents a sample point within the structure of interest. In actual modeling computations, a much denser grid is used in order to capture the current and voltage distribution at many points within the structure. A structure pattern is defined by open circuiting impedance branches (admittance values of 0). For example, the S shape, shown by diagonal hatching in Figure B.2-2 can be created by open circuiting the impedance branches coincident with the cross hatched and the diagonal hatched regions. 214 Figure B.2-1. Representative impedance grid. Each box represents and impedance. 215 Figure B.2-2. Definition of S-shaped region on impedance grid. B.2.1. Network Solution Methodology Once the structure has been defined, the next step is to generate the sparse modified nodal admittance (MNA) matrix [43]. This is accomplished using the element stamp method, in which each impedance is represented in the MNA matrix with 4 216 positive or negative admittances. For example, for an impedance Z connected between nodes i and j in an impedance grid, entries would be inserted into the overall MNA matrix as shown in the right hand side of Figure B.2-1. i Z j col Vi col Vj row i 1/Z -1/Z row j -1/Z 1/Z Figure B.2-1. Impedance and corresponding entries in MNA matrix. Once the admittance matrix is established, input and output connection terminals for the structure are defined. An ideal current source is inserted between these points, and the output point is grounded. The insertion of the current source allows us to set up a nonsingular set of equations which can be solved for nodal voltages throughout the structure. Current flow in the x and y directions can also be calculated from the voltage distribution by the computing the voltage gradients, since impedance is assumed to be spatially constant. The system is finally solved using lower-upper matrix decomposition (LU) techniques to compute the voltage and current distribution within the structure [52]. Remapping routines then take the results and remap them graphically to correspond with the drawn structure. Current flow can be analyzed by viewing the x and y gradients of the voltage, since the grid impedance is fixed. 217 B.2.2. Mathematical Implementation Matlab was used as the implementation environment. The generalized system formulation is given by [M ][V ] = [I ] (B-1) where M is the MNA matrix, V is the vector of node voltages, and I is the right hand side current vector. A LU decomposition formulation modifies the equation to [LM ][U M ][V ] = [I ] (B-2) which can be solved in two steps using an auxiliary vector Z as [LM ][Z ] = [I ] [U M ][V ] = [Z ] (B-3) Once the vector V has been computed, it must be then remapped back to the geometry of the structure under analysis, that is converting a 1 dimensional matrix into a 2 dimensional matrix. This can be accomplished fairly easily, if a left to right node numbering scheme is used in the generation of the MNA matrix, as is best illustrated by Figure B.2-1. 218 Voltage Vector 2D Voltage Mesh Matrix V1 V2 Vn V1 V n+1 V n+2 V 2n V2 V n 2 -n+1 V n 2 -n+2 Vn2 Vn V n+1 Vn2 Figure B.2-1. Mapping operation between computed voltage vector and 2D voltage matrix for actual geometrical structure being analyzed. The software is highly efficient in terms of both speed and memory requirements. Sparse matrix routines are used, which yield great savings in terms of memory. For example, a 60x60 grid would represent a 3600 element circuit and hence a tableau matrix size of 3600x3600. If sparsity were not used, this matrix would occupy over 100 MB of memory using IEEE 64-bit floating point precision. However, using sparse matrix techniques, a "virtual" 3600x3600 matrix is constructed using indexing, with most matrices having densities of less than 0.1% and memory requirements on the order of 100KB - a savings of 4 orders of magnitude. A sparsity plot of an MNA matrix developed for solving a resistor current distribution is shown in Figure B.2-2. This matrix is only 0.2122% dense. 219 Figure B.2-2. MNA matrix sparsity pattern for serpentine resistor analysis. 220 Figure B.2-3 Contour and indexed image plots of current distribution for two different geometry bends 221 B.3. Visualization Results An example of the output generated by the software can be seen in Figure B.2-3. The program shows the current flow contour (x and y current gradient) and color indexed (magnitude of current gradient) image plots of current flow through the structures considered in Figure B.1-1. As can be seen, the current distribution through the corners in both cases is considerably different. This implies that a single square corner is not an appropriate building block, since the current flow and thus impedance across the corner square is modified when additional material is attached to it. Visualization results for a gridded metal plate are presented. The input and output points are at the stubs on the left and right hand sides of the structure. Gridded plates occur in many technologies where it is not possible to achieve good meal coverage. The grid used in this case was 120 x 120, yielding a total of 14,400 grid points. Several output plots for this case are shown, including indexed color plots which show current intensity, contour plots which show current crowding, as well as some cross section current gradient profiles. Figure B.3-1 shows the current intensity plot through the structure. Actual current profiles through sections A-A’, B-B’ , and C-C’ are shown in Figure B.3-4, Figure B.3-5, and Figure B.3-6 respectively. The magnitude of the current gradient is shown in a contour plot in Figure B.3-2, and a overlay plot of X and Y directed current gradients illustrating the current crowding effect is shown in Figure B.3-3. 222 Figure B.3-1. Indexed current intensity plot of gridded structure. 223 Figure B.3-2. Current gradient magnitude contour plot. 224 Figure B.3-3. Contour plots of X and Y directed current gradients showing current crowding effects. 225 Figure B.3-4. Current profile plot through axis A-A’. 226 Figure B.3-5. Current profile plot through axis B-B’. Figure B.3-6. Current profile plot through axis C-C’. 227 B.4. Source Code B.4.1. Fundamental Structure Geometry Input and Matrix Generator % % The following matrix shows the basic shape of the required mesh. Exp will % expand it by its factor. B MUST BE SQUARE B = [ 0 0 0 0 0 0 0 0 0 1 1 1; 1 1 1 1 1 1 1 1 1 1 1 1; 1 0 0 0 0 0 0 0 0 1 1 1 ; 1 1 1 1 1 1 1 1 1 0 0 0; 0 0 0 0 0 0 0 0 1 0 0 0; 1 1 1 1 1 1 1 1 1 0 0 0; 1 0 0 0 0 0 0 0 0 0 0 0; 1 1 1 1 1 1 1 1 1 1 1 1; 0 0 0 0 0 0 0 0 0 0 0 1; 1 1 1 0 0 0 0 0 0 0 0 1; 1 1 1 1 1 1 1 1 1 1 1 1; 1 1 1 0 0 0 0 0 0 0 0 0 ] B = flipud(B); Exp = 1; x_dim = size(B,2); y_dim = size(B,1); ones_matrix = ones(Exp,Exp); A = sparse(Exp*x_dim+2,Exp*y_dim+2); for j=1:y_dim for i=1:x_dim beg_x = Exp*(i-1)+2; beg_y = Exp*(j-1)+2; A(beg_x:beg_x+Exp-1,beg_y:beg_y+Exp-1) = B(i,j) .* ones_matrix; end end % This matrix has a pad of zeros around it to aid in matrix building x_dim = size(A,2); y_dim = size(A,1); M = sparse(x_dim*y_dim,x_dim*y_dim); % Resistive element to insert in matrix build L=0; C = 0; RE = 1; for i=2:(y_dim-1) for j=2:(x_dim-1); index = j + x_dim*(i-1); % Set up horizontal resistor between node j and j+1 228 % if (A(i,j) & A(i,j+1)) ~= 0 M(index,index) = M(index,index) + RE + C; M(index,index+1) = M(index,index+1) - RE; M(index+1,index) = M(index+1,index) - RE; M(index+1,index+1) = M(index+1,index+1) + RE; end % Set up vertical resistor between node j and j+x_dim % if (A(i,j) & A(i+1,j)) ~= 0 M(index,index) = M(index,index) + RE; M(index+x_dim,index+x_dim) = M(index+x_dim,index+x_dim) + RE; M(index,index+x_dim) = M(index,index+x_dim) - RE; M(index+x_dim,index) = M(index+x_dim,index) - RE; end end end % Now set up the connection matrix in MC R = RE; MC=sparse(x_dim*y_dim,x_dim*y_dim); M(index+x_dim,index) = M(index+x_dim,index) - RE; end end end % Now set up the connection matrix in MC R = RE; MC=sparse(x_dim*y_dim,x_dim*y_dim); for i = 2:(y_dim-1) for j=2:(x_dim -1) node_num = j + x_dim*(i-1); if (A(i,j) & A(i,j-1)) ~= 0 MC(node_num-1,node_num) = R; end if (A(i,j) & A(i,j+1)) ~= 0 MC(node_num+1,node_num) = R; end if (A(i,j) & A(i-1,j)) ~= 0 MC(node_num-x_dim,node_num) = R; end if (A(i,j) & A(i+1,j)) ~= 0 MC(node_num+x_dim,node_num) = R; end end end 229 B.4.2. Input and Output Point Definition Routine and Solver % Program to compute Admittance matrix for resistive mesh disp(' Node numbering is done linearly, starting at the top left') disp(' and increasing left-right, top-bottom, with the bottom') disp(' rightmost node being x_dimension*y_dimension'); % %x_dim = input('Enter x dimension of resistive mesh '); %y_dim = input('Enter y dimension of resistive mesh '); %node_in = input('Enter positive node of current source '); %node_out = input('Enter negative node of current source '); figure surface(full(A)); colormap(hot); [x,y] = ginput(1); x = floor(x); y=floor(y); node_in = length(A)*(x-1)+y [x,y] = ginput(1); x = floor(x); y=floor(y); node_out = length(A)*(x-1)+y % Set up ground M2 = [M(1:(node_out-1),1:(node_out-1)) M(1:(node_out1),(node_out+1):x_dim*y_dim); ... M((node_out+1):x_dim*y_dim,1:(node_out-1)) M((node_out+1):x_dim*y_dim,(node_out+1):x_dim*y_dim)]; disp('Finished ground calculations\n'); % % Set up RHS vector, let ground be the node where the output is taken % RHS = zeros(size(M2,2),1); RHS(node_in) = 1; RHS(node_out) = 0; % % Solve for node voltages % VM contains the voltages % [VM] = linsolve(M2,RHS); % %Set up voltage mesh so we can graphically see the voltages for i=1:1*(y_dim-1) for j=1:1*(x_dim-1) 230 index = j + x_dim*(i-1); V_mesh(i,j) = VM(index); end end [px,py] =gradient(V_mesh,0.1,0.1); B.4.3. Linear Solver Routine function [x] = linsolve(A,b) % % This function will solve a linear system of the % form Ax=b % [L,U,P] = lu(A); y = zeros(length(A),1); x=y; % First - solve Ly=b; then Ux=y y(1) = b(1); for i = 2:length(L) y(i) = b(i) - L(i,1:(i-1)) * y(1:(i-1)); end L1 = length(L); x(L1) = y(L1)/U(L1,L1); for i=(L1-1):-1:1 x(i) = (y(i) - U(i,i+1:L1) * x(i+1:L1))/U(i,i); end flipud(x); B.4.4. Nodal Elimination Routine function [MC]=nelim_v2(A,M,node_in,node_out) % program to create connection matrix for wire mesh disp(' Node elimination program ') disp(' Node numbering is done linearly, starting at the top left') disp(' and increasing left-right, top-bottom, with the bottom') disp(' rightmost node being x_dimension*y_dimension'); % x_dim = size(A,2); y_dim = size(A,1); xy = x_dim*y_dim; % attempt the nodal elimination inv_col_sum = 0; 231 for i=2:xy if (i~=node_in)&(i~=node_out) % Locate non zero entries columnwise [R,C] = find(M(:,i)); for z = 1:length(R) inv_col_sum = inv_col_sum + 1./M(R(z),i); end; inv_col_sum; for j=1:(length(R)-1) Colselect = R(j); for k=(j+1):(length(R)) if (M(R(k),Colselect)==0) M(R(k),Colselect) = M(R(j),i).*M(R(k),i).*inv_col_sum; M(Colselect,R(k)) = M(R(k),Colselect); else M(R(k),Colselect) = 1/(1/M(R(k),Colselect) + 1/(M(R(j),i).*M(R(k),i).*inv_col_sum)); M(Colselect,R(k)) = M(R(k),Colselect); end if (M(R(k), Colselect)>900) %disp('problem'); disp(M(R(k),Colselect)); %pause end end end % eliminate row and column clear R; M(i,:) = zeros(1,size(M,1)); M(:,i) = zeros(size(M,2),1); spy(M) %pause inv_col_sum = 0; % end big if end end MC = M; 232 APPENDIX C HSPICE CIRCUIT OPTIMIZATION C.1. Introduction The Hspice circuit simulator contains a very useful circuit optimizer which is able to modify circuit parameters in order to meet user defined constraints. The optimizer is capable of goal-based or curve-fit optimization, with no limitations upon the number or range of parameters to be optimized. The optimization method used by the simulator is Leavenberg-Marquardt, which gives good results in a wide variety of problems. In addition the users manual, details of the Hspice optimization procedures can be found in [53] and [54]. In this appendix, a brief description of the optimization process is given, with the goal of allowing the reader to attempt such optimization runs themselves. It is important to point out that this appendix is only a guideline, and for detailed information regarding various options and procedures, the reader is directed to the Hspice User’s Manual. 233 C.2. Input File Parameters In order to set up an optimization run, there are essentially three sets of parameters that must be defined; definitions of the desired goals, such as minimization of curve fit error or 3-dB bandwidth goals; definitions of the parameters that are allowed to vary along with their valid ranges; and finally a definition of optimization success – the level of error that must be achieved to be deemed successful. The optimization runs in this thesis were all curve fit optimization runs, since S-parameters were obtained through measurements and circuits were required to fit those parameters over a frequency band. C.2.1. Desired Goal Definition The desired goals are defined in terms of a .Measure statement. The .Measure statement can be used for both goal based and curve fit optimization runs. Multiple .Measure statements can be used simultaneously to reach several goals at the same time. The .Measure statements are very versatile, and can contain a number of different parameters for measuring many different circuit characteristics, such as rise and fall times, bandwidths, RMS errors between parameters, or any user-defined functions defined in .PARAM statements. For curve fit optimizations, the keyword ERR1 is used within the .Measure statement to determine the RMS error between a measured and calculated value. A segment of an input file in which a .Measure statement is used is given below. .measure ac comp7 err1 par(sa11r) vr(1100) .measure ac comp8 err1 par(sa11i) vi(1100) 234 .measure ac comp9 err1 par(sa21r) vr(2100) .measure ac comp10 err1 par(sa21i) vi(2100) .data measured mer file= ‘all.txt' freq=1 sa11r=2 sa11i=3 sa21r=4 sa21i=5 out = ‘all.out' .enddata In this section of code four .Measure statements are used. Each of the statements compute the RMS error between the measured value (par(…)) and simulated value (v[r/I](…)), and stores the result in a variable (comp[x]). The measured data is included in the input file by use of the .data statement. In this example, the measured data is stored in the file “all.txt” and is 5 columns, with the first column being frequency data (freq) and the last column being sa21i. As mentioned earlier, the .Measure statements can take many forms, and can be used to determine a wide variety of circuit performance parameters, and this capability gives the optimization method considerable flexibility. In addition, for multi-goal optimizations, each goal can be defined relative importance, so that priorities can be set. An example of such a statement is .Measure tot_power avg power goal=10mW weight=5. In this statement, the average power is measured and stored in tot_power, with a power goal set to 10mW and a weighting of 5 with respect to other .Measure statements within the deck. 235 C.2.2. Definition of Circuit Parameters for Optimization There are usually several variables which are desired to be optimized with respect to defined goals. Increasing the number of parameters increases the complexity of the problem. For a one parameter optimization, the optimizer only needs to search a onedimensional parameter space, but for a n-parameter optimization, the search space becomes n-dimensional. While it is true that a multi-parameter optimization is computationally expensive, it is not necessary that it requires a proportionally long time to converge. Under experience gathered under this research, it is clear that for fairly complex problems, the single most important factor for convergence success is a good choice of the initial starting vector. If the initial starting vector is close to the global minimum, only a few iterations may be required to converge successfully, regardless of the number of parameters involved. The parameters to be optimized are defined using the .Param statement over an interval, with an initial staring point. A section of input code illustrating the use of this statement is given below. … R1 2 3 C1 3 0 … .Param .Param Rx Cx Rx = OPT1(5,0.1,10) Cx = OPT1(1e-9,1e-12,1e-7) In this code segment, Rx and Cx are defined as parameters requiring optimization. Rx is defined to have an initial starting guess of 5Ω, with a possible range of 0.1Ω to 10Ω. Cx 236 is defined similarly to have a starting guess of 1e-9F, and a range of 1e-12F to 1e-7F. Both parameters are designated to have the optimization name OPT1, and will thus be optimized together when the optimization run using OPT1 is invoked. Any number of circuit parameters can be defined as parameters for optimization, including parameters within subcircuits and transistor model parameters. C.2.3. Criteria for Successful Optimization In order to define convergence success, optimization criteria must be defined These definitions are made in the optimization model statement. An example of a model statement is .MODEL OPTIM OPT RELIN=1E-3 RELOUT=1E-2 ITROPT=100. In this statement the optimization model name is OPTIM, and convergence success is defined when the input parameters change less than 0.1% or when the output parameters change less than 1% between successive iterations. Additionally, an iteration limit of 100 is set. Additional parameters can be defined in this statement to control step sizes, gradients, etc. The reader is directed to the user manuals for more information. C.2.4. Optimization Execution The actual invocation of optimization occurs in the analysis sweep statement. For example, for an AC analysis optimization, a statement may have the form of … .ac data=measured optimize=opt1 237 + results=comp7,comp8,comp9,comp10 + model=converge … .measure ac comp7 err1 par(sa11r) vr(1100) .measure ac comp8 err1 par(sa11i) vi(1100) .measure ac comp9 err1 par(sa21r) vr(2100) .measure ac comp10 err1 par(sa21i) vi(2100) In this case, the optimize=opt1 segment of the .ac statement calls the optimization run using the opt1 named optimization circuit parameters. The .measure results comp7, comp8, comp9, and comp10 are used as optimization goals, and the optimization model converge is used to determine the convergence criteria. The actual frequency points used for the run are taken from the measured data. C.3. Complete Optimization Control File Example Using all the individual input parts described above, a sample optimization control deck is shown. … * Circuit Here … .PARAM + c_cou2 = opt1(1.4e-15,0.01f,1n) + r2sq = opt1(5.7e-3,1e-5,10) + l2sq = opt1(1.4e-11,1f,1u) + c2sq = opt1(4.8e-15,0.01f,1n) .ac data=measured optimize=opt1 + results=comp7,comp8,comp9,comp10 + model=converge 238 .model converge opt relin=1e-5 relout=1e-5 close=200 itropt=300 .measure .measure .measure .measure ac ac ac ac comp7 err1 par(sa11r) vr(1100) comp8 err1 par(sa11i) vi(1100) comp9 err1 par(sa21r) vr(2100) comp10 err1 par(sa21i) vi(2100) .print par(sa11r) vr(1100) par(sa11i) vi(1100) .print par(sa21r) vr(2100) par(sa21i) vi(2100) .data measured mer file= ‘all.txt' freq=1 sa11r=2 sa11i=3 sa21r=4 sa21i=5 out = ‘all.out' .enddata .param freq=500M, sa11r=0, sa11i=0, sa21r=0, sa21i=0 .end In this control deck, a simultaneous curve fit optimization is performed with respect to 4 curves defined by the measured data in the variables sa11r, sa11i, sa21r, and sa21i. The circuit parameters c_cou2, r2sq, l2sq, and c2sq are varied in order to reach the optimization goals (the actual circuit itself is not shown). The optimization goals are to minimize the RMS error of each measured parameter with respect to certain circuit voltages (comp7 – comp10) , that is, the circuit voltages must mimic the measured parameters over the frequency band to within the specified error (as defined in the .model statement). For example, vr(1100)(circuit) is desired to track sa11r (measured). Successful completion of this optimization run will yield values for c_cou2, r2sq, l2sq, and c2sq such that all goals are met; that is, the circuit voltages duplicate the behavior of the measured data over the measured frequency range such that the convergence criteria are satisfied. 239 C.4. Simultaneous S-parameter Circuit Optimization It has been mentioned earlier in this thesis that simultaneous optimization of more than one circuit utilizing the same circuit parameters but different measured S-parameters could prove to be very valuable in validating extracted building block circuit models. This occurs because the models are tested in entirely different situations and are constrained to be the same for all cases. A method of achieving this has been devised for an arbitrary number of simultaneous circuit optimizations with respect to measured S11 and S21 parameters for each circuit. The simultaneous optimization method is based upon the construction of a circuit to measure S11 and S21 parameters of any other circuit. This S-parameter measuring circuit is defined below. va1i 100 0 dc 0 ac 1 ra1i 100 200 50 xa1 200 300 ***Circuit to be Measured Here*** ra1o 300 400 50 va1o 400 0 dc 6 ac 0 ea11 500 0 (200,0) 2 va11 500 1100 ac 1 ra11 1100 0 1g ea21 2100 0 (300,0) 2 ra21 2100 0 1g In this segment, the circuit whose S-parameters are to be measured is defined as a subcircuit and is then referenced in the xa1 statement. The two connection points are the input and output ports of the circuit under consideration. S11 is represented as V(1100) 240 and S21 is represented as V(2100). An additional circuit can have it’s S-parameters measured similarly by redefining the S-parameter measuring circuit with different node names. An input file with two such circuits is shown below. ****************************** * measuring circuit for 1st subcircuit ****************************** v1i 1 0 dc 0 ac 1 r1i 1 2 50 x1 2 3 line10 r1o 3 4 50 v1o 4 0 dc 6 ac 0 e11 5 0 (2,0) 2 v11 5 11 ac 1 r11 11 0 1g e21 21 0 (3,0) 2 r21 21 0 1g ***************************** * measuring circuit for 2nd subcircuit **************************** va1i 100 0 dc 0 ac 1 ra1i 100 200 50 xa1 200 300 line40 ra1o 300 400 50 va1o 400 0 dc 6 ac 0 ea11 500 0 (200,0) 2 va11 500 1100 ac 1 ra11 1100 0 1g ea21 2100 0 (300,0) 2 ra21 2100 0 1g In this segment, the first measuring circuit measures the S-parameters of the subcircuit line10 and outputs S11 in V(11) and S21 in V(21). In the second circuit, the Sparameters of subcircuit line40 are measured and S11 is output in V(1100) and S21 in 241 V(2100). It is easy to see that any number of measuring circuits can be used to simultaneously measure the S-parameters of several different circuits. Since S-parameters of different circuits can be measured simultaneously, simultaneous optimizations can also be performed. A sample control deck for the optimization of two circuits is given below. * Circuit definitions here .param + c_cou = opt1(6.4e-12,1f,1n) + rl = opt1(1e4,1,1e8) + r2 = opt1(4.7e-1,0.00001,10) + l2 = opt1(1.2e-11,.01p,1u) + c2 = opt1(9.2e-15,0.1f,1n) + rsq = opt1(0.30,0.01,10) + lsq = opt1(0.4e-11,1f,1u) + csq = opt1(2.1e-15,0.01f,1n) ****************************** * circuit for 1st subcircuit ****************************** v1i 1 0 dc 0 ac 1 r1i 1 2 50 x1 2 3 line10 r1o 3 4 50 v1o 4 0 dc 6 ac 0 e11 5 0 (2,0) 2 v11 5 11 ac 1 r11 11 0 1g e21 21 0 (3,0) 2 r21 21 0 1g ***************************** * circuit for 2nd subcircuit **************************** va1i 100 0 dc 0 ac 1 ra1i 100 200 50 242 xa1 200 300 line40 ra1o 300 400 50 va1o 400 0 dc 6 ac 0 ea11 500 0 (200,0) 2 va11 500 1100 ac 1 ra11 1100 0 1g ea21 2100 0 (300,0) 2 ra21 2100 0 1g .ac data=measured optimize=opt1 + results=comp1,comp2,comp5,comp6,comp7,comp8,comp9,comp10 + model=converge .model converge opt relin=1e-4 relout=1e-3 close=100 + itropt=500 .measure .measure .measure .measure .measure .measure .measure .measure .print .print .print .print ac ac ac ac ac ac ac ac comp1 err1 par(s11r) vr(11) comp2 err1 par(s11i) vi(11) comp5 err1 par(s21r) vr(21) comp6 err1 par(s21i) vi(21) comp7 err1 par(sa11r) vr(1100) comp8 err1 par(sa11i) vi(1100) comp9 err1 par(sa21r) vr(2100) comp10 err1 par(sa21i) vi(2100) par(s11r) vr(11) par(s11i) vi(11) par(s21r) vr(21) par(s21i) vi(21) par(sa11r) vr(1100) par(sa11i) vi(1100) par(sa21r) vr(2100) par(sa21i) vi(2100) .data measured mer file= 'c2c14' freq=1 s11r=2 s11i=3 s21r=4 s21i=5 s12r=6 + s12i=7 s22r=8 s22i=9 + sa11r=11 sa11i=12 sa21r=13 sa21i=14 sa12r=15 sa12i=16 + sa22r=17 sa22i=18 out = 'c2c14_data.txt' .enddata .param freq=500m, s11r=0, s11i=0, s21r=0, s21i=0, s12r=0, + s12i=0, s22r=0, s22i=0 .param sa11r=0, sa11i=0, sa21r=0, sa21i=0, sa12r=0, 243 + sa12i=0, sa22r=0, + sa22i=0 .end In this example, the line10 and line40 circuits are simultaneously curve fit optimized to their respective measured data sets, as contained in the c2c14 data file. The s11r, s11i, etc. measured data correspond to line10, and the sa11r, sa11i, etc. measured data correspond to line40. RMS errors are minimized by use of the .measure statements, and all of them are used in the analysis optimization run (.ac statement). The circuit parameters being optimized are defined in the .param statement, and these may be used in both line10 and line40 circuits. In order to achieve successful optimization, the computed values for the circuit components must satisfy convergence criteria for both circuits, and thus are more likely to be correct than if only one circuit was used. 244 APPENDIX D CIRCUITS AND DATA FOR SERPENTINE RESISTOR MODELING D.1. Introduction Input files and measured S-parameter data for test structure optimization for the serpentine resistor modeling study described earlier in this thesis are presented in this appendix. In addition, the circuit file representing the complete model of the 9 segment resistor is also show, with associated measured S-parameters. All circuit files are written for the Star-Hspice circuit simulator. It should be noted that in some cases, certain subcircuit (.subckt) calls are defined but are never used in the actual optimization runs. Additionally, only S11 and S21 results are shown for the measured data, since S22 and S11 are equal, and S12 and S21 are also equivalent for these structures. 245 D.2. Test Structure 1 D.2.1. Circuit Optimization Input File Hspice input circuit for optimization and parameter extraction of test structure 1 building blocks is shown below. .option accurate node nopage ingold=2 post acct=1 probe .subckt mstl30u_sq 1 5 r1 1 2 r2sq l1 2 3 l2sq c1 3 0 c2sq r2 3 0 10g r1r 3 4 r2sq l2r 4 5 l2sq cc1 1 3 c_cou2 cc2 3 5 c_cou2 .ends .subckt mstl_pad 1 5 r1 1 2 rpad l1 2 3 lpad c1 3 0 cpad r2 3 0 10g r1r 3 4 rpad l2r 4 5 lpad cc1 1 3 c_cou_pad cc2 3 5 c_cou_pad .ends .subckt mstl30u_sq5 1 6 x1 1 2 mstl30u_sq x2 2 3 mstl30u_sq x3 3 4 mstl30u_sq x4 4 5 mstl30u_sq x5 5 6 mstl30u_sq .ends .subckt mstl30u_sq3 1 4 x1 1 2 mstl30u_sq x2 2 3 mstl30u_sq x3 3 4 mstl30u_sq .ends ********************************* .subckt line30u 1 7 x1 x2 x3 x4 x5 x6 1 2 3 4 5 6 2 mstl_pad 3 mstl30u_sq5 4 mstl30u_sq5 5 mstl30u_sq5 6 mstl30u_sq5 7 mstl_pad 246 ro 7 0 1t .ends .param ****************************************** * 30 u line specs ***************************************** + c_cou2 = opt1(1.4e-15,0.01f,1n) + r2sq = opt1(5.7e-3,1e-5,10) + l2sq = opt1(1.4e-11,1f,1u) + c2sq = opt1(4.8e-15,0.01f,1n) ************************************** * pad specs ************************************** + c_cou_pad = 1.8e-15 + rpad = 0.08 + lpad = 1.2e-11 + cpad = 1.8e-15 ***************************** * circuit for line **************************** va1i 100 0 dc 0 ac 1 ra1i 100 200 50 xa1 200 300 line ra1o 300 400 50 va1o 400 0 dc 6 ac 0 ea11 500 0 (200,0) 2 va11 500 1100 ac 1 ra11 1100 0 1g ea21 2100 0 (300,0) 2 ra21 2100 0 1g .ac data=measured optimize=opt1 + results=comp7,comp8,comp9,comp10 + model=converge .model converge opt relin=1e-5 relout=1e-5 close=200 itropt=300 .measure .measure .measure .measure ac ac ac ac comp7 err1 par(sa11r) vr(1100) comp8 err1 par(sa11i) vi(1100) comp9 err1 par(sa21r) vr(2100) comp10 err1 par(sa21i) vi(2100) .print par(sa11r) vr(1100) par(sa11i) vi(1100) .print par(sa21r) vr(2100) par(sa21i) vi(2100) .data measured mer file= ‘all.txt' freq=1 sa11r=2 sa11i=3 sa21r=4 sa21i=5 sa12r=6 sa12i=7 sa22r=8 sa22i=9 out = ‘all.out' .enddata .param freq=100m, s11r=0, s11i=0, s21r=0, s21i=0, s12r=0, s12i=0, s22r=0, s22i=0 .param sa11r=0, sa11i=0, sa21r=0, sa21i=0, sa12r=0, sa12i=0, sa22r=0, + sa22i=0 .end 247 D.2.2. Measured S-Parameter Data Frequency S11(R) Measured S11(I) Measured S21(R) Measured S21(I) Measured 5.00E+08 5.98E+08 6.95E+08 7.93E+08 8.90E+08 9.88E+08 1.09E+09 1.28E+09 1.48E+09 1.67E+09 1.87E+09 2.06E+09 2.26E+09 2.45E+09 2.65E+09 2.84E+09 3.04E+09 3.23E+09 3.43E+09 3.62E+09 3.82E+09 4.01E+09 4.21E+09 4.40E+09 4.60E+09 4.79E+09 4.99E+09 5.18E+09 5.38E+09 5.57E+09 5.77E+09 5.96E+09 6.16E+09 6.35E+09 6.55E+09 6.74E+09 6.94E+09 7.13E+09 7.33E+09 7.52E+09 7.72E+09 7.91E+09 8.11E+09 8.30E+09 8.50E+09 3.70E-02 3.65E-02 3.59E-02 3.64E-02 3.63E-02 3.57E-02 3.64E-02 3.64E-02 3.64E-02 3.70E-02 3.83E-02 3.71E-02 3.92E-02 3.93E-02 4.03E-02 4.14E-02 4.22E-02 4.37E-02 4.34E-02 4.22E-02 4.32E-02 4.45E-02 4.60E-02 4.68E-02 4.82E-02 4.93E-02 5.17E-02 5.40E-02 5.42E-02 5.52E-02 5.81E-02 5.68E-02 5.94E-02 6.13E-02 6.32E-02 6.48E-02 6.72E-02 6.93E-02 7.10E-02 7.14E-02 7.58E-02 7.48E-02 7.80E-02 7.73E-02 8.01E-02 8.48E-03 9.87E-03 1.19E-02 1.31E-02 1.52E-02 1.68E-02 1.93E-02 2.19E-02 2.56E-02 2.85E-02 3.27E-02 3.67E-02 3.87E-02 4.03E-02 4.53E-02 4.93E-02 5.17E-02 5.40E-02 5.95E-02 6.25E-02 6.53E-02 6.90E-02 7.37E-02 7.72E-02 8.03E-02 8.19E-02 8.55E-02 8.85E-02 9.38E-02 9.59E-02 9.83E-02 1.02E-01 1.05E-01 1.07E-01 1.10E-01 1.14E-01 1.15E-01 1.15E-01 1.19E-01 1.23E-01 1.25E-01 1.27E-01 1.32E-01 1.33E-01 1.36E-01 9.65E-01 9.63E-01 9.63E-01 9.63E-01 9.63E-01 9.61E-01 9.63E-01 9.64E-01 9.62E-01 9.61E-01 9.61E-01 9.58E-01 9.59E-01 9.59E-01 9.54E-01 9.54E-01 9.54E-01 9.55E-01 9.52E-01 9.50E-01 9.49E-01 9.48E-01 9.47E-01 9.46E-01 9.46E-01 9.45E-01 9.41E-01 9.40E-01 9.39E-01 9.36E-01 9.34E-01 9.32E-01 9.31E-01 9.28E-01 9.25E-01 9.25E-01 9.20E-01 9.20E-01 9.19E-01 9.11E-01 9.10E-01 9.08E-01 9.06E-01 9.07E-01 9.02E-01 -1.64E-02 -2.04E-02 -2.35E-02 -2.73E-02 -3.07E-02 -3.31E-02 -3.70E-02 -4.27E-02 -4.87E-02 -5.45E-02 -6.26E-02 -6.93E-02 -7.35E-02 -8.17E-02 -8.72E-02 -9.42E-02 -9.86E-02 -1.07E-01 -1.12E-01 -1.18E-01 -1.23E-01 -1.30E-01 -1.35E-01 -1.40E-01 -1.45E-01 -1.53E-01 -1.58E-01 -1.64E-01 -1.70E-01 -1.75E-01 -1.83E-01 -1.89E-01 -1.93E-01 -1.98E-01 -2.05E-01 -2.10E-01 -2.15E-01 -2.21E-01 -2.27E-01 -2.32E-01 -2.37E-01 -2.41E-01 -2.47E-01 -2.53E-01 -2.58E-01 248 8.69E+09 8.89E+09 9.08E+09 9.28E+09 9.47E+09 9.67E+09 9.86E+09 1.03E+10 1.06E+10 1.10E+10 1.14E+10 1.18E+10 1.22E+10 1.26E+10 1.30E+10 1.34E+10 1.38E+10 1.42E+10 1.45E+10 1.49E+10 1.53E+10 1.57E+10 1.61E+10 1.65E+10 1.69E+10 1.73E+10 1.77E+10 1.81E+10 1.84E+10 1.88E+10 1.92E+10 1.96E+10 2.00E+10 8.09E-02 8.25E-02 8.55E-02 8.81E-02 9.15E-02 9.35E-02 9.51E-02 9.81E-02 9.92E-02 1.05E-01 1.08E-01 1.16E-01 1.20E-01 1.26E-01 1.24E-01 1.23E-01 1.37E-01 1.40E-01 1.49E-01 1.47E-01 1.52E-01 1.48E-01 1.53E-01 1.65E-01 1.73E-01 1.77E-01 1.74E-01 1.69E-01 1.71E-01 1.79E-01 1.96E-01 2.11E-01 2.18E-01 1.37E-01 1.38E-01 1.42E-01 1.45E-01 1.49E-01 1.51E-01 1.55E-01 1.60E-01 1.64E-01 1.67E-01 1.68E-01 1.73E-01 1.78E-01 1.86E-01 1.87E-01 1.92E-01 1.92E-01 1.91E-01 1.99E-01 2.10E-01 2.12E-01 2.14E-01 2.12E-01 2.10E-01 2.19E-01 2.31E-01 2.41E-01 2.42E-01 2.26E-01 2.22E-01 2.39E-01 2.42E-01 2.44E-01 9.01E-01 8.98E-01 8.96E-01 8.92E-01 8.90E-01 8.89E-01 8.85E-01 8.82E-01 8.75E-01 8.71E-01 8.67E-01 8.61E-01 8.55E-01 8.51E-01 8.47E-01 8.37E-01 8.32E-01 8.30E-01 8.23E-01 8.19E-01 8.09E-01 8.00E-01 7.98E-01 7.99E-01 7.89E-01 7.83E-01 7.69E-01 7.63E-01 7.62E-01 7.61E-01 7.48E-01 7.39E-01 7.34E-01 D.3. Test Structure 2 D.3.1. Circuit Optimization Input File .option accurate node nopage ingold=2 post acct=1 probe numdgt=10 opts * 3 coupled lines .subckt mstl_c3 1 6 11 r1l 1 2 rsq 5 10 15 249 -2.62E-01 -2.68E-01 -2.71E-01 -2.76E-01 -2.80E-01 -2.87E-01 -2.93E-01 -3.00E-01 -3.13E-01 -3.21E-01 -3.29E-01 -3.39E-01 -3.44E-01 -3.55E-01 -3.65E-01 -3.77E-01 -3.81E-01 -3.89E-01 -3.98E-01 -4.09E-01 -4.14E-01 -4.28E-01 -4.37E-01 -4.38E-01 -4.48E-01 -4.57E-01 -4.65E-01 -4.83E-01 -4.88E-01 -4.97E-01 -4.98E-01 -5.03E-01 -5.14E-01 l1l 2 3 lsq c1 3 0 csq r1r 3 4 rsq l1r 4 5 lsq rg1 3 0 10mega cca 1 3 ccsq ccb 3 5 ccsq cc12a 3 8 c_cou r2l 6 7 rsq l2l 7 8 lsq c2 8 0 csq rg2 8 0 10mega l2r 8 9 lsq r2r 9 10 rsq ccb1 6 8 ccsq ccb2 8 10 ccsq k1 l1l l2l k=cou_l k2 l2r l1r k=cou_l r3l 11 12 rsq l3l 12 13 lsq c3 13 0 csq rg3 13 0 10mega l3r 13 14 lsq r3r 14 15 rsq ccc1 11 13 ccsq ccc2 13 15 ccsq cc23b 8 13 c_cou k23a l2l l3l k=cou_l k23b l2r l3r k=cou_l .param + rsq = .0957 + lsq = 1.04e-11 *+ csq = 4.04e-15 + ccsq= 8.9e-17 .ends * microstrip coupled 3 lines set of 5 .subckt mstl_c3_5 1 2 3 16 17 18 x1 1 2 3 4 5 6 mstl_c3 x2 4 5 6 7 8 9 mstl_c3 x3 7 8 9 10 11 12 mstl_c3 x4 10 11 12 13 14 15 mstl_c3 x5 13 14 15 16 17 18 mstl_c3 .ends * microstrip coupled 3 lines set of 4 .subckt mstl_c3_4 1 2 3 13 14 15 x1 1 2 3 4 5 6 mstl_c3 x2 4 5 6 7 8 9 mstl_c3 x3 7 8 9 10 11 12 mstl_c3 x4 10 11 12 13 14 15 mstl_c3 .ends * microstrip coupled 3 lines set of 3 .subckt mstl_c3_3 1 2 3 10 11 12 x1 1 2 3 4 5 6 mstl_c3 x2 4 5 6 7 8 9 mstl_c3 x3 7 8 9 10 11 12 mstl_c3 .ends * microstrip coupled 3 lines set of 2 .subckt mstl_c3_2 1 2 3 7 8 9 x1 1 2 3 4 5 6 mstl_c3 x2 4 5 6 7 8 9 mstl_c3 .ends 250 ************************************************************* *microstrip coupled 2 lines .subckt mstl_c2 1 6 5 10 r1l 1 2 rsq l1l 2 3 lsq c1 3 0 csq r1r 3 4 rsq l1r 4 5 lsq rg1 3 0 10mega ccs1 1 3 ccsq ccs2 3 5 ccsq *ccouple1 1 6 c_cou ccouple2 3 8 c_cou r2l 6 7 rsq l2l 7 8 lsq c2 8 0 csq rg2 8 0 10mega l2r 8 9 lsq r2r 9 10 rsq ccs3 6 8 ccsq ccs4 8 10 ccsq k1 l1l l2l k=cou_l k2 l2r l1r k=cou_l .param + rsq = 5.7e-2 + lsq = 7.14e-11 *+ 3.11e-14 + csq = 2.71e-14 + ccsq = 3.53e-17 + c_cou = 7.3e-16 + cou_l = 0.48 .ends *set of 5 microstrip coupled 2 lines .subckt mstl_c2_5 1 2 11 12 x1 1 2 3 4 mstl_c2 x2 3 4 5 6 mstl_c2 x3 5 6 7 8 mstl_c2 x4 7 8 9 10 mstl_c2 x5 9 10 11 12 mstl_c2 .ends *set of 6 microstrip coupled 2 lines .subckt mstl_c2_6 1 2 13 14 x1 1 2 3 4 mstl_c2 x2 3 4 5 6 mstl_c2 x3 5 6 7 8 mstl_c2 x4 7 8 9 10 mstl_c2 x5 9 10 11 12 mstl_c2 x6 11 12 13 14 mstl_c2 .ends *set of 4 microstrip coupled 2 lines .subckt mstl_c2_4 1 2 9 10 x1 1 2 3 4 mstl_c2 x2 3 4 5 6 mstl_c2 x3 5 6 7 8 mstl_c2 x4 7 8 9 10 mstl_c2 .ends *set of 3 microstrip coupled 2 lines .subckt mstl_c2_3 1 2 7 8 x1 1 2 3 4 mstl_c2 x2 3 4 5 6 mstl_c2 x3 5 6 7 8 mstl_c2 251 .ends *set of 2 microstrip coupled 2 lines .subckt mstl_c2_2 1 2 5 6 x1 1 2 3 4 mstl_c2 x2 3 4 5 6 mstl_c2 .ends ************************************************************* *pads .subckt mstl_pad 1 5 r1 1 2 rsq l1 2 3 lsq c1 3 0 csq r2 3 0 10g r1r 3 4 rsq l2r 4 5 lsq cc1 1 3 ccsq cc2 3 5 ccsq .param + rsq = 8.9e-3 + lsq = 2.9e-13 + csq = 1e-17 + ccsq = 1.0e-15 .ends *microstrip striaght line .subckt mstl_sq 1 5 r1 1 2 rsq l1 2 3 lsq c1 3 0 csq r2 3 0 10g r1r 3 4 rsq l2r 4 5 lsq cc1 1 3 ccsq cc2 3 5 ccsq .param + rsq = .0957 + lsq = 1.04e-11 *+ csq = 4.04e-15 + ccsq= 8.9e-17 .ends *set of 5 microstrip blocks .subckt mstl_sq_5 1 6 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq x4 4 5 mstl_sq x5 5 6 mstl_sq .ends *set of 4 microstrip blocks .subckt mstl_sq_4 1 5 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq x4 4 5 mstl_sq .ends *set of 3 microstrip blocks .subckt mstl_sq_3 1 4 x1 1 2 mstl_sq x2 2 3 mstl_sq 252 x3 3 4 mstl_sq .ends *set of 2 microstrip blocks .subckt mstl_sq_2 1 3 x1 1 2 mstl_sq x2 2 3 mstl_sq .ends .subckt res3seg 1 13 x1 a1 2 3 4 5 6 mstl_c3_4 x2 4 5 6 7 8 9 mstl_c3_4 x3 7 8 9 10 11 12 mstl_c3 x4 10 11 mstl_corn_u x5 2 3 mstl_corn_u x6 b1 a1 mstl_sq_5 x8 1 b1 mstl_pad x7 12 a13 mstl_sq_5 x9 a13 13 mstl_pad ro 13 0 1g .ends v1i 1 0 dc 0 ac 1 r1i 1 2 50 x1 2 3 res3seg r1o 3 4 50 v1o 4 0 dc 6 ac 0 e11 5 0 (2,0) 2 v11 5 11 ac 1 r11 11 0 1g e21 21 0 (3,0) 2 r21 21 0 1g .param + cou_l = opt1(0.1,0,.7) + c_cou = opt1(1e-15,1e-17,1e-11) + r2 = opt1(.8,1e-6,10) + c2 = opt1(5e-15,1e-17,1e-12) + csq = opt1(4e-15,1e-17,1e-12) + l2 = opt1(1e-11,1e-13,1e-7) + c_cou2cr = opt1(1e-15,1e-17,1e-12) *+ ccsq = 1e-17 *+ rsq = opt1(1e-4,1e-7,1) .ac data=measured optimize=opt1 + results=comp1,comp2,comp5,comp6 + model=converge .model converge opt relin=1e-5 relout=1e-5 close=200 itropt=300 .measure .measure .measure .measure ac ac ac ac comp1 comp2 comp5 comp6 err1 err1 err1 err1 par(s11r) par(s11i) par(s21r) par(s21i) vr(11) vi(11) vr(21) vi(21) .print par(s11r) vr(11) par(s11i) vi(11) .print par(s21r) vr(21) par(s21i) vi(21) .data measured mer file= 'bll.txt' freq=1 s11r=2 s11i=3 s21r=4 s21i=5 s12r=6 s12i=7 s22r=8 s22i=9 out = 'bll.out' 253 .enddata .param freq=100m, s11r=0, s11i=0, s21r=0, s21i=0, s12r=0, s12i=0, s22r=0, s22i=0 .param sa11r=0, sa11i=0, sa21r=0, sa21i=0, sa12r=0, sa12i=0, sa22r=0, + sa22i=0 .end D.3.2. Measured S-Parameter Data Frequency S11(R) Measured S11(I) Measured S21(R) Measured S21(I) Measured 5.00E+08 5.98E+08 6.95E+08 7.93E+08 8.90E+08 9.88E+08 1.09E+09 1.18E+09 1.28E+09 1.38E+09 1.48E+09 1.57E+09 1.67E+09 1.77E+09 1.87E+09 1.96E+09 2.06E+09 2.16E+09 2.26E+09 2.35E+09 2.45E+09 2.55E+09 2.65E+09 2.74E+09 2.84E+09 2.94E+09 3.04E+09 3.13E+09 3.23E+09 3.33E+09 3.43E+09 3.52E+09 3.62E+09 3.72E+09 3.82E+09 3.91E+09 4.01E+09 4.11E+09 4.21E+09 6.90E-02 6.90E-02 6.84E-02 6.89E-02 6.97E-02 6.87E-02 6.89E-02 7.01E-02 7.05E-02 7.05E-02 7.06E-02 7.04E-02 7.20E-02 7.35E-02 7.33E-02 7.41E-02 7.40E-02 7.42E-02 7.58E-02 7.65E-02 7.57E-02 7.69E-02 7.72E-02 7.99E-02 8.05E-02 8.08E-02 7.96E-02 8.21E-02 8.35E-02 8.45E-02 8.37E-02 8.28E-02 8.39E-02 8.54E-02 8.63E-02 8.62E-02 8.76E-02 8.97E-02 9.22E-02 1.30E-02 1.53E-02 1.75E-02 1.97E-02 2.14E-02 2.66E-02 2.81E-02 3.03E-02 3.33E-02 3.59E-02 3.77E-02 4.08E-02 4.34E-02 4.57E-02 4.73E-02 5.12E-02 5.40E-02 5.50E-02 5.82E-02 6.09E-02 6.22E-02 6.37E-02 6.66E-02 6.84E-02 7.12E-02 7.41E-02 7.51E-02 7.99E-02 8.05E-02 8.31E-02 8.59E-02 8.95E-02 9.05E-02 9.23E-02 9.68E-02 9.78E-02 1.02E-01 1.03E-01 1.06E-01 9.30E-01 9.30E-01 9.29E-01 9.30E-01 9.30E-01 9.29E-01 9.28E-01 9.28E-01 9.28E-01 9.25E-01 9.26E-01 9.28E-01 9.25E-01 9.25E-01 9.22E-01 9.24E-01 9.23E-01 9.22E-01 9.21E-01 9.19E-01 9.18E-01 9.18E-01 9.16E-01 9.16E-01 9.15E-01 9.14E-01 9.12E-01 9.12E-01 9.12E-01 9.10E-01 9.09E-01 9.05E-01 9.04E-01 9.04E-01 9.03E-01 9.01E-01 8.99E-01 8.98E-01 8.95E-01 -2.50E-02 -2.86E-02 -3.65E-02 -3.94E-02 -4.49E-02 -4.86E-02 -5.35E-02 -5.88E-02 -6.17E-02 -6.74E-02 -7.22E-02 -7.75E-02 -7.93E-02 -8.63E-02 -9.07E-02 -9.51E-02 -9.87E-02 -1.02E-01 -1.07E-01 -1.12E-01 -1.20E-01 -1.25E-01 -1.27E-01 -1.31E-01 -1.38E-01 -1.40E-01 -1.43E-01 -1.50E-01 -1.55E-01 -1.56E-01 -1.61E-01 -1.65E-01 -1.71E-01 -1.75E-01 -1.78E-01 -1.82E-01 -1.84E-01 -1.90E-01 -1.94E-01 254 4.30E+09 4.40E+09 4.50E+09 4.60E+09 4.69E+09 4.79E+09 4.89E+09 4.99E+09 5.08E+09 5.18E+09 5.28E+09 5.38E+09 5.47E+09 5.57E+09 5.67E+09 5.77E+09 5.86E+09 5.96E+09 6.06E+09 6.16E+09 6.25E+09 6.35E+09 6.45E+09 6.55E+09 6.64E+09 6.74E+09 6.84E+09 6.94E+09 7.03E+09 7.13E+09 7.23E+09 7.33E+09 7.42E+09 7.52E+09 7.62E+09 7.72E+09 7.81E+09 7.91E+09 8.01E+09 8.11E+09 8.20E+09 8.30E+09 8.40E+09 8.50E+09 8.59E+09 8.69E+09 8.79E+09 8.89E+09 8.98E+09 9.08E+09 9.18E+09 9.25E-02 9.29E-02 9.37E-02 9.61E-02 9.70E-02 9.84E-02 1.01E-01 1.03E-01 1.04E-01 1.05E-01 1.07E-01 1.08E-01 1.09E-01 1.10E-01 1.13E-01 1.13E-01 1.15E-01 1.15E-01 1.18E-01 1.19E-01 1.21E-01 1.22E-01 1.23E-01 1.25E-01 1.27E-01 1.28E-01 1.30E-01 1.31E-01 1.33E-01 1.35E-01 1.37E-01 1.40E-01 1.40E-01 1.42E-01 1.43E-01 1.48E-01 1.48E-01 1.48E-01 1.50E-01 1.52E-01 1.53E-01 1.53E-01 1.57E-01 1.58E-01 1.60E-01 1.61E-01 1.62E-01 1.65E-01 1.66E-01 1.67E-01 1.68E-01 1.09E-01 1.11E-01 1.12E-01 1.15E-01 1.17E-01 1.20E-01 1.20E-01 1.22E-01 1.25E-01 1.27E-01 1.28E-01 1.33E-01 1.34E-01 1.36E-01 1.39E-01 1.39E-01 1.42E-01 1.42E-01 1.43E-01 1.46E-01 1.49E-01 1.50E-01 1.51E-01 1.54E-01 1.57E-01 1.58E-01 1.59E-01 1.61E-01 1.61E-01 1.62E-01 1.63E-01 1.66E-01 1.68E-01 1.69E-01 1.71E-01 1.72E-01 1.74E-01 1.75E-01 1.79E-01 1.82E-01 1.80E-01 1.82E-01 1.86E-01 1.84E-01 1.90E-01 1.90E-01 1.90E-01 1.90E-01 1.93E-01 1.93E-01 1.93E-01 255 8.95E-01 8.94E-01 8.91E-01 8.92E-01 8.89E-01 8.89E-01 8.86E-01 8.86E-01 8.84E-01 8.82E-01 8.81E-01 8.79E-01 8.76E-01 8.75E-01 8.74E-01 8.72E-01 8.70E-01 8.67E-01 8.66E-01 8.64E-01 8.64E-01 8.61E-01 8.58E-01 8.57E-01 8.54E-01 8.52E-01 8.50E-01 8.46E-01 8.46E-01 8.44E-01 8.42E-01 8.41E-01 8.39E-01 8.34E-01 8.31E-01 8.28E-01 8.29E-01 8.25E-01 8.24E-01 8.21E-01 8.20E-01 8.19E-01 8.16E-01 8.13E-01 8.12E-01 8.08E-01 8.06E-01 8.03E-01 8.02E-01 8.01E-01 7.94E-01 -1.98E-01 -2.03E-01 -2.06E-01 -2.10E-01 -2.13E-01 -2.18E-01 -2.23E-01 -2.25E-01 -2.31E-01 -2.34E-01 -2.40E-01 -2.42E-01 -2.46E-01 -2.51E-01 -2.55E-01 -2.59E-01 -2.63E-01 -2.68E-01 -2.71E-01 -2.73E-01 -2.77E-01 -2.81E-01 -2.84E-01 -2.89E-01 -2.92E-01 -2.98E-01 -2.99E-01 -3.04E-01 -3.09E-01 -3.12E-01 -3.13E-01 -3.19E-01 -3.22E-01 -3.26E-01 -3.27E-01 -3.33E-01 -3.35E-01 -3.38E-01 -3.40E-01 -3.46E-01 -3.48E-01 -3.53E-01 -3.55E-01 -3.60E-01 -3.61E-01 -3.64E-01 -3.66E-01 -3.71E-01 -3.74E-01 -3.77E-01 -3.80E-01 9.28E+09 9.37E+09 9.47E+09 9.57E+09 9.67E+09 9.76E+09 9.86E+09 9.96E+09 1.01E+10 1.73E-01 1.74E-01 1.77E-01 1.81E-01 1.82E-01 1.84E-01 1.86E-01 1.86E-01 1.91E-01 1.95E-01 1.96E-01 1.98E-01 2.01E-01 2.02E-01 2.03E-01 2.03E-01 2.06E-01 2.07E-01 7.94E-01 7.93E-01 7.91E-01 7.89E-01 7.88E-01 7.83E-01 7.79E-01 7.79E-01 7.78E-01 D.4. 9-Segment Resistor D.4.1. Circuit File .option accurate node nopage ingold=2 post acct=2 probe * 9 coupled lines .subckt mstl_c9 1 6 11 16 21 26 31 36 41 r1l 1 2 rsq l1l 2 3 lsq c1 3 0 csq r1r 3 4 rsq l1r 4 5 lsq rg1 3 0 10mega cca 1 3 ccsq ccb 3 5 ccsq cc12a 3 8 c_cou r2l 6 7 rsq l2l 7 8 lsq c2 8 0 csq rg2 8 0 10mega l2r 8 9 lsq r2r 9 10 rsq ccb1 6 8 ccsq ccb2 8 10 ccsq k1 l1l l2l k=cou_l k2 l2r l1r k=cou_l r3l 11 12 rsq l3l 12 13 lsq c3 13 0 csq rg3 13 0 10mega l3r 13 14 lsq r3r 14 15 rsq ccc1 11 13 ccsq ccc2 13 15 ccsq 5 10 15 20 25 30 35 40 45 r4l 16 17 rsq l4l 17 18 lsq c4 18 0 csq l4r 18 19 lsq r4r 19 20 rsq cc4a 16 18 ccsq cc4b 18 20 ccsq cc34 13 18 c_cou 256 -3.83E-01 -3.84E-01 -3.88E-01 -3.93E-01 -3.93E-01 -3.99E-01 -3.99E-01 -4.04E-01 -4.09E-01 k34a l4l l3l k=cou_l k34b l4r l3r k=cou_l r5l 21 22 rsq l5l 22 23 lsq c5 23 0 csq l5r 23 24 lsq r5r 24 25 rsq cc5a 21 23 ccsq cc5b 23 25 ccsq cc45 18 23 c_cou k45a l4l l5l k=cou_l k45b l4r l5r k=cou_l r6l 26 27 rsq l6l 27 28 lsq c6 28 0 csq l6r 28 29 lsq r6r 29 30 rsq cc6a 26 28 ccsq cc6b 28 30 ccsq cc56 23 28 c_cou k56a l5l l6l k=cou_l k56b l5r l6r k=cou_l r7l 31 32 rsq l7l 32 33 lsq c7 33 0 csq l7r 33 34 lsq r7r 34 35 rsq cc7a 31 33 ccsq cc7b 33 35 ccsq cc67 28 33 c_cou k67a l6l l7l k=cou_l k67b l6r l7r k=cou_l r8l 36 37 rsq l8l 37 38 lsq c8 38 0 csq l8r 38 39 lsq r8r 39 40 rsq cc8a 36 38 ccsq cc8b 38 40 ccsq cc78 33 38 c_cou k78a l7l l8l k=cou_l k78b l7r l8r k=cou_l r9l 41 42 rsq l9l 42 43 lsq c9 43 0 csq l9r 43 44 lsq r9r 44 45 rsq cc9a 41 43 ccsq cc9b 43 45 ccsq cc89 38 43 c_cou k89a l8l l9l k=cou_l k89b l8r l9r k=cou_l cc23b 8 13 c_cou k23a l2l l3l k=cou_l k23b l2r l3r k=cou_l .param + rsq = .0957 + lsq = 1.04e-11 *+ csq = 4.04e-15 + ccsq= 8.9e-17 257 .ends * microstrip coupled .subckt mstl_c9_5 1 x1 1 2 3 4 5 6 7 8 9 x2 10 11 12 13 14 15 x3 19 20 21 22 23 24 x4 28 29 30 31 32 33 x5 37 38 39 40 41 42 .ends 9 lines set of 5 2 3 4 5 6 7 8 9 46 47 48 49 50 51 52 53 54 10 11 12 13 14 15 16 17 18 mstl_c9 16 17 18 19 20 21 22 23 24 25 26 27 mstl_c9 25 26 27 28 29 30 31 32 33 34 35 36 mstl_c9 34 35 36 37 38 39 40 41 42 43 44 45 mstl_c9 43 44 45 46 47 48 49 50 51 52 53 54 mstl_c9 * microstrip coupled .subckt mstl_c9_4 1 x1 1 2 3 4 5 6 7 8 9 x2 10 11 12 13 14 15 x3 19 20 21 22 23 24 x4 28 29 30 31 32 33 .ends 9 lines set of 4 2 3 4 5 6 7 8 9 37 38 39 40 41 42 43 44 45 10 11 12 13 14 15 16 17 18 mstl_c9 16 17 18 19 20 21 22 23 24 25 26 27 mstl_c9 25 26 27 28 29 30 31 32 33 34 35 36 mstl_c9 34 35 36 37 38 39 40 41 42 43 44 45 mstl_c9 * microstrip coupled .subckt mstl_c9_3 1 x1 1 2 3 4 5 6 7 8 9 x2 10 11 12 13 14 15 x3 19 20 21 22 23 24 .ends 9 lines set of 3 2 3 4 5 6 7 8 9 28 29 30 31 32 33 34 35 36 10 11 12 13 14 15 16 17 18 mstl_c9 16 17 18 19 20 21 22 23 24 25 26 27 mstl_c9 25 26 27 28 29 30 31 32 33 34 35 36 mstl_c9 * microstrip coupled 9 lines set of 2 .subckt mstl_c9_2 1 2 3 4 5 6 7 8 9 19 20 21 22 23 24 25 26 27 x1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 mstl_c9 x2 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 mstl_c9 .ends * microstrip coupled 9 lines set of 1 .subckt mstl_c9_1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 x1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 mstl_c9 .ends * microstrip coupled 3 lines set of 5 .subckt mstl_c3_5 1 2 3 16 17 18 x1 1 2 3 4 5 6 mstl_c3 x2 4 5 6 7 8 9 mstl_c3 x3 7 8 9 10 11 12 mstl_c3 x4 10 11 12 13 14 15 mstl_c3 x5 13 14 15 16 17 18 mstl_c3 .ends * microstrip coupled 3 lines set of 4 .subckt mstl_c3_4 1 2 3 13 14 15 x1 1 2 3 4 5 6 mstl_c3 x2 4 5 6 7 8 9 mstl_c3 x3 7 8 9 10 11 12 mstl_c3 x4 10 11 12 13 14 15 mstl_c3 .ends * microstrip coupled 3 lines set of 3 .subckt mstl_c3_3 1 2 3 10 11 12 x1 1 2 3 4 5 6 mstl_c3 x2 4 5 6 7 8 9 mstl_c3 x3 7 8 9 10 11 12 mstl_c3 .ends * microstrip coupled 3 lines set of 2 .subckt mstl_c3_2 1 2 3 7 8 9 258 x1 1 2 3 4 5 6 x2 4 5 6 7 8 9 .ends mstl_c3 mstl_c3 ************************************************************* *microstrip coupled 2 lines .subckt mstl_c2 1 6 5 10 r1l 1 2 rsq l1l 2 3 lsq c1 3 0 csq r1r 3 4 rsq l1r 4 5 lsq rg1 3 0 10mega ccs1 1 3 ccsq ccs2 3 5 ccsq *ccouple1 1 6 c_cou ccouple2 3 8 c_cou r2l 6 7 rsq l2l 7 8 lsq c2 8 0 csq rg2 8 0 10mega l2r 8 9 lsq r2r 9 10 rsq ccs3 6 8 ccsq ccs4 8 10 ccsq k1 l1l l2l k=cou_l k2 l2r l1r k=cou_l .param + rsq = 5.7e-2 + lsq = 7.14e-11 *+ 3.11e-14 + csq = 2.71e-14 + ccsq = 3.53e-17 + c_cou = 7.3e-16 + cou_l = 0.48 .ends *set of 5 microstrip coupled 2 lines .subckt mstl_c2_5 1 2 11 12 x1 1 2 3 4 mstl_c2 x2 3 4 5 6 mstl_c2 x3 5 6 7 8 mstl_c2 x4 7 8 9 10 mstl_c2 x5 9 10 11 12 mstl_c2 .ends *set of 6 microstrip coupled 2 lines .subckt mstl_c2_6 1 2 13 14 x1 1 2 3 4 mstl_c2 x2 3 4 5 6 mstl_c2 x3 5 6 7 8 mstl_c2 x4 7 8 9 10 mstl_c2 x5 9 10 11 12 mstl_c2 x6 11 12 13 14 mstl_c2 .ends *set of 4 microstrip coupled 2 lines .subckt mstl_c2_4 1 2 9 10 x1 1 2 3 4 mstl_c2 x2 3 4 5 6 mstl_c2 x3 5 6 7 8 mstl_c2 x4 7 8 9 10 mstl_c2 .ends *set of 3 microstrip coupled 2 lines .subckt mstl_c2_3 1 2 7 8 259 x1 1 2 3 4 mstl_c2 x2 3 4 5 6 mstl_c2 x3 5 6 7 8 mstl_c2 .ends *set of 2 microstrip coupled 2 lines .subckt mstl_c2_2 1 2 5 6 x1 1 2 3 4 mstl_c2 x2 3 4 5 6 mstl_c2 .ends ************************************************************* *microstrip striaght line .subckt mstl_sq 1 5 r1 1 2 rsq l1 2 3 lsq c1 3 0 csq r2 3 0 10g r1r 3 4 rsq l2r 4 5 lsq cc1 1 3 ccsq cc2 3 5 ccsq .param + rsq = .0957 + lsq = 1.04e-11 *+ csq = 4.04e-15 + ccsq= 8.9e-17 .ends *set of 5 microstrip blocks .subckt mstl_sq_5 1 6 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq x4 4 5 mstl_sq x5 5 6 mstl_sq .ends *set of 4 microstrip blocks .subckt mstl_sq_4 1 5 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq x4 4 5 mstl_sq .ends *set of 3 microstrip blocks .subckt mstl_sq_3 1 4 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq .ends *set of 2 microstrip blocks .subckt mstl_sq_2 1 3 x1 1 2 mstl_sq x2 2 3 mstl_sq .ends ********************************************************* *corner bend (shape of l with 3 blocks) * corner l split in half .subckt mstl_corn_l_half 1 5 r1 1 2 r2 260 l1 2 3 l2 c1 3 0 c2 r2 3 0 10g r2r 3 4 r2 l2r 4 5 l2 cc1 1 3 c_cou2cr cc2 3 5 c_cou2cr .param + r2 = 6.2e-2 + l2 = 1.453e-10 + c2 = 4.558e-14 + c_cou2cr = 1.238e-16 .ends *corner bend (shape of l with 3 blocks) .subckt mstl_corn_l 1 3 x1 1 2 mstl_corn_l_half x2 3 2 mstl_corn_l_half .ends *composite corner (u shaped made of 5 squares) .subckt mstl_corn_u 1 5 r1 1 2 r2 l1 2 3 l2 c1 3 0 c2 r2 3 0 10g r2r 3 4 r2 l2r 4 5 l2 cc1 1 3 c_cou2cr cc2 3 5 c_cou2cr .ends *corner (single square - not very good) .subckt mstl_corn_1 1 5 r1 1 2 r2 l1 2 3 l2 c1 3 0 c2 r2 3 0 10g r2r 3 4 r2 l2r 4 5 l2 cc1 1 3 c_cou2cr cc2 3 5 c_cou2cr .param + r2 = 1e-6 + l2 = 1.311e-11 + c2 = 4.594e-14 + c_cou2cr = 1.033e-17 .ends .subckt corn_st 1 6 5 10 r1l 1 2 rsq l1l 2 3 lsq c1 3 0 csq r1r 3 4 rsq l1r 4 5 lsq rg1 3 0 10mega ccs1 1 3 ccsq ccs2 3 5 ccsq ccouple2 3 8 c_cou k1 l1l l2l k=cou_l k2 l1r l2r k=cou_l r2l 6 7 r2 l2l 7 8 l2 c2 8 0 c2 261 rg2 8 0 10mega l2r 8 9 l2 r2r 9 10 r2 ccs3 6 8 c_cou2cr ccs4 8 10 c_cou2cr .param + rsq = .1111 + lsq = 1.448e-10 + csq = 6.154e-14 + ccsq = 9.786e-15 + r2 = 6.6e-2 + l2 = 1.143e-10 + c2 = 4.668e-14 + c_cou2cr = 1e-17 .ends .subckt corn4_corn2 1 6 15 10 r1l 1 2 rsq l1l 2 3 lsq c1 3 0 csq r1r 3 4 rsq l1r 4 5 lsq rg1 3 0 10mega ccs1 1 3 ccsq ccs2 3 5 ccsq ccouple2 3 8 c_cou k1 l1l l2l k=cou_l k2 l1r l2r k=cou_l r2l 6 7 r2 l2l 7 8 l2 c2 8 0 c2 rg2 8 0 10mega l2r 8 9 l2 r2r 9 10 r2 ccs3 6 8 c_cou2cr ccs4 8 10 c_cou2cr r3l 5 12 r2 l3l 12 13 l2 c3 13 0 c2 rg3 13 0 10mega l3r 13 14 l2 r3r 14 15 r2 ccs6 5 13 c_cou2cr ccs7 13 15 c_cou2cr *k3 l3r l2r k=cou_l_2 lshunt1 5 15 ls1 lshunt2 6 10 ls2 kshunt lshunt1 lshunt2 k=cou_l_2 .param + rsq = .107 + lsq = 1.428e-10 + csq = 6.206e-14 + ccsq = 4.47e-14 + r2 = 6.2e-2 + l2 = 1.443e-10 + c2 = 4.668e-14 + c_cou2cr = 1e-17 .ends .subckt corn6_corn3 41 1 6 70 15 10 r1l 1 2 rsq l1l 2 3 lsq c1 3 0 csq r1r 3 4 rsq l1r 4 5 lsq rg1 3 0 10mega 262 ccs1 1 3 ccsq ccs2 3 5 ccsq ccouple2 3 8 c_cou k1 l1l l2l k=cou_l k2 l1r l2r k=cou_l r2l 6 7 r2 l2l 7 8 l2 c2 8 0 c2 rg2 8 0 10mega l2r 8 9 l2 r2r 9 10 r2 ccs3 6 8 c_cou2cr ccs4 8 10 c_cou2cr r3l 5 12 r2 l3l 12 13 l2 c3 13 0 c2 rg3 13 0 10mega l3r 13 14 l2 r3r 14 15 r2 ccs6 5 13 c_cou2cr ccs7 13 15 c_cou2cr *k3 l3r l2r k=cou_l_2 lshunt1 5 15 ls1 lshunt2 6 10 ls2 kshunt lshunt1 lshunt2 k=cou_l_2 r4l 41 42 rsq l4l 42 43 lsq c4 43 0 csq r4r 43 44 rsq l4r 44 45 lsq rg4 43 0 10mega ccs41 41 43 ccsq ccs42 43 45 ccsq ccouple4_1 43 3 c_cou_line r5l 45 52 rsq l5l 52 53 lsq c5 53 0 csq r5r 53 54 rsq l5r 54 55 lsq rg5 53 0 10mega ccs51 45 53 ccsq ccs52 53 55 ccsq r6l 55 67 r2 l6l 67 68 l2 c6 68 0 c2 rg6 68 0 10mega l6r 68 69 l2 r6r 69 70 r2 ccs61 55 68 c_cou2cr ccs62 68 70 c_cou2cr ccouple5_2 53 13 c_cou lshunt3 55 70 ls1 kshunt3 lshunt2 lshunt3 k=cou_l_2 k14a k14b k35a k35b l1l l1r l3l l3r l4l l4r l5l l5r k=cou_line k=cou_line k=cou_l k=cou_l .param + rsq = .107 + lsq = 1.428e-10 + csq = 6.206e-14 263 + + + + + + + ccsq = 4.47e-14 r2 = 6.2e-2 l2 = 1.443e-10 c2 = 4.668e-14 c_cou2cr = 1e-17 cou_line = 0.48 c_cou_line = 7.3e-16 .ends *pads .subckt mstl_pad 1 5 r1 1 2 rsq l1 2 3 lsq c1 3 0 csq r2 3 0 10g r1r 3 4 rsq l2r 4 5 lsq cc1 1 3 ccsq cc2 3 5 ccsq .param + rsq = 8.9e-3 + lsq = 2.9e-13 + csq = 1.1e-17 + ccsq = 1.0e-15 .ends .subckt mstl_corn_lcomp 1 2 8 7 x1 1 2 3 4 corn4_corn2 x4 8 7 3 4 corn4_corn2 .ends .subckt mstl_corn_6 1 2 3 9 10 11 x1 1 2 3 4 5 6 corn6_corn3 x4 9 10 11 4 5 6 corn6_corn3 .ends x1 a1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 mstl_c9_4 x2 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 mstl_c9_4 x4 19 20 21 22 23 24 25 26 27 37 38 39 40 41 42 43 44 45 mstl_c9_1 x5 2 3 mstl_corn_u x6 4 5 mstl_corn_u x7 6 7 mstl_corn_u x8 8 9 mstl_corn_u x9 37 38 mstl_corn_u x10 39 40 mstl_corn_u x11 41 42 mstl_corn_u x12 43 44 mstl_corn_u x13 45 46 mstl_sq_4 x14 b1 a1 mstl_sq_4 x15 46 47 mstl_pad x16 1 b1 mstl_pad ro 47 0 1g vpl 1 0 pulse(0 2.5 2n 0.1n 0.2n 10n 20n) ac 1 *.tran 1p 40n 1p .ac dec 60 1mega 20giga .net v(47) vpl rin=50 rout=50 .param cou_l = 1.942499581e-01 264 .param .param .param .param .param .param c_cou r2 c2 csq l2 c_cou2cr .measure .measure .measure .measure .measure .measure .measure .measure ac ac ac ac ac ac ac ac comp1 comp2 comp3 comp4 comp5 comp6 comp7 comp8 = = = = = = err1 err1 err1 err1 err1 err1 err1 err1 3.426036373e-16 1.000000000e-06 1.000000000e-17 1.589017165e-15 1.000000000e-13 1.739727694e-15 par(s11r) par(s11i) par(s12r) par(s12i) par(s21r) par(s21i) par(s22r) par(s22i) s11(r) s11(i) s12(r) s12(i) s21(r) s21(i) s22(r) s22(i) .ac data=measured .print par(s11r) s11(r) par(s11i) s11(i) .print par(s12r) s12(r) par(s12i) s12(i) .print par(s21r) s21(r) par(s21i) s21(i) .print par(s22r) s22(r) par(s22i) s22(i) *.param s11db='20*log(sqrt((par(s11r))^2+(par(s11i))^2))' *.param s21db='20*log(sqrt((par(s21r))^2+(par(s21i))^2))' .print s11(db) .print s12(db) .print s21(db) .print s22(db) .data measured mer file= 'ill.txt' freq=1 s11r=2 s11i=3 s21r=4 s21i=5 s12r=6 s12i=7 s22r=8 s22i=9 out = 'ill.out' .enddata .param freq=100m, s11r=0, s11i=0, s21r=0, s21i=0, s12r=0, s12i=0, s22r=0, s22i=0 .param sa11r=0, sa11i=0, sa21r=0, sa21i=0, sa12r=0, sa12i=0, sa22r=0, + sa22i=0 .end D.4.2. Measured S-Parameter Data Frequency S11(R) Measured S11(I) Measured S21(R) Measured S21(I) Measured 5.00E+08 5.98E+08 6.95E+08 7.93E+08 8.90E+08 9.88E+08 1.09E+09 1.18E+09 1.28E+09 1.38E+09 1.48E+09 1.44E-01 1.44E-01 1.44E-01 1.45E-01 1.45E-01 1.45E-01 1.47E-01 1.48E-01 1.49E-01 1.51E-01 1.50E-01 2.30E-02 2.68E-02 3.18E-02 3.59E-02 4.04E-02 4.47E-02 4.96E-02 5.34E-02 5.73E-02 6.28E-02 6.73E-02 8.56E-01 8.55E-01 8.55E-01 8.53E-01 8.51E-01 8.49E-01 8.49E-01 8.48E-01 8.47E-01 8.46E-01 8.43E-01 -4.32E-02 -5.20E-02 -6.09E-02 -6.81E-02 -7.73E-02 -8.39E-02 -9.20E-02 -1.01E-01 -1.08E-01 -1.16E-01 -1.24E-01 265 1.57E+09 1.67E+09 1.77E+09 1.87E+09 1.96E+09 2.06E+09 2.16E+09 2.26E+09 2.35E+09 2.45E+09 2.55E+09 2.65E+09 2.74E+09 2.84E+09 2.94E+09 3.04E+09 3.13E+09 3.23E+09 3.33E+09 3.43E+09 3.52E+09 3.62E+09 3.72E+09 3.82E+09 3.91E+09 4.01E+09 4.11E+09 4.21E+09 4.30E+09 4.40E+09 4.50E+09 4.60E+09 4.69E+09 4.79E+09 4.89E+09 4.99E+09 5.08E+09 5.18E+09 5.28E+09 5.38E+09 5.47E+09 5.57E+09 5.67E+09 5.77E+09 5.86E+09 5.96E+09 1.53E-01 1.56E-01 1.57E-01 1.59E-01 1.61E-01 1.62E-01 1.64E-01 1.67E-01 1.69E-01 1.69E-01 1.73E-01 1.73E-01 1.77E-01 1.81E-01 1.82E-01 1.83E-01 1.87E-01 1.89E-01 1.91E-01 1.93E-01 1.94E-01 1.97E-01 2.00E-01 2.04E-01 2.06E-01 2.09E-01 2.11E-01 2.17E-01 2.20E-01 2.22E-01 2.26E-01 2.30E-01 2.34E-01 2.35E-01 2.42E-01 2.46E-01 2.49E-01 2.52E-01 2.54E-01 2.59E-01 2.63E-01 2.66E-01 2.71E-01 2.74E-01 2.79E-01 2.81E-01 7.18E-02 7.66E-02 7.99E-02 8.35E-02 8.86E-02 9.26E-02 9.56E-02 9.85E-02 1.02E-01 1.05E-01 1.09E-01 1.14E-01 1.16E-01 1.20E-01 1.23E-01 1.27E-01 1.32E-01 1.32E-01 1.37E-01 1.41E-01 1.45E-01 1.49E-01 1.50E-01 1.55E-01 1.59E-01 1.63E-01 1.66E-01 1.68E-01 1.71E-01 1.75E-01 1.78E-01 1.80E-01 1.81E-01 1.84E-01 1.85E-01 1.89E-01 1.91E-01 1.92E-01 1.93E-01 1.96E-01 1.99E-01 1.98E-01 2.02E-01 2.03E-01 2.04E-01 2.06E-01 266 8.42E-01 8.41E-01 8.37E-01 8.35E-01 8.34E-01 8.31E-01 8.28E-01 8.26E-01 8.24E-01 8.21E-01 8.17E-01 8.15E-01 8.12E-01 8.07E-01 8.05E-01 8.03E-01 7.98E-01 7.95E-01 7.90E-01 7.87E-01 7.81E-01 7.79E-01 7.73E-01 7.72E-01 7.66E-01 7.63E-01 7.60E-01 7.54E-01 7.52E-01 7.48E-01 7.46E-01 7.42E-01 7.35E-01 7.34E-01 7.30E-01 7.26E-01 7.22E-01 7.17E-01 7.11E-01 7.08E-01 7.02E-01 6.97E-01 6.93E-01 6.89E-01 6.85E-01 6.79E-01 -1.33E-01 -1.39E-01 -1.48E-01 -1.55E-01 -1.64E-01 -1.71E-01 -1.77E-01 -1.85E-01 -1.93E-01 -2.00E-01 -2.12E-01 -2.14E-01 -2.20E-01 -2.31E-01 -2.37E-01 -2.44E-01 -2.50E-01 -2.59E-01 -2.63E-01 -2.70E-01 -2.77E-01 -2.84E-01 -2.90E-01 -2.96E-01 -3.01E-01 -3.08E-01 -3.13E-01 -3.17E-01 -3.24E-01 -3.31E-01 -3.36E-01 -3.40E-01 -3.48E-01 -3.54E-01 -3.59E-01 -3.65E-01 -3.70E-01 -3.75E-01 -3.81E-01 -3.86E-01 -3.91E-01 -3.96E-01 -4.02E-01 -4.06E-01 -4.11E-01 -4.18E-01 6.06E+09 6.16E+09 6.25E+09 6.35E+09 6.45E+09 6.55E+09 6.64E+09 6.74E+09 6.84E+09 6.94E+09 7.03E+09 7.13E+09 7.23E+09 7.33E+09 7.42E+09 7.52E+09 7.62E+09 7.72E+09 7.81E+09 7.91E+09 8.01E+09 8.11E+09 8.20E+09 8.30E+09 8.40E+09 8.50E+09 8.59E+09 8.69E+09 8.79E+09 8.89E+09 8.98E+09 9.08E+09 9.18E+09 9.28E+09 9.37E+09 9.47E+09 9.57E+09 9.67E+09 9.76E+09 9.86E+09 9.96E+09 1.01E+10 1.02E+10 1.03E+10 1.03E+10 1.04E+10 2.86E-01 2.88E-01 2.91E-01 2.96E-01 3.00E-01 3.04E-01 3.08E-01 3.12E-01 3.15E-01 3.19E-01 3.22E-01 3.26E-01 3.29E-01 3.35E-01 3.34E-01 3.40E-01 3.46E-01 3.50E-01 3.53E-01 3.54E-01 3.56E-01 3.64E-01 3.64E-01 3.68E-01 3.74E-01 3.77E-01 3.81E-01 3.82E-01 3.87E-01 3.90E-01 3.94E-01 3.97E-01 3.99E-01 4.05E-01 4.07E-01 4.13E-01 4.17E-01 4.19E-01 4.23E-01 4.28E-01 4.32E-01 4.34E-01 4.39E-01 4.41E-01 4.44E-01 4.46E-01 2.08E-01 2.10E-01 2.09E-01 2.11E-01 2.13E-01 2.13E-01 2.15E-01 2.15E-01 2.16E-01 2.17E-01 2.15E-01 2.16E-01 2.17E-01 2.17E-01 2.18E-01 2.19E-01 2.19E-01 2.19E-01 2.22E-01 2.19E-01 2.21E-01 2.23E-01 2.22E-01 2.24E-01 2.24E-01 2.20E-01 2.23E-01 2.23E-01 2.23E-01 2.21E-01 2.22E-01 2.22E-01 2.21E-01 2.20E-01 2.21E-01 2.20E-01 2.19E-01 2.20E-01 2.19E-01 2.16E-01 2.15E-01 2.14E-01 2.15E-01 2.15E-01 2.13E-01 2.14E-01 267 6.75E-01 6.69E-01 6.65E-01 6.63E-01 6.58E-01 6.51E-01 6.47E-01 6.41E-01 6.38E-01 6.33E-01 6.27E-01 6.22E-01 6.16E-01 6.13E-01 6.06E-01 6.01E-01 5.97E-01 5.91E-01 5.88E-01 5.82E-01 5.77E-01 5.71E-01 5.69E-01 5.64E-01 5.58E-01 5.52E-01 5.49E-01 5.44E-01 5.37E-01 5.34E-01 5.29E-01 5.25E-01 5.19E-01 5.13E-01 5.08E-01 5.03E-01 5.00E-01 4.95E-01 4.87E-01 4.84E-01 4.78E-01 4.75E-01 4.70E-01 4.66E-01 4.59E-01 4.55E-01 -4.22E-01 -4.25E-01 -4.30E-01 -4.35E-01 -4.39E-01 -4.44E-01 -4.46E-01 -4.53E-01 -4.56E-01 -4.60E-01 -4.67E-01 -4.68E-01 -4.72E-01 -4.78E-01 -4.82E-01 -4.82E-01 -4.86E-01 -4.89E-01 -4.94E-01 -4.98E-01 -5.00E-01 -5.04E-01 -5.08E-01 -5.12E-01 -5.15E-01 -5.19E-01 -5.18E-01 -5.23E-01 -5.25E-01 -5.28E-01 -5.34E-01 -5.35E-01 -5.40E-01 -5.42E-01 -5.44E-01 -5.44E-01 -5.49E-01 -5.50E-01 -5.54E-01 -5.53E-01 -5.56E-01 -5.59E-01 -5.62E-01 -5.63E-01 -5.66E-01 -5.65E-01 1.05E+10 1.06E+10 1.07E+10 1.08E+10 1.09E+10 1.10E+10 1.11E+10 1.12E+10 1.13E+10 1.14E+10 1.15E+10 1.16E+10 1.17E+10 1.18E+10 1.19E+10 1.20E+10 1.21E+10 1.22E+10 1.23E+10 1.24E+10 1.25E+10 1.26E+10 1.27E+10 1.28E+10 1.29E+10 1.30E+10 1.31E+10 1.32E+10 1.33E+10 1.34E+10 1.35E+10 1.36E+10 1.37E+10 1.38E+10 1.39E+10 1.40E+10 1.41E+10 1.42E+10 1.42E+10 1.43E+10 1.44E+10 1.45E+10 1.46E+10 1.47E+10 1.48E+10 1.49E+10 4.48E-01 4.53E-01 4.58E-01 4.57E-01 4.62E-01 4.65E-01 4.69E-01 4.73E-01 4.75E-01 4.76E-01 4.82E-01 4.85E-01 4.88E-01 4.92E-01 4.93E-01 4.96E-01 4.99E-01 5.02E-01 5.07E-01 5.07E-01 5.11E-01 5.17E-01 5.16E-01 5.20E-01 5.26E-01 5.25E-01 5.27E-01 5.29E-01 5.33E-01 5.37E-01 5.39E-01 5.36E-01 5.41E-01 5.43E-01 5.43E-01 5.47E-01 5.52E-01 5.54E-01 5.52E-01 5.58E-01 5.59E-01 5.64E-01 5.65E-01 5.70E-01 5.74E-01 5.74E-01 2.12E-01 2.11E-01 2.09E-01 2.07E-01 2.05E-01 2.06E-01 2.05E-01 2.00E-01 2.00E-01 1.98E-01 1.98E-01 1.93E-01 1.95E-01 1.93E-01 1.90E-01 1.89E-01 1.87E-01 1.85E-01 1.84E-01 1.82E-01 1.82E-01 1.81E-01 1.78E-01 1.78E-01 1.79E-01 1.74E-01 1.72E-01 1.69E-01 1.66E-01 1.63E-01 1.61E-01 1.60E-01 1.57E-01 1.53E-01 1.54E-01 1.53E-01 1.51E-01 1.48E-01 1.45E-01 1.41E-01 1.40E-01 1.33E-01 1.35E-01 1.33E-01 1.30E-01 1.27E-01 268 4.52E-01 4.45E-01 4.40E-01 4.36E-01 4.33E-01 4.29E-01 4.20E-01 4.16E-01 4.14E-01 4.09E-01 4.03E-01 3.99E-01 3.94E-01 3.90E-01 3.87E-01 3.80E-01 3.76E-01 3.73E-01 3.68E-01 3.63E-01 3.55E-01 3.53E-01 3.48E-01 3.43E-01 3.37E-01 3.34E-01 3.28E-01 3.24E-01 3.18E-01 3.13E-01 3.08E-01 3.04E-01 2.98E-01 2.95E-01 2.93E-01 2.87E-01 2.80E-01 2.79E-01 2.75E-01 2.71E-01 2.65E-01 2.63E-01 2.59E-01 2.52E-01 2.49E-01 2.46E-01 -5.69E-01 -5.72E-01 -5.73E-01 -5.77E-01 -5.78E-01 -5.81E-01 -5.83E-01 -5.83E-01 -5.89E-01 -5.88E-01 -5.93E-01 -5.93E-01 -5.96E-01 -5.98E-01 -5.98E-01 -6.00E-01 -5.99E-01 -5.98E-01 -6.03E-01 -6.01E-01 -6.04E-01 -6.07E-01 -6.08E-01 -6.08E-01 -6.10E-01 -6.14E-01 -6.15E-01 -6.17E-01 -6.18E-01 -6.19E-01 -6.21E-01 -6.23E-01 -6.19E-01 -6.25E-01 -6.24E-01 -6.28E-01 -6.26E-01 -6.31E-01 -6.28E-01 -6.31E-01 -6.28E-01 -6.33E-01 -6.31E-01 -6.30E-01 -6.30E-01 -6.33E-01 1.50E+10 1.51E+10 1.52E+10 1.53E+10 1.54E+10 1.55E+10 1.56E+10 1.57E+10 1.58E+10 1.59E+10 1.60E+10 1.61E+10 1.62E+10 1.63E+10 1.64E+10 1.65E+10 1.66E+10 1.67E+10 1.68E+10 1.69E+10 1.70E+10 1.71E+10 1.72E+10 1.73E+10 1.74E+10 1.75E+10 1.76E+10 1.77E+10 1.78E+10 1.79E+10 1.80E+10 1.81E+10 1.81E+10 1.82E+10 1.83E+10 1.84E+10 1.85E+10 1.86E+10 1.87E+10 1.88E+10 1.89E+10 1.90E+10 1.91E+10 1.92E+10 1.93E+10 1.94E+10 5.81E-01 5.82E-01 5.83E-01 5.85E-01 5.87E-01 5.90E-01 5.87E-01 5.90E-01 5.91E-01 5.89E-01 5.95E-01 5.93E-01 5.95E-01 5.97E-01 5.99E-01 6.00E-01 6.02E-01 6.06E-01 6.08E-01 6.08E-01 6.12E-01 6.13E-01 6.17E-01 6.16E-01 6.19E-01 6.20E-01 6.26E-01 6.24E-01 6.29E-01 6.31E-01 6.25E-01 6.35E-01 6.30E-01 6.30E-01 6.29E-01 6.30E-01 6.30E-01 6.29E-01 6.30E-01 6.34E-01 6.33E-01 6.32E-01 6.36E-01 6.38E-01 6.40E-01 6.37E-01 1.23E-01 1.19E-01 1.14E-01 1.15E-01 1.14E-01 1.10E-01 1.07E-01 1.05E-01 1.04E-01 1.01E-01 9.72E-02 9.83E-02 9.32E-02 9.13E-02 8.75E-02 8.34E-02 8.37E-02 7.65E-02 7.42E-02 7.07E-02 6.80E-02 6.29E-02 6.31E-02 5.85E-02 5.17E-02 4.93E-02 4.90E-02 4.69E-02 4.59E-02 4.54E-02 4.01E-02 3.66E-02 3.28E-02 3.39E-02 2.42E-02 2.68E-02 2.71E-02 1.61E-02 1.58E-02 1.14E-02 9.09E-03 1.62E-03 -2.75E-03 -8.97E-03 -1.05E-02 -1.97E-02 269 2.40E-01 2.33E-01 2.30E-01 2.25E-01 2.18E-01 2.13E-01 2.11E-01 2.00E-01 1.96E-01 1.91E-01 1.89E-01 1.79E-01 1.78E-01 1.76E-01 1.71E-01 1.68E-01 1.64E-01 1.62E-01 1.57E-01 1.52E-01 1.50E-01 1.48E-01 1.41E-01 1.40E-01 1.35E-01 1.27E-01 1.25E-01 1.16E-01 1.11E-01 1.07E-01 9.93E-02 9.27E-02 9.13E-02 8.38E-02 8.28E-02 7.43E-02 6.80E-02 6.42E-02 6.51E-02 5.90E-02 5.72E-02 5.50E-02 4.94E-02 5.07E-02 3.79E-02 4.01E-02 -6.35E-01 -6.33E-01 -6.36E-01 -6.36E-01 -6.32E-01 -6.33E-01 -6.37E-01 -6.35E-01 -6.37E-01 -6.37E-01 -6.36E-01 -6.38E-01 -6.40E-01 -6.43E-01 -6.45E-01 -6.48E-01 -6.40E-01 -6.44E-01 -6.43E-01 -6.43E-01 -6.39E-01 -6.42E-01 -6.37E-01 -6.38E-01 -6.35E-01 -6.37E-01 -6.34E-01 -6.34E-01 -6.32E-01 -6.32E-01 -6.29E-01 -6.33E-01 -6.31E-01 -6.33E-01 -6.43E-01 -6.37E-01 -6.38E-01 -6.42E-01 -6.47E-01 -6.44E-01 -6.38E-01 -6.46E-01 -6.41E-01 -6.38E-01 -6.36E-01 -6.32E-01 1.95E+10 1.96E+10 1.97E+10 1.98E+10 1.99E+10 2.00E+10 6.41E-01 6.42E-01 6.42E-01 6.46E-01 6.44E-01 6.41E-01 -2.21E-02 -2.69E-02 -3.41E-02 -4.03E-02 -4.52E-02 -4.58E-02 270 4.18E-02 3.73E-02 3.61E-02 3.24E-02 2.83E-02 2.51E-02 -6.34E-01 -6.35E-01 -6.32E-01 -6.33E-01 -6.37E-01 -6.37E-01 APPENDIX E CIRCUITS AND DATA FOR INTERDIGITAL CAPACITOR MODELING E.1. Introduction Input files and measured S-parameter data for test structure optimization for the interdigital capacitor modeling study described earlier in this thesis are presented in this appendix. In addition, the circuit file representing the complete model of the 9 segment resistor is also show, with associated measured S-parameters. All circuit files are written for the Star-Hspice circuit simulator. It should be noted that in some cases, certain subcircuit (.subckt) calls are defined but are never used in the actual optimization runs. E.2. Test Structure 1 E.2.1. Circuit Optimization Input File .option accurate dccap=1 node nopage ingold=2 post acct=2 .subckt mstl_pad 1 5 r1 1 2 r2 271 l1 2 3 l2 c1 3 0 c2 r2 3 0 10g r1r 3 4 r2 l2r 4 5 l2 .ends .subckt mstl_sq 1 5 r1 1 2 rsq l1 2 3 lsq c1 3 0 csq r1r 3 4 rsq l2r 4 5 lsq cc1 1 3 c_cou cc2 3 5 c_cou .ends .subckt mstl_sq5 1 6 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq x4 4 5 mstl_sq x5 5 6 mstl_sq .ends .subckt mstl_sq10 1 3 x1 1 2 mstl_sq5 x2 2 3 mstl_sq5 .ends .subckt line10 1 4 x1 1 2 mstl_pad x2 2 3 mstl_sq10 x3 3 4 mstl_pad ro 4 0 1g .ends *vpl 1 0 pulse(0 2.5 2n 0.1n 0.2n 10n 20n) ac 1 *.tran 1p 40n 1p *.ac dec 60 1mega 15giga *.net v(8) vpl rin=50 rout=50 .param + c_cou = opt1(6.4e-12,1f,1n) + rl = opt1(1e4,1,1e8) + r2 = opt1(4.7e-1,0.00001,10) + l2 = opt1(1.2e-11,.01p,1u) + c2 = opt1(9.2e-15,0.1f,1n) + rsq = opt1(0.30,0.01,10) + lsq = opt1(0.4e-11,1f,1u) + csq = opt1(2.1e-15,0.01f,1n) ****************************** * circuit for 1st subcircuit ****************************** v1i 1 0 dc 0 ac 1 r1i 1 2 50 x1 2 3 line10 r1o 3 4 50 v1o 4 0 dc 6 ac 0 e11 5 0 (2,0) 2 v11 5 11 ac 1 r11 11 0 1g e21 21 0 (3,0) 2 r21 21 0 1g .ac data=measured optimize=opt1 + results=comp1,comp2,comp5,comp6 272 + model=converge .model converge opt relin=1e-4 relout=1e-3 close=100 itropt=500 .measure .measure .measure .measure ac ac ac ac comp1 comp2 comp5 comp6 err1 err1 err1 err1 par(s11r) par(s11i) par(s21r) par(s21i) vr(11) vi(11) vr(21) vi(21) .print par(s11r) vr(11) par(s11i) vi(11) .print par(s21r) vr(21) par(s21i) vi(21) .data measured mer file= 'c2c14' freq=1 s11r=2 s11i=3 s21r=4 s21i=5 s12r=6 s12i=7 s22r=8 s22i=9 out = 'c2c14_data.txt' .enddata .param freq=500m, s11r=0, s11i=0, s21r=0, s21i=0, s12r=0, s12i=0, s22r=0, s22i=0 .param sa11r=0, sa11i=0, sa21r=0, sa21i=0, sa12r=0, sa12i=0, sa22r=0, + sa22i=0 .end E.2.2. Measured S-Parameter Data Freq 4.50E+07 1.45E+08 2.45E+08 3.44E+08 4.44E+08 5.44E+08 6.44E+08 7.43E+08 8.43E+08 9.43E+08 1.04E+09 1.14E+09 1.24E+09 1.34E+09 1.44E+09 1.54E+09 1.64E+09 1.74E+09 1.84E+09 1.94E+09 2.04E+09 2.14E+09 2.24E+09 2.34E+09 2.44E+09 2.54E+09 Meas. S11(R) 6.89E-02 6.90E-02 6.94E-02 6.96E-02 6.97E-02 7.04E-02 7.08E-02 7.14E-02 7.18E-02 7.24E-02 7.28E-02 7.36E-02 7.49E-02 7.56E-02 7.67E-02 7.80E-02 7.90E-02 8.02E-02 8.13E-02 8.27E-02 8.39E-02 8.50E-02 8.66E-02 8.76E-02 8.90E-02 9.07E-02 Meas. S11(I) 1.97E-03 5.08E-03 8.50E-03 1.17E-02 1.52E-02 1.86E-02 2.19E-02 2.51E-02 2.85E-02 3.16E-02 3.52E-02 3.88E-02 4.21E-02 4.54E-02 4.84E-02 5.16E-02 5.46E-02 5.75E-02 6.03E-02 6.33E-02 6.66E-02 6.91E-02 7.21E-02 7.52E-02 7.79E-02 8.05E-02 273 Meas. S21(R) 9.30E-01 9.30E-01 9.30E-01 9.30E-01 9.29E-01 9.29E-01 9.28E-01 9.28E-01 9.27E-01 9.26E-01 9.24E-01 9.23E-01 9.24E-01 9.22E-01 9.22E-01 9.18E-01 9.17E-01 9.15E-01 9.13E-01 9.11E-01 9.09E-01 9.08E-01 9.06E-01 9.04E-01 9.01E-01 8.99E-01 Meas. S21(I) -3.20E-03 -1.08E-02 -1.88E-02 -2.63E-02 -3.40E-02 -4.18E-02 -4.95E-02 -5.73E-02 -6.48E-02 -7.25E-02 -8.02E-02 -8.70E-02 -9.44E-02 -1.02E-01 -1.09E-01 -1.17E-01 -1.24E-01 -1.31E-01 -1.39E-01 -1.46E-01 -1.53E-01 -1.60E-01 -1.67E-01 -1.74E-01 -1.81E-01 -1.88E-01 2.64E+09 2.74E+09 2.84E+09 2.94E+09 3.04E+09 3.14E+09 3.24E+09 3.34E+09 3.44E+09 3.54E+09 3.64E+09 3.74E+09 3.84E+09 3.94E+09 4.04E+09 4.14E+09 4.24E+09 4.34E+09 4.44E+09 4.53E+09 4.63E+09 4.73E+09 4.83E+09 4.93E+09 5.03E+09 5.13E+09 5.23E+09 5.33E+09 5.43E+09 5.53E+09 5.63E+09 5.73E+09 5.83E+09 5.93E+09 6.03E+09 6.13E+09 6.23E+09 6.33E+09 6.43E+09 6.53E+09 6.63E+09 6.73E+09 6.83E+09 6.93E+09 7.03E+09 7.13E+09 7.23E+09 9.20E-02 9.33E-02 9.48E-02 9.65E-02 9.79E-02 9.94E-02 1.01E-01 1.03E-01 1.04E-01 1.07E-01 1.09E-01 1.10E-01 1.12E-01 1.15E-01 1.17E-01 1.19E-01 1.21E-01 1.23E-01 1.25E-01 1.28E-01 1.30E-01 1.32E-01 1.34E-01 1.37E-01 1.39E-01 1.42E-01 1.44E-01 1.47E-01 1.49E-01 1.51E-01 1.54E-01 1.56E-01 1.59E-01 1.61E-01 1.63E-01 1.66E-01 1.68E-01 1.71E-01 1.73E-01 1.75E-01 1.78E-01 1.80E-01 1.83E-01 1.86E-01 1.89E-01 1.91E-01 1.94E-01 8.34E-02 8.62E-02 8.89E-02 9.18E-02 9.47E-02 9.71E-02 9.95E-02 1.03E-01 1.05E-01 1.08E-01 1.11E-01 1.13E-01 1.15E-01 1.18E-01 1.20E-01 1.22E-01 1.24E-01 1.27E-01 1.29E-01 1.31E-01 1.33E-01 1.35E-01 1.37E-01 1.39E-01 1.42E-01 1.43E-01 1.44E-01 1.46E-01 1.48E-01 1.50E-01 1.51E-01 1.53E-01 1.54E-01 1.56E-01 1.57E-01 1.59E-01 1.60E-01 1.61E-01 1.63E-01 1.64E-01 1.65E-01 1.66E-01 1.68E-01 1.69E-01 1.70E-01 1.71E-01 1.72E-01 274 8.97E-01 8.95E-01 8.93E-01 8.90E-01 8.88E-01 8.86E-01 8.84E-01 8.81E-01 8.78E-01 8.76E-01 8.74E-01 8.71E-01 8.68E-01 8.66E-01 8.63E-01 8.60E-01 8.57E-01 8.54E-01 8.51E-01 8.48E-01 8.45E-01 8.42E-01 8.38E-01 8.35E-01 8.32E-01 8.29E-01 8.26E-01 8.22E-01 8.18E-01 8.15E-01 8.12E-01 8.08E-01 8.05E-01 8.01E-01 7.98E-01 7.94E-01 7.90E-01 7.87E-01 7.83E-01 7.79E-01 7.75E-01 7.71E-01 7.67E-01 7.64E-01 7.59E-01 7.54E-01 7.51E-01 -1.95E-01 -2.02E-01 -2.09E-01 -2.15E-01 -2.22E-01 -2.29E-01 -2.36E-01 -2.42E-01 -2.49E-01 -2.56E-01 -2.62E-01 -2.69E-01 -2.75E-01 -2.82E-01 -2.88E-01 -2.95E-01 -3.01E-01 -3.08E-01 -3.14E-01 -3.20E-01 -3.26E-01 -3.32E-01 -3.39E-01 -3.44E-01 -3.50E-01 -3.57E-01 -3.63E-01 -3.68E-01 -3.74E-01 -3.80E-01 -3.85E-01 -3.91E-01 -3.97E-01 -4.03E-01 -4.08E-01 -4.14E-01 -4.20E-01 -4.25E-01 -4.31E-01 -4.36E-01 -4.41E-01 -4.46E-01 -4.51E-01 -4.57E-01 -4.62E-01 -4.67E-01 -4.72E-01 7.33E+09 7.43E+09 7.53E+09 7.63E+09 7.73E+09 7.83E+09 7.93E+09 8.03E+09 8.13E+09 8.23E+09 8.33E+09 8.43E+09 8.53E+09 8.63E+09 8.73E+09 8.83E+09 8.92E+09 9.02E+09 9.12E+09 9.22E+09 9.32E+09 9.42E+09 9.52E+09 9.62E+09 9.72E+09 9.82E+09 9.92E+09 1.00E+10 1.01E+10 1.02E+10 1.03E+10 1.04E+10 1.05E+10 1.06E+10 1.07E+10 1.08E+10 1.09E+10 1.10E+10 1.11E+10 1.12E+10 1.13E+10 1.14E+10 1.15E+10 1.16E+10 1.17E+10 1.18E+10 1.19E+10 1.97E-01 1.99E-01 2.02E-01 2.05E-01 2.08E-01 2.10E-01 2.13E-01 2.16E-01 2.18E-01 2.21E-01 2.25E-01 2.28E-01 2.31E-01 2.35E-01 2.39E-01 2.42E-01 2.44E-01 2.48E-01 2.50E-01 2.53E-01 2.56E-01 2.59E-01 2.61E-01 2.64E-01 2.66E-01 2.69E-01 2.71E-01 2.74E-01 2.77E-01 2.80E-01 2.82E-01 2.85E-01 2.87E-01 2.90E-01 2.92E-01 2.94E-01 2.97E-01 2.99E-01 3.02E-01 3.04E-01 3.07E-01 3.09E-01 3.11E-01 3.13E-01 3.14E-01 3.16E-01 3.18E-01 1.74E-01 1.75E-01 1.76E-01 1.77E-01 1.78E-01 1.79E-01 1.80E-01 1.81E-01 1.82E-01 1.83E-01 1.84E-01 1.85E-01 1.85E-01 1.86E-01 1.86E-01 1.84E-01 1.84E-01 1.84E-01 1.84E-01 1.83E-01 1.83E-01 1.82E-01 1.82E-01 1.81E-01 1.81E-01 1.81E-01 1.80E-01 1.80E-01 1.80E-01 1.79E-01 1.79E-01 1.78E-01 1.78E-01 1.77E-01 1.77E-01 1.77E-01 1.76E-01 1.75E-01 1.75E-01 1.74E-01 1.72E-01 1.72E-01 1.71E-01 1.69E-01 1.69E-01 1.68E-01 1.67E-01 275 7.47E-01 7.43E-01 7.39E-01 7.35E-01 7.30E-01 7.27E-01 7.23E-01 7.19E-01 7.14E-01 7.10E-01 7.06E-01 7.01E-01 6.95E-01 6.90E-01 6.85E-01 6.81E-01 6.77E-01 6.73E-01 6.67E-01 6.63E-01 6.59E-01 6.54E-01 6.49E-01 6.45E-01 6.41E-01 6.36E-01 6.32E-01 6.28E-01 6.23E-01 6.18E-01 6.14E-01 6.09E-01 6.04E-01 6.00E-01 5.95E-01 5.91E-01 5.86E-01 5.82E-01 5.77E-01 5.72E-01 5.67E-01 5.63E-01 5.58E-01 5.54E-01 5.49E-01 5.45E-01 5.41E-01 -4.77E-01 -4.83E-01 -4.88E-01 -4.92E-01 -4.97E-01 -5.02E-01 -5.08E-01 -5.12E-01 -5.17E-01 -5.23E-01 -5.27E-01 -5.32E-01 -5.37E-01 -5.41E-01 -5.45E-01 -5.48E-01 -5.53E-01 -5.57E-01 -5.61E-01 -5.66E-01 -5.69E-01 -5.73E-01 -5.77E-01 -5.81E-01 -5.84E-01 -5.88E-01 -5.92E-01 -5.96E-01 -6.00E-01 -6.03E-01 -6.07E-01 -6.10E-01 -6.14E-01 -6.18E-01 -6.21E-01 -6.25E-01 -6.28E-01 -6.32E-01 -6.35E-01 -6.38E-01 -6.41E-01 -6.45E-01 -6.48E-01 -6.51E-01 -6.54E-01 -6.57E-01 -6.61E-01 1.20E+10 1.21E+10 1.22E+10 1.23E+10 1.24E+10 1.25E+10 1.26E+10 1.27E+10 1.28E+10 1.29E+10 1.30E+10 1.31E+10 1.32E+10 1.33E+10 1.34E+10 1.35E+10 1.36E+10 1.37E+10 1.38E+10 1.39E+10 1.40E+10 1.41E+10 1.42E+10 1.43E+10 1.44E+10 1.45E+10 1.46E+10 1.47E+10 1.48E+10 1.49E+10 1.50E+10 1.51E+10 1.52E+10 1.53E+10 1.54E+10 1.55E+10 1.56E+10 1.57E+10 1.58E+10 1.59E+10 1.60E+10 1.61E+10 1.62E+10 1.63E+10 1.64E+10 1.65E+10 1.66E+10 3.20E-01 3.22E-01 3.24E-01 3.27E-01 3.29E-01 3.31E-01 3.33E-01 3.35E-01 3.37E-01 3.40E-01 3.42E-01 3.45E-01 3.47E-01 3.49E-01 3.51E-01 3.54E-01 3.55E-01 3.58E-01 3.60E-01 3.61E-01 3.63E-01 3.65E-01 3.66E-01 3.68E-01 3.70E-01 3.72E-01 3.74E-01 3.76E-01 3.78E-01 3.80E-01 3.82E-01 3.84E-01 3.86E-01 3.89E-01 3.91E-01 3.93E-01 3.96E-01 3.98E-01 4.00E-01 4.02E-01 4.04E-01 4.06E-01 4.08E-01 4.09E-01 4.11E-01 4.12E-01 4.14E-01 1.66E-01 1.65E-01 1.64E-01 1.63E-01 1.63E-01 1.62E-01 1.61E-01 1.61E-01 1.60E-01 1.59E-01 1.59E-01 1.58E-01 1.56E-01 1.55E-01 1.54E-01 1.53E-01 1.52E-01 1.51E-01 1.50E-01 1.48E-01 1.47E-01 1.46E-01 1.45E-01 1.43E-01 1.42E-01 1.41E-01 1.39E-01 1.38E-01 1.36E-01 1.35E-01 1.34E-01 1.32E-01 1.31E-01 1.29E-01 1.28E-01 1.27E-01 1.25E-01 1.23E-01 1.21E-01 1.19E-01 1.17E-01 1.16E-01 1.14E-01 1.11E-01 1.09E-01 1.08E-01 1.06E-01 276 5.36E-01 5.32E-01 5.27E-01 5.23E-01 5.18E-01 5.14E-01 5.09E-01 5.04E-01 5.00E-01 4.94E-01 4.89E-01 4.84E-01 4.79E-01 4.74E-01 4.69E-01 4.64E-01 4.60E-01 4.55E-01 4.50E-01 4.44E-01 4.40E-01 4.35E-01 4.30E-01 4.26E-01 4.20E-01 4.15E-01 4.11E-01 4.05E-01 4.01E-01 3.95E-01 3.90E-01 3.85E-01 3.80E-01 3.75E-01 3.69E-01 3.64E-01 3.58E-01 3.53E-01 3.47E-01 3.42E-01 3.37E-01 3.31E-01 3.26E-01 3.21E-01 3.16E-01 3.10E-01 3.05E-01 -6.64E-01 -6.67E-01 -6.70E-01 -6.73E-01 -6.77E-01 -6.80E-01 -6.83E-01 -6.87E-01 -6.90E-01 -6.93E-01 -6.96E-01 -6.99E-01 -7.02E-01 -7.05E-01 -7.07E-01 -7.10E-01 -7.13E-01 -7.15E-01 -7.18E-01 -7.21E-01 -7.23E-01 -7.26E-01 -7.29E-01 -7.31E-01 -7.34E-01 -7.37E-01 -7.39E-01 -7.42E-01 -7.44E-01 -7.47E-01 -7.49E-01 -7.52E-01 -7.54E-01 -7.57E-01 -7.59E-01 -7.61E-01 -7.63E-01 -7.65E-01 -7.68E-01 -7.70E-01 -7.71E-01 -7.73E-01 -7.75E-01 -7.77E-01 -7.79E-01 -7.81E-01 -7.82E-01 1.67E+10 1.68E+10 1.69E+10 1.70E+10 1.71E+10 1.72E+10 1.73E+10 1.74E+10 1.75E+10 1.76E+10 1.77E+10 1.78E+10 1.79E+10 1.80E+10 1.81E+10 1.82E+10 1.83E+10 1.84E+10 1.85E+10 1.86E+10 1.87E+10 1.88E+10 1.89E+10 1.90E+10 1.91E+10 1.92E+10 1.93E+10 1.94E+10 1.95E+10 1.96E+10 1.97E+10 1.98E+10 1.99E+10 2.00E+10 4.16E-01 4.18E-01 4.19E-01 4.21E-01 4.22E-01 4.24E-01 4.25E-01 4.27E-01 4.28E-01 4.29E-01 4.30E-01 4.32E-01 4.33E-01 4.35E-01 4.35E-01 4.37E-01 4.38E-01 4.41E-01 4.43E-01 4.44E-01 4.51E-01 4.59E-01 4.48E-01 4.50E-01 4.50E-01 4.50E-01 4.49E-01 4.48E-01 4.46E-01 4.44E-01 4.43E-01 4.41E-01 4.40E-01 4.40E-01 1.04E-01 1.02E-01 9.91E-02 9.69E-02 9.42E-02 9.19E-02 8.85E-02 8.61E-02 8.41E-02 8.11E-02 7.89E-02 7.65E-02 7.40E-02 7.17E-02 6.91E-02 6.74E-02 6.50E-02 6.29E-02 5.97E-02 5.76E-02 5.55E-02 4.95E-02 4.74E-02 4.45E-02 4.03E-02 3.64E-02 3.23E-02 2.85E-02 2.49E-02 2.16E-02 1.98E-02 1.71E-02 1.57E-02 1.48E-02 2.99E-01 2.94E-01 2.89E-01 2.83E-01 2.77E-01 2.72E-01 2.67E-01 2.61E-01 2.56E-01 2.51E-01 2.45E-01 2.40E-01 2.35E-01 2.29E-01 2.24E-01 2.18E-01 2.13E-01 2.07E-01 2.01E-01 1.96E-01 1.90E-01 1.84E-01 1.79E-01 1.74E-01 1.68E-01 1.63E-01 1.59E-01 1.55E-01 1.51E-01 1.48E-01 1.44E-01 1.40E-01 1.38E-01 1.33E-01 E.3. Test Structure 2 E.3.1. Circuit Optimization Input File .option accurate dccap=1 node nopage ingold=2 post acct=2 probe * u circuit 10 mil in length .subckt mstlc1 1 6 11 5 10 15 277 -7.84E-01 -7.86E-01 -7.87E-01 -7.89E-01 -7.91E-01 -7.92E-01 -7.93E-01 -7.95E-01 -7.97E-01 -7.97E-01 -7.98E-01 -8.00E-01 -8.01E-01 -8.02E-01 -8.03E-01 -8.04E-01 -8.06E-01 -8.07E-01 -8.08E-01 -8.08E-01 -8.08E-01 -8.08E-01 -8.08E-01 -8.08E-01 -8.08E-01 -8.08E-01 -8.07E-01 -8.07E-01 -8.07E-01 -8.08E-01 -8.08E-01 -8.09E-01 -8.11E-01 -8.12E-01 r1l 1 2 r l1l 2 3 l c1 3 0 c r1r 3 4 r l1r 4 5 l rg1 3 0 10mega ccouple 3 8 c_cou r2l 6 7 r l2l 7 8 l c2 8 0 c rg2 8 0 10mega l2r 8 9 l r2r 9 10 r k1 l1l l2l k=cou_l k2 l2r l1r k=cou_l r3l 11 12 r l3l 12 13 l c3 13 0 c rg3 13 0 10mega r3r 13 14 r l3r 14 15 l cc23 8 13 c_cou k23l l2l l3l k=cou_l k23r l2r l3r k=cou_l .ends .subckt mstl_pad 1 5 r1 1 2 r2 l1 2 3 l2 c1 3 0 c2 r2 3 0 10g r1r 3 4 r2 l2r 4 5 l2 .ends .subckt mstl_sq 1 5 r1 1 2 rsq l1 2 3 lsq c1 3 0 csq *r2 3 0 rl r1r 3 4 rsq l2r 4 5 lsq cc1 1 3 ccsq cc2 3 5 ccsq .ends .subckt mstl_corner 1 5 r1 1 2 rc l1 2 3 lc c1 3 0 cc r1r 3 4 rc l2r 4 5 lc cc1 1 3 ccsq cc2 3 5 ccsq .ends .subckt mstlc5 x1 1 2 3 4 5 6 x2 4 5 6 7 8 9 x3 7 8 9 10 11 x4 10 11 12 13 x5 13 14 15 16 1 2 3 16 17 18 mstlc1 mstlc1 12 mstlc1 14 15 mstlc1 17 18 mstlc1 278 .ends .subckt mstlc4 x1 1 2 3 4 5 6 x2 4 5 6 7 8 9 x3 7 8 9 10 11 x4 10 11 12 13 .ends 1 2 3 13 14 15 mstlc1 mstlc1 12 mstlc1 14 15 mstlc1 .subckt mstl_sq5 1 6 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq x4 4 5 mstl_sq x5 5 6 mstl_sq .ends .subckt mstl_sq4 1 5 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq x4 4 5 mstl_sq .ends x1 x2 x3 x4 x5 x6 x7 x8 x9 r0 1 2 mstl_pad 2 3 mstl_sq4 3 4 5 6 7 8 mstlc5 6 7 8 9 10 11 mstlc5 9 10 11 12 13 14 mstlc5 4 5 mstl_corner 12 13 mstl_corner 14 15 mstl_sq4 15 16 mstl_pad 16 0 1g vpl 1 0 pulse(0 2.5 2n 0.1n 0.2n 10n 20n) ac 1 *.tran 1p 40n 1p *.ac dec 60 1mega 15giga .net v(16) vpl rin=50 rout=50 .param + c_cou = opt1(2.9e-15,0.1f,1n) + cou_l = opt1(0.2,0,1) + + + + csq = opt1(2.9e-15,0.01f,1n) rc = opt1(0.30,0.01,10) lc = opt1(2.4e-11,1f,1u) cc = opt1(2.1e-15,0.01f,1n) .param .param .param .param .param .param ccsq r2 l2 c2 rsq lsq = 1.000e-15 = 0.081e+00 = 1.230e-11 = 3.603e-15 = 6.684e-02 = 1.009e-11 $ $ $ $ $ $ 4.446e-03 5.731e+00 2.624e-03 1.027e+00 4.455e+00 5.633e+01 -2.409e+02 1.154e-03 6.587e+01 -3.999e-02 -4.904e-02 -1.497e-02 .param r=rsq l=lsq c=csq .ac data=measured optimize=opt1 + results=comp1,comp2,comp5,comp6 + model=converge .model converge opt relin=1e-3 relout=1e-4 close=200 itropt=300 .measure ac comp1 err1 par(s11r) s11(r) .measure ac comp2 err1 par(s11i) s11(i) .measure ac comp3 err1 par(s12r) s12(r) 279 .measure .measure .measure .measure .measure ac ac ac ac ac comp4 comp5 comp6 comp7 comp8 err1 err1 err1 err1 err1 par(s12i) par(s21r) par(s21i) par(s22r) par(s22i) s12(i) s21(r) s21(i) s22(r) s22(i) .ac data=measured .print par(s11r) s11(r) par(s11i) s11(i) .print par(s12r) s12(r) par(s12i) s12(i) .print par(s21r) s21(r) par(s21i) s21(i) .print par(s22r) s22(r) par(s22i) s22(i) *.param s11db='20*log(sqrt((par(s11r))^2+(par(s11i))^2))' *.param s21db='20*log(sqrt((par(s21r))^2+(par(s21i))^2))' .print s11(db) .print s12(db) .print s21(db) .print s22(db) .data measured freq s11r s11i s21r s21i s12r s12i s22r s22I E.3.2. Measured S-Parameter Data Freq Meas. S11(R) Meas. S11(I) Meas. S21(R) Meas. S21(I) 4.50E+07 1.45E+08 2.45E+08 3.44E+08 4.44E+08 5.44E+08 6.44E+08 7.43E+08 8.43E+08 9.43E+08 1.04E+09 1.14E+09 1.24E+09 1.34E+09 1.44E+09 1.54E+09 1.64E+09 1.74E+09 1.84E+09 1.94E+09 2.04E+09 2.14E+09 2.24E+09 2.34E+09 2.44E+09 8.37E-02 8.44E-02 8.49E-02 8.53E-02 8.56E-02 8.64E-02 8.69E-02 8.75E-02 8.81E-02 8.88E-02 8.97E-02 9.07E-02 9.19E-02 9.28E-02 9.40E-02 9.55E-02 9.66E-02 9.80E-02 9.93E-02 1.01E-01 1.02E-01 1.04E-01 1.06E-01 1.07E-01 1.09E-01 2.48E-03 6.60E-03 1.11E-02 1.53E-02 1.96E-02 2.40E-02 2.82E-02 3.24E-02 3.67E-02 4.08E-02 4.54E-02 4.95E-02 5.36E-02 5.78E-02 6.18E-02 6.58E-02 6.97E-02 7.37E-02 7.75E-02 8.14E-02 8.57E-02 8.92E-02 9.30E-02 9.72E-02 1.01E-01 9.15E-01 9.15E-01 9.15E-01 9.14E-01 9.14E-01 9.13E-01 9.12E-01 9.12E-01 9.11E-01 9.10E-01 9.09E-01 9.08E-01 9.09E-01 9.07E-01 9.06E-01 9.03E-01 9.01E-01 8.99E-01 8.97E-01 8.95E-01 8.93E-01 8.92E-01 8.90E-01 8.88E-01 8.85E-01 -3.30E-03 -1.08E-02 -1.83E-02 -2.57E-02 -3.30E-02 -4.03E-02 -4.77E-02 -5.53E-02 -6.23E-02 -6.96E-02 -7.68E-02 -8.37E-02 -9.10E-02 -9.76E-02 -1.05E-01 -1.12E-01 -1.19E-01 -1.26E-01 -1.33E-01 -1.40E-01 -1.47E-01 -1.54E-01 -1.61E-01 -1.68E-01 -1.74E-01 280 2.54E+09 2.64E+09 2.74E+09 2.84E+09 2.94E+09 3.04E+09 3.14E+09 3.24E+09 3.34E+09 3.44E+09 3.54E+09 3.64E+09 3.74E+09 3.84E+09 3.94E+09 4.04E+09 4.14E+09 4.24E+09 4.34E+09 4.44E+09 4.53E+09 4.63E+09 4.73E+09 4.83E+09 4.93E+09 5.03E+09 5.13E+09 5.23E+09 5.33E+09 5.43E+09 5.53E+09 5.63E+09 5.73E+09 5.83E+09 5.93E+09 6.03E+09 6.13E+09 6.23E+09 6.33E+09 6.43E+09 6.53E+09 6.63E+09 6.73E+09 6.83E+09 6.93E+09 7.03E+09 7.13E+09 1.11E-01 1.13E-01 1.15E-01 1.17E-01 1.19E-01 1.21E-01 1.23E-01 1.25E-01 1.27E-01 1.29E-01 1.32E-01 1.35E-01 1.37E-01 1.39E-01 1.42E-01 1.45E-01 1.47E-01 1.50E-01 1.53E-01 1.55E-01 1.58E-01 1.61E-01 1.64E-01 1.67E-01 1.70E-01 1.73E-01 1.76E-01 1.79E-01 1.82E-01 1.85E-01 1.88E-01 1.92E-01 1.95E-01 1.98E-01 2.01E-01 2.04E-01 2.07E-01 2.11E-01 2.14E-01 2.17E-01 2.20E-01 2.24E-01 2.27E-01 2.30E-01 2.34E-01 2.38E-01 2.41E-01 1.04E-01 1.08E-01 1.12E-01 1.16E-01 1.19E-01 1.23E-01 1.26E-01 1.29E-01 1.33E-01 1.37E-01 1.39E-01 1.43E-01 1.46E-01 1.49E-01 1.52E-01 1.55E-01 1.58E-01 1.61E-01 1.64E-01 1.67E-01 1.69E-01 1.72E-01 1.75E-01 1.78E-01 1.80E-01 1.83E-01 1.85E-01 1.88E-01 1.90E-01 1.93E-01 1.95E-01 1.97E-01 1.99E-01 2.01E-01 2.03E-01 2.05E-01 2.07E-01 2.09E-01 2.11E-01 2.12E-01 2.14E-01 2.16E-01 2.18E-01 2.19E-01 2.20E-01 2.22E-01 2.23E-01 281 8.83E-01 8.81E-01 8.78E-01 8.76E-01 8.74E-01 8.72E-01 8.69E-01 8.67E-01 8.64E-01 8.62E-01 8.59E-01 8.57E-01 8.54E-01 8.51E-01 8.48E-01 8.45E-01 8.42E-01 8.39E-01 8.36E-01 8.32E-01 8.29E-01 8.26E-01 8.23E-01 8.20E-01 8.16E-01 8.13E-01 8.10E-01 8.06E-01 8.02E-01 7.99E-01 7.95E-01 7.92E-01 7.88E-01 7.85E-01 7.81E-01 7.77E-01 7.74E-01 7.70E-01 7.66E-01 7.62E-01 7.58E-01 7.54E-01 7.50E-01 7.46E-01 7.42E-01 7.38E-01 7.32E-01 -1.81E-01 -1.87E-01 -1.94E-01 -2.00E-01 -2.07E-01 -2.13E-01 -2.20E-01 -2.26E-01 -2.33E-01 -2.39E-01 -2.45E-01 -2.51E-01 -2.58E-01 -2.64E-01 -2.70E-01 -2.76E-01 -2.83E-01 -2.89E-01 -2.95E-01 -3.00E-01 -3.06E-01 -3.12E-01 -3.18E-01 -3.24E-01 -3.29E-01 -3.34E-01 -3.40E-01 -3.46E-01 -3.51E-01 -3.56E-01 -3.62E-01 -3.67E-01 -3.72E-01 -3.78E-01 -3.83E-01 -3.88E-01 -3.93E-01 -3.98E-01 -4.03E-01 -4.08E-01 -4.13E-01 -4.17E-01 -4.22E-01 -4.27E-01 -4.32E-01 -4.36E-01 -4.41E-01 7.23E+09 7.33E+09 7.43E+09 7.53E+09 7.63E+09 7.73E+09 7.83E+09 7.93E+09 8.03E+09 8.13E+09 8.23E+09 8.33E+09 8.43E+09 8.53E+09 8.63E+09 8.73E+09 8.83E+09 8.92E+09 9.02E+09 9.12E+09 9.22E+09 9.32E+09 9.42E+09 9.52E+09 9.62E+09 9.72E+09 9.82E+09 9.92E+09 1.00E+10 2.44E-01 2.49E-01 2.52E-01 2.55E-01 2.59E-01 2.62E-01 2.66E-01 2.69E-01 2.72E-01 2.76E-01 2.79E-01 2.83E-01 2.86E-01 2.89E-01 2.92E-01 2.95E-01 2.99E-01 3.03E-01 3.07E-01 3.09E-01 3.13E-01 3.17E-01 3.20E-01 3.24E-01 3.27E-01 3.31E-01 3.34E-01 3.38E-01 3.41E-01 2.24E-01 2.26E-01 2.27E-01 2.28E-01 2.29E-01 2.30E-01 2.31E-01 2.32E-01 2.33E-01 2.34E-01 2.34E-01 2.35E-01 2.36E-01 2.36E-01 2.37E-01 2.38E-01 2.38E-01 2.38E-01 2.39E-01 2.40E-01 2.40E-01 2.40E-01 2.40E-01 2.40E-01 2.40E-01 2.40E-01 2.40E-01 2.39E-01 2.39E-01 7.29E-01 7.24E-01 7.20E-01 7.17E-01 7.13E-01 7.07E-01 7.04E-01 7.00E-01 6.96E-01 6.92E-01 6.87E-01 6.84E-01 6.79E-01 6.75E-01 6.71E-01 6.66E-01 6.62E-01 6.58E-01 6.53E-01 6.49E-01 6.44E-01 6.40E-01 6.36E-01 6.31E-01 6.27E-01 6.22E-01 6.18E-01 6.13E-01 6.09E-01 E.4. Test Structure 3 E.4.1. Circuit Optimization Input File .option accurate dccap=1 node nopage ingold=2 post acct=2 probe * u circuit 10 mil in length .subckt mstlc1 1 6 11 16 21 26 5 10 15 20 25 30 r1l 1 2 r l1l 2 3 l c1 3 0 c r1r 3 4 r l1r 4 5 l 282 -4.45E-01 -4.49E-01 -4.54E-01 -4.58E-01 -4.62E-01 -4.67E-01 -4.71E-01 -4.75E-01 -4.79E-01 -4.83E-01 -4.87E-01 -4.91E-01 -4.95E-01 -4.99E-01 -5.03E-01 -5.07E-01 -5.11E-01 -5.14E-01 -5.18E-01 -5.22E-01 -5.25E-01 -5.29E-01 -5.32E-01 -5.36E-01 -5.39E-01 -5.42E-01 -5.46E-01 -5.49E-01 -5.52E-01 rg1 3 0 10mega ccouple 3 8 c_cou r2l 6 7 r l2l 7 8 l c2 8 0 c rg2 8 0 10mega l2r 8 9 l r2r 9 10 r k1 l1l l2l k=cou_l k2 l2r l1r k=cou_l r3l 11 12 r l3l 12 13 l c3 13 0 c rg3 13 0 10mega r3r 13 14 r l3r 14 15 l cc23 8 13 c_cou k23l l2l l3l k=cou_l k23r l2r l3r k=cou_l r4l 16 17 r l4l 17 18 l c4 18 0 c rg4 18 0 10mega r4r 18 19 r l4r 19 20 l cc34 13 18 c_cou k34l l3l l4l k=cou_l k34r l3r l4r k=cou_l r5l 21 22 r l5l 22 23 l c5 23 0 c rg5 23 0 10mega r5r 23 24 r l5r 24 25 l cc45 18 23 c_cou k45l l4l l5l k=cou_l k45r l4r l5r k=cou_l r6l 26 27 r l6l 27 28 l c6 28 0 c r6r 28 29 r l6r 29 30 l cc56 23 28 c_cou k56l l5l l6l k=cou_l k56r l5r l6r k=cou_l .ends .subckt mstl_pad 1 5 r1 1 2 r2 l1 2 3 l2 c1 3 0 c2 r2 3 0 10g r1r 3 4 r2 l2r 4 5 l2 .ends .subckt mstl_corner 1 6 5 r1 1 2 rc l1 2 3 lc c1 3 0 cc r2 3 0 10g 283 r1r 3 4 rc l2r 4 5 lc r6 6 0 1g c6 6 3 cc2 .ends .subckt mstl_sq 1 5 r1 1 2 rsq l1 2 3 lsq c1 3 0 csq r1r 3 4 rsq l2r 4 5 lsq cc1 1 3 ccsq cc2 3 5 ccsq .ends .subckt mstlc5 1 2 3 4 5 6 31 32 33 34 35 36 x1 1 2 3 4 5 6 7 8 9 10 11 12 mstlc1 x2 7 8 9 10 11 12 13 14 15 16 17 18 mstlc1 x3 13 14 15 16 17 18 19 20 21 22 23 24 mstlc1 x4 19 20 21 22 23 24 25 26 27 28 29 30 mstlc1 x5 25 26 27 28 29 30 31 32 33 34 35 36 mstlc1 .ends .subckt mstlc4 x1 1 2 3 4 5 6 x2 4 5 6 7 8 9 x3 7 8 9 10 11 x4 10 11 12 13 .ends 1 2 3 13 14 15 mstlc1 mstlc1 12 mstlc1 14 15 mstlc1 .subckt mstl_sq5 1 6 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq x4 4 5 mstl_sq x5 5 6 mstl_sq .ends .subckt mstl_sq4 1 5 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq x4 4 5 mstl_sq .ends *.subckt br10 1 6 *x1 1 2 br5 *x5 2 6 br5 *.ends x1 1 2 mstl_pad x2 2 5 mstl_sq4 x3 3 4 5 6 7 8 9 10 11 12 13 14 mstlc5 x5 9 10 11 12 13 14 21 22 23 24 25 26 mstlc1 x6 3 4 5 mstl_corner x10 5 6 7 mstl_corner x12 22 23 24 mstl_corner x13 24 25 26 mstl_corner x8 27 24 mstl_sq4 x9 28 27 mstl_pad rs1 8 0 1g rs2 21 0 1g r0 28 0 1g 284 vpl 1 0 pulse(0 2.5 2n 0.1n 0.2n 10n 20n) ac 1 *.tran 1p 40n 1p *.ac dec 60 1mega 15giga .net v(28) vpl rin=50 rout=50 .param + c_cou = opt1(2.2e-15,1f,1n) + cou_l = opt1(0.5,0,1) *+ csq = opt1(1.9e-15,0.1f,1n) + csq = 1.5e-15 + rc = opt1(0.30,0.01,10) + lc = opt1(14.4e-11,1f,1u) + cc = opt1(2.1e-15,0.01f,1n) + cc2 = opt1(8.1e-15,0.01f,1n) .param ccsq .param r2 .param l2 .param c2 .param rsq .param lsq *.param csq = 1.000e-15 = 1.071e+00 = 2.930e-14 = 2.603e-15 = 7.684e-02 = 1.009e-11 = 2.969e-15 $ $ $ $ $ $ 4.446e-03 5.731e+00 2.624e-03 1.027e+00 4.455e+00 5.633e+01 $ 3.244e+01 -2.409e+02 1.154e-03 6.587e+01 -3.999e-02 -4.904e-02 -1.497e-02 -7.008e-03 .param r=rsq l=lsq c=csq .ac data=measured optimize=opt1 + results=comp1,comp2,comp5,comp6 + model=converge .model converge opt relin=1e-5 relout=1e-4 close=100 itropt=200 .measure .measure .measure .measure .measure .measure .measure .measure ac ac ac ac ac ac ac ac comp1 comp2 comp3 comp4 comp5 comp6 comp7 comp8 err1 err1 err1 err1 err1 err1 err1 err1 .ac data=measured .print par(s11r) s11(r) .print par(s12r) s12(r) .print par(s21r) s21(r) .print par(s22r) s22(r) par(s11r) par(s11i) par(s12r) par(s12i) par(s21r) par(s21i) par(s22r) par(s22i) s11(r) s11(i) s12(r) s12(i) s21(r) s21(i) s22(r) s22(i) par(s11i) par(s12i) par(s21i) par(s22i) s11(i) s12(i) s21(i) s22(i) .data measured file=’c9.txt’ freq s11r s11i s21r s21i s12r s12i s22r s22i .enddata .param freq=100m, s11r=0, s11i=0, s21r=0, s21i=0, s12r=0, s12i=0, s22r=0, s22i=0 .end E.4.2. Measured S-Parameter Data Freq 4.50E+07 1.45E+08 2.45E+08 3.44E+08 Meas. S11(R) 1.00E+00 9.99E-01 9.99E-01 9.99E-01 Meas. S11(I) -2.08E-03 -1.00E-02 -1.84E-02 -2.67E-02 285 Meas. S21(R) -3.85E-05 9.08E-05 2.70E-04 5.26E-04 Meas. S21(I) 2.04E-03 7.26E-03 1.25E-02 1.76E-02 4.44E+08 5.44E+08 6.44E+08 7.43E+08 8.43E+08 9.43E+08 1.04E+09 1.14E+09 1.24E+09 1.34E+09 1.44E+09 1.54E+09 1.64E+09 1.74E+09 1.84E+09 1.94E+09 2.04E+09 2.14E+09 2.24E+09 2.34E+09 2.44E+09 2.54E+09 2.64E+09 2.74E+09 2.84E+09 2.94E+09 3.04E+09 3.14E+09 3.24E+09 3.34E+09 3.44E+09 3.54E+09 3.64E+09 3.74E+09 3.84E+09 3.94E+09 4.04E+09 4.14E+09 4.24E+09 4.34E+09 4.44E+09 4.53E+09 4.63E+09 4.73E+09 4.83E+09 4.93E+09 5.03E+09 9.98E-01 9.98E-01 9.97E-01 9.96E-01 9.96E-01 9.95E-01 9.94E-01 9.93E-01 9.93E-01 9.91E-01 9.90E-01 9.88E-01 9.86E-01 9.84E-01 9.83E-01 9.80E-01 9.79E-01 9.76E-01 9.75E-01 9.72E-01 9.69E-01 9.67E-01 9.65E-01 9.62E-01 9.60E-01 9.57E-01 9.54E-01 9.52E-01 9.49E-01 9.46E-01 9.42E-01 9.40E-01 9.36E-01 9.33E-01 9.29E-01 9.25E-01 9.21E-01 9.17E-01 9.14E-01 9.09E-01 9.05E-01 9.00E-01 8.96E-01 8.92E-01 8.87E-01 8.82E-01 8.78E-01 -3.46E-02 -4.29E-02 -5.06E-02 -5.90E-02 -6.69E-02 -7.48E-02 -8.26E-02 -9.09E-02 -9.91E-02 -1.06E-01 -1.15E-01 -1.23E-01 -1.31E-01 -1.39E-01 -1.47E-01 -1.55E-01 -1.62E-01 -1.70E-01 -1.78E-01 -1.85E-01 -1.93E-01 -2.01E-01 -2.09E-01 -2.16E-01 -2.23E-01 -2.31E-01 -2.38E-01 -2.46E-01 -2.54E-01 -2.61E-01 -2.68E-01 -2.76E-01 -2.83E-01 -2.90E-01 -2.98E-01 -3.06E-01 -3.13E-01 -3.20E-01 -3.28E-01 -3.35E-01 -3.41E-01 -3.49E-01 -3.56E-01 -3.62E-01 -3.68E-01 -3.75E-01 -3.81E-01 286 8.56E-04 1.26E-03 1.76E-03 2.32E-03 2.95E-03 3.66E-03 4.55E-03 5.49E-03 6.48E-03 7.55E-03 8.72E-03 9.93E-03 1.13E-02 1.27E-02 1.42E-02 1.57E-02 1.74E-02 1.91E-02 2.09E-02 2.28E-02 2.46E-02 2.67E-02 2.88E-02 3.10E-02 3.33E-02 3.57E-02 3.81E-02 4.06E-02 4.33E-02 4.59E-02 4.86E-02 5.16E-02 5.44E-02 5.74E-02 6.06E-02 6.37E-02 6.70E-02 7.03E-02 7.37E-02 7.70E-02 8.04E-02 8.39E-02 8.74E-02 9.09E-02 9.44E-02 9.83E-02 1.02E-01 2.26E-02 2.77E-02 3.27E-02 3.78E-02 4.29E-02 4.79E-02 5.31E-02 5.80E-02 6.31E-02 6.80E-02 7.31E-02 7.78E-02 8.28E-02 8.76E-02 9.24E-02 9.73E-02 1.02E-01 1.07E-01 1.12E-01 1.17E-01 1.21E-01 1.26E-01 1.31E-01 1.35E-01 1.40E-01 1.45E-01 1.49E-01 1.54E-01 1.58E-01 1.63E-01 1.67E-01 1.72E-01 1.76E-01 1.80E-01 1.85E-01 1.89E-01 1.93E-01 1.97E-01 2.01E-01 2.05E-01 2.09E-01 2.13E-01 2.17E-01 2.21E-01 2.25E-01 2.29E-01 2.33E-01 5.13E+09 5.23E+09 5.33E+09 5.43E+09 5.53E+09 5.63E+09 5.73E+09 5.83E+09 5.93E+09 6.03E+09 6.13E+09 6.23E+09 6.33E+09 6.43E+09 6.53E+09 6.63E+09 6.73E+09 6.83E+09 6.93E+09 7.03E+09 7.13E+09 7.23E+09 7.33E+09 7.43E+09 7.53E+09 7.63E+09 7.73E+09 7.83E+09 7.93E+09 8.03E+09 8.13E+09 8.23E+09 8.33E+09 8.43E+09 8.53E+09 8.63E+09 8.73E+09 8.83E+09 8.92E+09 9.02E+09 9.12E+09 9.22E+09 9.32E+09 9.42E+09 9.52E+09 9.62E+09 9.72E+09 8.74E-01 8.69E-01 8.64E-01 8.60E-01 8.56E-01 8.51E-01 8.46E-01 8.41E-01 8.35E-01 8.30E-01 8.25E-01 8.19E-01 8.13E-01 8.07E-01 8.01E-01 7.96E-01 7.90E-01 7.83E-01 7.77E-01 7.71E-01 7.62E-01 7.58E-01 7.53E-01 7.45E-01 7.40E-01 7.33E-01 7.27E-01 7.19E-01 7.12E-01 7.06E-01 6.99E-01 6.93E-01 6.88E-01 6.81E-01 6.74E-01 6.69E-01 6.63E-01 6.55E-01 6.49E-01 6.42E-01 6.33E-01 6.26E-01 6.19E-01 6.12E-01 6.05E-01 5.98E-01 5.91E-01 -3.87E-01 -3.93E-01 -3.99E-01 -4.06E-01 -4.12E-01 -4.18E-01 -4.25E-01 -4.32E-01 -4.38E-01 -4.44E-01 -4.50E-01 -4.56E-01 -4.62E-01 -4.68E-01 -4.74E-01 -4.79E-01 -4.85E-01 -4.91E-01 -4.96E-01 -5.00E-01 -5.04E-01 -5.10E-01 -5.09E-01 -5.16E-01 -5.22E-01 -5.28E-01 -5.30E-01 -5.36E-01 -5.40E-01 -5.44E-01 -5.48E-01 -5.51E-01 -5.55E-01 -5.59E-01 -5.58E-01 -5.63E-01 -5.70E-01 -5.75E-01 -5.79E-01 -5.83E-01 -5.87E-01 -5.90E-01 -5.93E-01 -5.95E-01 -5.99E-01 -6.02E-01 -6.05E-01 287 1.06E-01 1.10E-01 1.15E-01 1.19E-01 1.23E-01 1.28E-01 1.33E-01 1.37E-01 1.42E-01 1.47E-01 1.52E-01 1.56E-01 1.61E-01 1.66E-01 1.71E-01 1.76E-01 1.81E-01 1.86E-01 1.91E-01 1.96E-01 2.01E-01 2.06E-01 2.09E-01 2.16E-01 2.22E-01 2.28E-01 2.31E-01 2.38E-01 2.43E-01 2.49E-01 2.54E-01 2.60E-01 2.66E-01 2.72E-01 2.78E-01 2.84E-01 2.90E-01 2.97E-01 3.03E-01 3.09E-01 3.15E-01 3.21E-01 3.27E-01 3.33E-01 3.40E-01 3.47E-01 3.54E-01 2.37E-01 2.40E-01 2.44E-01 2.48E-01 2.51E-01 2.55E-01 2.58E-01 2.61E-01 2.64E-01 2.67E-01 2.70E-01 2.73E-01 2.76E-01 2.79E-01 2.82E-01 2.84E-01 2.87E-01 2.89E-01 2.91E-01 2.94E-01 2.95E-01 2.97E-01 2.99E-01 3.01E-01 3.04E-01 3.06E-01 3.07E-01 3.09E-01 3.11E-01 3.13E-01 3.14E-01 3.16E-01 3.18E-01 3.19E-01 3.18E-01 3.21E-01 3.23E-01 3.23E-01 3.24E-01 3.25E-01 3.25E-01 3.26E-01 3.27E-01 3.28E-01 3.29E-01 3.29E-01 3.30E-01 9.82E+09 9.92E+09 1.00E+10 1.01E+10 1.02E+10 1.03E+10 1.04E+10 1.05E+10 1.06E+10 1.07E+10 1.08E+10 1.09E+10 1.10E+10 1.11E+10 1.12E+10 1.13E+10 1.14E+10 1.15E+10 1.16E+10 1.17E+10 1.18E+10 1.19E+10 1.20E+10 1.21E+10 1.22E+10 1.23E+10 1.24E+10 1.25E+10 1.26E+10 1.27E+10 1.28E+10 1.29E+10 1.30E+10 1.31E+10 1.32E+10 1.33E+10 1.34E+10 1.35E+10 1.36E+10 1.37E+10 1.38E+10 1.39E+10 1.40E+10 1.41E+10 1.42E+10 1.43E+10 1.44E+10 5.84E-01 5.76E-01 5.69E-01 5.61E-01 5.53E-01 5.45E-01 5.38E-01 5.29E-01 5.22E-01 5.14E-01 5.05E-01 4.98E-01 4.89E-01 4.82E-01 4.73E-01 4.65E-01 4.57E-01 4.50E-01 4.43E-01 4.36E-01 4.29E-01 4.23E-01 4.17E-01 4.11E-01 4.05E-01 3.98E-01 3.92E-01 3.86E-01 3.80E-01 3.75E-01 3.69E-01 3.64E-01 3.58E-01 3.51E-01 3.45E-01 3.37E-01 3.30E-01 3.24E-01 3.17E-01 3.11E-01 3.04E-01 2.98E-01 2.89E-01 2.82E-01 2.75E-01 2.68E-01 2.60E-01 -6.08E-01 -6.11E-01 -6.14E-01 -6.17E-01 -6.19E-01 -6.22E-01 -6.23E-01 -6.25E-01 -6.27E-01 -6.28E-01 -6.29E-01 -6.30E-01 -6.31E-01 -6.32E-01 -6.32E-01 -6.32E-01 -6.31E-01 -6.30E-01 -6.30E-01 -6.30E-01 -6.29E-01 -6.29E-01 -6.29E-01 -6.29E-01 -6.29E-01 -6.29E-01 -6.28E-01 -6.27E-01 -6.27E-01 -6.27E-01 -6.27E-01 -6.27E-01 -6.28E-01 -6.28E-01 -6.29E-01 -6.29E-01 -6.28E-01 -6.27E-01 -6.26E-01 -6.26E-01 -6.26E-01 -6.26E-01 -6.25E-01 -6.25E-01 -6.23E-01 -6.22E-01 -6.21E-01 288 3.61E-01 3.68E-01 3.75E-01 3.81E-01 3.88E-01 3.95E-01 4.01E-01 4.07E-01 4.14E-01 4.20E-01 4.26E-01 4.33E-01 4.38E-01 4.44E-01 4.51E-01 4.57E-01 4.62E-01 4.67E-01 4.73E-01 4.78E-01 4.82E-01 4.87E-01 4.92E-01 4.97E-01 5.02E-01 5.07E-01 5.12E-01 5.16E-01 5.22E-01 5.28E-01 5.34E-01 5.40E-01 5.48E-01 5.54E-01 5.61E-01 5.68E-01 5.74E-01 5.80E-01 5.87E-01 5.93E-01 6.00E-01 6.06E-01 6.13E-01 6.18E-01 6.24E-01 6.29E-01 6.35E-01 3.29E-01 3.28E-01 3.28E-01 3.27E-01 3.26E-01 3.25E-01 3.24E-01 3.22E-01 3.21E-01 3.19E-01 3.17E-01 3.15E-01 3.13E-01 3.12E-01 3.09E-01 3.06E-01 3.04E-01 3.01E-01 2.99E-01 2.97E-01 2.94E-01 2.92E-01 2.90E-01 2.88E-01 2.86E-01 2.84E-01 2.83E-01 2.81E-01 2.79E-01 2.79E-01 2.78E-01 2.76E-01 2.74E-01 2.71E-01 2.68E-01 2.65E-01 2.61E-01 2.58E-01 2.54E-01 2.51E-01 2.47E-01 2.42E-01 2.37E-01 2.32E-01 2.27E-01 2.22E-01 2.16E-01 1.45E+10 1.46E+10 1.47E+10 1.48E+10 1.49E+10 1.50E+10 1.51E+10 1.52E+10 1.53E+10 1.54E+10 1.55E+10 1.56E+10 1.57E+10 1.58E+10 1.59E+10 1.60E+10 1.61E+10 1.62E+10 1.63E+10 1.64E+10 1.65E+10 1.66E+10 1.67E+10 1.68E+10 1.69E+10 1.70E+10 1.71E+10 1.72E+10 1.73E+10 1.74E+10 1.75E+10 1.76E+10 1.77E+10 1.78E+10 1.79E+10 1.80E+10 1.81E+10 1.82E+10 1.83E+10 1.84E+10 1.85E+10 1.86E+10 1.87E+10 1.88E+10 1.89E+10 1.90E+10 1.91E+10 2.53E-01 2.46E-01 2.40E-01 2.33E-01 2.27E-01 2.20E-01 2.14E-01 2.08E-01 2.03E-01 1.97E-01 1.92E-01 1.86E-01 1.80E-01 1.74E-01 1.69E-01 1.64E-01 1.58E-01 1.54E-01 1.50E-01 1.44E-01 1.40E-01 1.34E-01 1.28E-01 1.23E-01 1.19E-01 1.14E-01 1.09E-01 1.04E-01 1.00E-01 9.56E-02 9.18E-02 8.72E-02 8.24E-02 7.73E-02 7.24E-02 6.83E-02 6.25E-02 5.81E-02 5.39E-02 4.91E-02 4.61E-02 4.31E-02 3.70E-02 2.84E-02 3.34E-02 3.02E-02 2.81E-02 -6.19E-01 -6.17E-01 -6.15E-01 -6.12E-01 -6.10E-01 -6.08E-01 -6.05E-01 -6.02E-01 -6.00E-01 -5.97E-01 -5.94E-01 -5.91E-01 -5.88E-01 -5.86E-01 -5.82E-01 -5.78E-01 -5.76E-01 -5.72E-01 -5.69E-01 -5.67E-01 -5.64E-01 -5.61E-01 -5.58E-01 -5.55E-01 -5.52E-01 -5.48E-01 -5.46E-01 -5.43E-01 -5.40E-01 -5.36E-01 -5.34E-01 -5.31E-01 -5.28E-01 -5.25E-01 -5.22E-01 -5.18E-01 -5.14E-01 -5.10E-01 -5.05E-01 -5.02E-01 -4.98E-01 -4.93E-01 -4.93E-01 -4.96E-01 -4.79E-01 -4.75E-01 -4.71E-01 289 6.39E-01 6.44E-01 6.49E-01 6.53E-01 6.57E-01 6.62E-01 6.65E-01 6.69E-01 6.72E-01 6.75E-01 6.78E-01 6.82E-01 6.86E-01 6.89E-01 6.93E-01 6.96E-01 7.00E-01 7.04E-01 7.09E-01 7.14E-01 7.18E-01 7.23E-01 7.27E-01 7.31E-01 7.33E-01 7.36E-01 7.39E-01 7.43E-01 7.45E-01 7.48E-01 7.50E-01 7.52E-01 7.55E-01 7.57E-01 7.59E-01 7.61E-01 7.63E-01 7.65E-01 7.67E-01 7.68E-01 7.69E-01 7.70E-01 7.71E-01 7.72E-01 7.72E-01 7.72E-01 7.73E-01 2.11E-01 2.05E-01 1.99E-01 1.94E-01 1.89E-01 1.83E-01 1.77E-01 1.72E-01 1.67E-01 1.62E-01 1.57E-01 1.52E-01 1.48E-01 1.43E-01 1.38E-01 1.33E-01 1.29E-01 1.25E-01 1.20E-01 1.15E-01 1.10E-01 1.03E-01 9.64E-02 8.96E-02 8.37E-02 7.70E-02 7.10E-02 6.46E-02 5.83E-02 5.15E-02 4.55E-02 3.95E-02 3.34E-02 2.72E-02 2.05E-02 1.38E-02 6.93E-03 7.94E-04 -6.04E-03 -1.23E-02 -1.81E-02 -2.46E-02 -3.04E-02 -3.71E-02 -4.29E-02 -4.83E-02 -5.38E-02 1.92E+10 1.93E+10 1.94E+10 1.95E+10 1.96E+10 1.97E+10 1.98E+10 1.99E+10 2.00E+10 2.62E-02 2.35E-02 2.09E-02 1.85E-02 1.44E-02 1.10E-02 6.85E-03 3.69E-03 3.51E-04 -4.68E-01 -4.65E-01 -4.62E-01 -4.60E-01 -4.57E-01 -4.53E-01 -4.51E-01 -4.47E-01 -4.43E-01 7.74E-01 7.74E-01 7.76E-01 7.77E-01 7.80E-01 7.82E-01 7.83E-01 7.85E-01 7.86E-01 -5.92E-02 -6.35E-02 -6.82E-02 -7.32E-02 -7.83E-02 -8.37E-02 -9.05E-02 -9.60E-02 -1.02E-01 E.5. 10-Segment Interdigital Capacitor E.5.1. 9-Segment Resistor and 10-Segment Interdigital Capacitor Series Equivalent Circuit .option accurate dccap=1 node nopage ingold=2 post acct=2 probe * u circuit 10 mil in length .subckt mstlc1 1 6 11 16 21 26 31 36 41 46 5 10 15 20 25 30 35 40 45 50 r1l 1 2 r l1l 2 3 l c1 3 0 c r1r 3 4 r l1r 4 5 l rg1 3 0 10mega ccouple 3 8 c_cou cc1a 1 3 cc cc1b 3 5 cc r2l 6 7 r l2l 7 8 l c2 8 0 c rg2 8 0 10mega l2r 8 9 l r2r 9 10 r k1 l1l l2l k=cou_l k2 l2r l1r k=cou_l cc2a 6 8 cc cc2b 6 8 cc r3l 11 12 r l3l 12 13 l c3 13 0 c rg3 13 0 10mega r3r 13 14 r l3r 14 15 l cc23 8 13 c_cou cc3a 11 13 cc cc3b 13 15 cc k23l l2l l3l k=cou_l 290 k23r l2r l3r k=cou_l r4l 16 17 r l4l 17 18 l c4 18 0 c rg4 18 0 10mega r4r 18 19 r l4r 19 20 l cc34 13 18 c_cou k34l l3l l4l k=cou_l k34r l3r l4r k=cou_l cc4a 16 18 cc cc4b 18 20 cc r5l 21 22 r l5l 22 23 l c5 23 0 c rg5 23 0 10mega r5r 23 24 r l5r 24 25 l cc45 18 23 c_cou k45l l4l l5l k=cou_l k45r l4r l5r k=cou_l cc5a 21 23 cc cc5b 23 25 cc r6l 26 27 r l6l 27 28 l c6 28 0 c rg6 28 0 10mega r6r 28 29 r l6r 29 30 l cc56 23 28 c_cou k56l l5l l6l k=cou_l k56r l5r l6r k=cou_l cc6a 26 28 cc cc6b 28 30 cc r7l 31 32 r l7l 32 33 l c7 33 0 c rg7 33 0 10mega r7r 33 34 r l7r 34 35 l cc67 28 33 c_cou k67l l6l l7l k=cou_l k67r l6r l7r k=cou_l cc7a 31 33 cc cc7b 33 35 cc r8l 36 37 r l8l 37 38 l c8 38 0 c rg8 38 0 10mega r8r 38 39 r l8r 39 40 l cc78 33 38 c_cou k78l l7l l8l k=cou_l k78r l7r l8r k=cou_l cc8a 36 38 cc cc8b 38 40 cc r9l 41 42 r l9l 42 43 l c9 43 0 c rg9 43 0 10mega r9r 43 44 r 291 l9r 44 45 l cc89 38 43 c_cou k89l l8l l9l k=cou_l k89r l8r l9r k=cou_l cc9a 41 43 cc cc9b 43 45 cc r10l 46 47 r l10l 47 48 l c10 48 0 c rg10 48 0 10mega r10r 48 49 r l10r 49 50 l cc910 43 48 c_cou k910l l9l l10l k=cou_l k910r l9r l10r k=cou_l cc10a 46 48 cc cc10b 48 50 cc .ends .subckt mstl_pad 1 5 r1 1 2 r2 l1 2 3 l2 c1 3 0 c2 r2 3 0 10g r1r 3 4 r2 l2r 4 5 l2 .ends .subckt mstl_corner 1 6 5 r1 1 2 rc l1 2 3 lc c1 3 0 cc r2 3 0 10g r1r 3 4 rc l2r 4 5 lc r6 6 0 1g c6 6 3 cc2 r6g 6 3 1g .ends .subckt mstl_sq 1 5 r1 1 2 rsq l1 2 3 lsq c1 3 0 csq r1r 3 4 rsq l2r 4 5 lsq cc1 1 3 ccsq cc2 3 5 ccsq .ends .subckt mstlc5 1 2 3 4 5 6 7 8 9 10 51 52 53 54 55 56 57 58 59 60 x1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 mstlc1 x2 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 mstlc1 x3 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 mstlc1 x4 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 mstlc1 x5 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 mstlc1 .ends .subckt mstlc16 1 2 3 4 5 6 7 8 9 10 41 42 43 44 45 46 47 48 49 50 x1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 mstlc5 x2 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 mstlc5 x3 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 mstlc5 x4 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 mstlc1 292 .ends .subckt mstlc4 x1 1 2 3 4 5 6 x2 4 5 6 7 8 9 x3 7 8 9 10 11 x4 10 11 12 13 .ends 1 2 3 13 14 15 mstlc1 mstlc1 12 mstlc1 14 15 mstlc1 .subckt mstl_sq5 1 6 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq x4 4 5 mstl_sq x5 5 6 mstl_sq .ends .subckt mstl_sq4 1 5 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq x4 4 5 mstl_sq .ends .subckt c7 1 28 x1 1 2 mstl_pad x2 2 7 mstl_sq4 x3 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 mstlc16 x6 3 4 5 mstl_corner x10 5 6 7 mstl_corner x14 7 8 9 mstl_corner x15 9 10 11 mstl_corner x12 14 15 16 mstl_corner x13 16 17 18 mstl_corner x16 18 19 20 mstl_corner x17 20 21 22 mstl_corner x8 18 27 mstl_sq4 x9 28 27 mstl_pad rs1 12 0 1g rs2 13 0 1g r0 28 0 1g .ends r1 x1 ls rt 5 1 1 3 1 3 3 0 0.1 c7 1n 50 vpl 5 0 pulse(0 2.5 2n 0.1n 0.2n 10n 20n) ac 1 *.tran 1p 40n 1p *.ac dec 60 1mega 15giga *.net v(28) vpl rin=50 rout=50 .param + cc = 1e-14 + c_cou = opt1(1.9e-15,1f,1n) + cou_l = opt1(0.4,0,1) + c_cou2 = opt1(0.1e-15,1f,1n) + cou_l2 = opt1(0.3,0,1) *+ csq = opt1(1.9e-15,0.1f,1n) + csq = 1.9e-15 + rc = opt1(0.02,0.01,10) + lc = opt1(10.4e-11,1f,1u) + cc = opt1(2.0e-15,0.01f,1n) 293 + cc2 = opt1(3.0e-15,0.01f,1n) .param * from .param .param .param .param .param .param .param r=rsq l=lsq c=csq snake_3_long_2_10g c_cou = 1.936e-15 cou_l = 3.175e-01 c_cou2 = 0.366e-15 cou_l2 = 2.921e-01 csq = 1.143e-15 lsq = 0.934e-11 c_cou3 = 0.5e-15 .measure .measure .measure .measure .measure .measure .measure .measure ac ac ac ac ac ac ac ac comp1 comp2 comp3 comp4 comp5 comp6 comp7 comp8 err1 err1 err1 err1 err1 err1 err1 err1 par(s11r) par(s11i) par(s12r) par(s12i) par(s21r) par(s21i) par(s22r) par(s22i) $ $ $ $ $ $ 4.435e+00 2.931e+01 3.297e+00 9.158e+00 2.655e+01 2.605e+01 6.177e-02 -2.063e-03 8.628e-02 -5.349e-03 -8.391e-03 6.879e-03 s11(r) s11(i) s12(r) s12(i) s21(r) s21(i) s22(r) s22(i) .ac data=measured .print vdb(3) vp(3) vm(3) vi(3) vr(3) .print par(vdb) par(vph) .data measured file=’reson.txt’ freq=1 vdb=2 vph=3 .enddata .param vdb=0, vph=0 .param freq=100m, s11r=0, s11i=0, s21r=0, s21i=0, s12r=0, s12i=0, s22r=0, s22i=0 .end E.5.2. Measured S-Parameter Data for 10-Segment Interdigital Capacitor Freq 4.50E+07 1.45E+08 2.45E+08 3.44E+08 4.44E+08 5.44E+08 6.44E+08 7.43E+08 8.43E+08 9.43E+08 1.04E+09 1.14E+09 1.24E+09 1.34E+09 1.44E+09 1.54E+09 1.64E+09 Meas. S11(R) 1.00E+00 9.98E-01 9.96E-01 9.93E-01 9.90E-01 9.85E-01 9.79E-01 9.73E-01 9.65E-01 9.57E-01 9.48E-01 9.38E-01 9.29E-01 9.17E-01 9.05E-01 8.91E-01 8.78E-01 Meas. S11(I) -9.52E-03 -3.39E-02 -5.85E-02 -8.27E-02 -1.07E-01 -1.31E-01 -1.54E-01 -1.77E-01 -2.00E-01 -2.22E-01 -2.44E-01 -2.66E-01 -2.87E-01 -3.07E-01 -3.27E-01 -3.46E-01 -3.65E-01 294 Meas. S21(R) -5.05E-05 9.88E-04 2.85E-03 5.60E-03 9.23E-03 1.37E-02 1.91E-02 2.54E-02 3.23E-02 4.01E-02 4.88E-02 5.83E-02 6.86E-02 7.98E-02 9.09E-02 1.03E-01 1.15E-01 Meas. S21(I) 8.48E-03 2.68E-02 4.52E-02 6.34E-02 8.14E-02 9.93E-02 1.17E-01 1.34E-01 1.51E-01 1.67E-01 1.84E-01 1.99E-01 2.15E-01 2.29E-01 2.43E-01 2.56E-01 2.68E-01 1.74E+09 1.84E+09 1.94E+09 2.04E+09 2.14E+09 2.24E+09 2.34E+09 2.44E+09 2.54E+09 2.64E+09 2.74E+09 2.84E+09 2.94E+09 3.04E+09 3.14E+09 3.24E+09 3.34E+09 3.44E+09 3.54E+09 3.64E+09 3.74E+09 3.84E+09 3.94E+09 4.04E+09 4.14E+09 4.24E+09 4.34E+09 4.44E+09 4.53E+09 4.63E+09 4.73E+09 4.83E+09 4.93E+09 5.03E+09 5.13E+09 5.23E+09 5.33E+09 5.43E+09 5.53E+09 5.63E+09 5.73E+09 5.83E+09 5.93E+09 6.03E+09 6.13E+09 6.23E+09 6.33E+09 8.64E-01 8.50E-01 8.34E-01 8.19E-01 8.03E-01 7.87E-01 7.71E-01 7.54E-01 7.37E-01 7.20E-01 7.03E-01 6.85E-01 6.68E-01 6.51E-01 6.33E-01 6.16E-01 5.98E-01 5.81E-01 5.63E-01 5.46E-01 5.29E-01 5.12E-01 4.95E-01 4.77E-01 4.62E-01 4.45E-01 4.29E-01 4.13E-01 3.97E-01 3.82E-01 3.68E-01 3.53E-01 3.38E-01 3.24E-01 3.10E-01 2.97E-01 2.84E-01 2.71E-01 2.59E-01 2.47E-01 2.34E-01 2.23E-01 2.12E-01 2.00E-01 1.90E-01 1.79E-01 1.69E-01 -3.82E-01 -3.99E-01 -4.16E-01 -4.31E-01 -4.46E-01 -4.61E-01 -4.73E-01 -4.86E-01 -4.98E-01 -5.10E-01 -5.20E-01 -5.30E-01 -5.38E-01 -5.47E-01 -5.55E-01 -5.62E-01 -5.69E-01 -5.74E-01 -5.80E-01 -5.84E-01 -5.88E-01 -5.92E-01 -5.95E-01 -5.97E-01 -5.99E-01 -6.00E-01 -6.01E-01 -6.00E-01 -6.00E-01 -6.00E-01 -5.99E-01 -5.97E-01 -5.95E-01 -5.93E-01 -5.91E-01 -5.88E-01 -5.84E-01 -5.81E-01 -5.78E-01 -5.74E-01 -5.69E-01 -5.66E-01 -5.61E-01 -5.56E-01 -5.52E-01 -5.47E-01 -5.41E-01 295 1.29E-01 1.42E-01 1.56E-01 1.70E-01 1.85E-01 2.00E-01 2.16E-01 2.31E-01 2.47E-01 2.63E-01 2.79E-01 2.95E-01 3.11E-01 3.27E-01 3.44E-01 3.60E-01 3.76E-01 3.92E-01 4.09E-01 4.25E-01 4.40E-01 4.56E-01 4.72E-01 4.87E-01 5.02E-01 5.16E-01 5.30E-01 5.44E-01 5.58E-01 5.72E-01 5.85E-01 5.98E-01 6.10E-01 6.23E-01 6.35E-01 6.46E-01 6.57E-01 6.68E-01 6.79E-01 6.89E-01 6.99E-01 7.08E-01 7.18E-01 7.26E-01 7.35E-01 7.43E-01 7.51E-01 2.80E-01 2.91E-01 3.01E-01 3.11E-01 3.20E-01 3.29E-01 3.37E-01 3.44E-01 3.50E-01 3.55E-01 3.60E-01 3.64E-01 3.68E-01 3.71E-01 3.73E-01 3.74E-01 3.75E-01 3.75E-01 3.75E-01 3.74E-01 3.72E-01 3.69E-01 3.66E-01 3.63E-01 3.59E-01 3.54E-01 3.49E-01 3.44E-01 3.38E-01 3.32E-01 3.25E-01 3.18E-01 3.11E-01 3.03E-01 2.95E-01 2.87E-01 2.78E-01 2.69E-01 2.60E-01 2.51E-01 2.41E-01 2.31E-01 2.21E-01 2.11E-01 2.00E-01 1.90E-01 1.79E-01 6.43E+09 6.53E+09 6.63E+09 6.73E+09 6.83E+09 6.93E+09 7.03E+09 7.13E+09 7.23E+09 7.33E+09 7.43E+09 7.53E+09 7.63E+09 7.73E+09 7.83E+09 7.93E+09 8.03E+09 8.13E+09 8.23E+09 8.33E+09 8.43E+09 8.53E+09 8.63E+09 8.73E+09 8.83E+09 8.92E+09 9.02E+09 9.12E+09 9.22E+09 9.32E+09 9.42E+09 9.52E+09 9.62E+09 9.72E+09 9.82E+09 9.92E+09 1.00E+10 1.59E-01 1.49E-01 1.40E-01 1.31E-01 1.22E-01 1.14E-01 1.06E-01 9.82E-02 9.05E-02 8.61E-02 7.70E-02 6.99E-02 6.27E-02 5.75E-02 5.04E-02 4.43E-02 3.86E-02 3.28E-02 2.78E-02 2.34E-02 1.88E-02 1.40E-02 9.60E-03 5.04E-03 1.37E-03 -3.66E-03 -7.29E-03 -1.22E-02 -1.76E-02 -2.27E-02 -2.80E-02 -3.14E-02 -3.31E-02 -3.44E-02 -3.57E-02 -3.65E-02 -3.73E-02 -5.36E-01 -5.31E-01 -5.26E-01 -5.20E-01 -5.15E-01 -5.09E-01 -5.03E-01 -4.96E-01 -4.91E-01 -4.86E-01 -4.79E-01 -4.73E-01 -4.66E-01 -4.61E-01 -4.54E-01 -4.48E-01 -4.41E-01 -4.34E-01 -4.28E-01 -4.22E-01 -4.16E-01 -4.11E-01 -4.05E-01 -3.99E-01 -3.94E-01 -3.90E-01 -3.84E-01 -3.79E-01 -3.74E-01 -3.67E-01 -3.59E-01 -3.51E-01 -3.43E-01 -3.36E-01 -3.30E-01 -3.24E-01 -3.19E-01 7.58E-01 7.65E-01 7.72E-01 7.78E-01 7.84E-01 7.91E-01 7.96E-01 8.01E-01 8.07E-01 8.11E-01 8.15E-01 8.19E-01 8.23E-01 8.27E-01 8.30E-01 8.33E-01 8.35E-01 8.36E-01 8.38E-01 8.39E-01 8.40E-01 8.41E-01 8.43E-01 8.43E-01 8.43E-01 8.44E-01 8.44E-01 8.42E-01 8.40E-01 8.40E-01 8.41E-01 8.43E-01 8.44E-01 8.44E-01 8.42E-01 8.41E-01 8.39E-01 1.69E-01 1.58E-01 1.47E-01 1.36E-01 1.25E-01 1.14E-01 1.04E-01 9.22E-02 7.97E-02 7.22E-02 5.76E-02 4.47E-02 3.26E-02 2.34E-02 9.58E-03 -2.99E-03 -1.47E-02 -2.68E-02 -3.91E-02 -5.01E-02 -6.12E-02 -7.25E-02 -8.36E-02 -9.47E-02 -1.06E-01 -1.17E-01 -1.28E-01 -1.39E-01 -1.49E-01 -1.57E-01 -1.65E-01 -1.76E-01 -1.88E-01 -1.99E-01 -2.11E-01 -2.23E-01 -2.34E-01 E.5.3. Voltage Magnitude and Phase of RC Series Circuit Terminated in 50 Ohm Resistor Frequency 4.50E+07 Measured Volatge -1.70E-02 Measured Phase -3.24E-01 296 1.45E+08 2.45E+08 3.44E+08 4.44E+08 5.44E+08 6.44E+08 7.43E+08 8.43E+08 9.43E+08 1.04E+09 1.14E+09 1.24E+09 1.34E+09 1.44E+09 1.54E+09 1.64E+09 1.74E+09 1.84E+09 1.94E+09 2.04E+09 2.14E+09 2.24E+09 2.34E+09 2.44E+09 2.54E+09 2.64E+09 2.74E+09 2.84E+09 2.94E+09 3.04E+09 3.14E+09 3.24E+09 3.34E+09 3.44E+09 3.54E+09 3.64E+09 3.74E+09 3.84E+09 3.94E+09 4.04E+09 4.14E+09 4.24E+09 4.34E+09 4.44E+09 4.53E+09 4.63E+09 4.73E+09 -1.80E-02 -1.90E-02 -2.10E-02 -2.40E-02 -2.80E-02 -3.20E-02 -3.70E-02 -4.30E-02 -5.00E-02 -5.80E-02 -6.70E-02 -7.60E-02 -8.50E-02 -9.80E-02 -1.10E-01 -1.24E-01 -1.39E-01 -1.54E-01 -1.71E-01 -1.90E-01 -2.12E-01 -2.32E-01 -2.57E-01 -2.83E-01 -3.10E-01 -3.39E-01 -3.72E-01 -4.06E-01 -4.44E-01 -4.84E-01 -5.28E-01 -5.73E-01 -6.23E-01 -6.79E-01 -7.38E-01 -8.01E-01 -8.73E-01 -9.47E-01 -1.03E+00 -1.12E+00 -1.22E+00 -1.33E+00 -1.44E+00 -1.57E+00 -1.72E+00 -1.87E+00 -2.04E+00 -1.04E+00 -1.76E+00 -2.48E+00 -3.20E+00 -3.93E+00 -4.65E+00 -5.38E+00 -6.11E+00 -6.85E+00 -7.59E+00 -8.33E+00 -9.08E+00 -9.83E+00 -1.06E+01 -1.14E+01 -1.21E+01 -1.29E+01 -1.37E+01 -1.45E+01 -1.53E+01 -1.62E+01 -1.70E+01 -1.78E+01 -1.87E+01 -1.96E+01 -2.05E+01 -2.14E+01 -2.23E+01 -2.32E+01 -2.42E+01 -2.52E+01 -2.62E+01 -2.72E+01 -2.82E+01 -2.93E+01 -3.04E+01 -3.15E+01 -3.27E+01 -3.39E+01 -3.51E+01 -3.64E+01 -3.77E+01 -3.90E+01 -4.04E+01 -4.18E+01 -4.32E+01 -4.47E+01 297 4.83E+09 4.93E+09 5.03E+09 5.13E+09 5.23E+09 5.33E+09 5.43E+09 5.53E+09 5.63E+09 5.73E+09 5.83E+09 5.93E+09 6.03E+09 6.13E+09 6.23E+09 6.33E+09 6.43E+09 6.53E+09 6.63E+09 6.73E+09 6.83E+09 6.93E+09 7.03E+09 7.13E+09 7.23E+09 7.33E+09 7.43E+09 7.53E+09 7.63E+09 7.73E+09 7.83E+09 7.93E+09 8.03E+09 8.13E+09 8.23E+09 8.33E+09 8.43E+09 8.53E+09 8.63E+09 8.73E+09 8.83E+09 8.92E+09 9.02E+09 9.12E+09 9.22E+09 9.32E+09 9.42E+09 -2.24E+00 -2.44E+00 -2.68E+00 -2.94E+00 -3.23E+00 -3.55E+00 -3.91E+00 -4.31E+00 -4.76E+00 -5.27E+00 -5.85E+00 -6.51E+00 -7.24E+00 -8.09E+00 -9.07E+00 -1.02E+01 -1.15E+01 -1.30E+01 -1.46E+01 -1.63E+01 -1.78E+01 -1.82E+01 -1.74E+01 -1.57E+01 -1.39E+01 -1.25E+01 -1.11E+01 -9.85E+00 -8.83E+00 -8.06E+00 -7.25E+00 -6.59E+00 -6.02E+00 -5.52E+00 -5.08E+00 -4.70E+00 -4.37E+00 -4.06E+00 -3.76E+00 -3.50E+00 -3.30E+00 -3.06E+00 -2.87E+00 -2.70E+00 -2.54E+00 -2.39E+00 -2.21E+00 -4.63E+01 -4.79E+01 -4.95E+01 -5.11E+01 -5.28E+01 -5.45E+01 -5.62E+01 -5.79E+01 -5.97E+01 -6.14E+01 -6.30E+01 -6.45E+01 -6.58E+01 -6.67E+01 -6.73E+01 -6.71E+01 -6.60E+01 -6.32E+01 -5.78E+01 -4.83E+01 -3.27E+01 -1.27E+01 5.56E+00 1.88E+01 2.56E+01 3.05E+01 3.12E+01 3.15E+01 3.13E+01 3.13E+01 2.97E+01 2.84E+01 2.70E+01 2.54E+01 2.39E+01 2.24E+01 2.10E+01 1.97E+01 1.84E+01 1.70E+01 1.57E+01 1.46E+01 1.32E+01 1.20E+01 1.09E+01 1.00E+01 9.05E+00 298 9.52E+09 9.62E+09 9.72E+09 9.82E+09 9.92E+09 1.00E+10 1.01E+10 1.02E+10 1.03E+10 1.04E+10 1.05E+10 1.06E+10 1.07E+10 1.08E+10 1.09E+10 1.10E+10 1.11E+10 1.12E+10 1.13E+10 1.14E+10 1.15E+10 1.16E+10 1.17E+10 1.18E+10 1.19E+10 1.20E+10 1.21E+10 1.22E+10 1.23E+10 1.24E+10 1.25E+10 1.26E+10 1.27E+10 1.28E+10 1.29E+10 1.30E+10 1.31E+10 1.32E+10 1.33E+10 1.34E+10 1.35E+10 1.36E+10 1.37E+10 1.38E+10 1.39E+10 1.40E+10 1.41E+10 -2.03E+00 -1.88E+00 -1.76E+00 -1.66E+00 -1.57E+00 -1.50E+00 -1.42E+00 -1.35E+00 -1.29E+00 -1.25E+00 -1.20E+00 -1.16E+00 -1.12E+00 -1.08E+00 -1.06E+00 -1.03E+00 -9.96E-01 -9.84E-01 -9.71E-01 -9.62E-01 -9.62E-01 -9.55E-01 -9.40E-01 -9.30E-01 -9.15E-01 -8.79E-01 -8.36E-01 -7.77E-01 -7.14E-01 -6.50E-01 -5.96E-01 -5.73E-01 -5.78E-01 -6.14E-01 -6.37E-01 -6.34E-01 -6.12E-01 -5.92E-01 -5.61E-01 -5.28E-01 -4.97E-01 -4.77E-01 -4.43E-01 -3.97E-01 -3.33E-01 -2.52E-01 -1.70E-01 7.84E+00 6.59E+00 5.36E+00 4.13E+00 2.94E+00 1.90E+00 8.16E-01 -2.64E-01 -1.27E+00 -2.21E+00 -3.12E+00 -4.00E+00 -4.85E+00 -5.71E+00 -6.54E+00 -7.28E+00 -8.02E+00 -8.82E+00 -9.52E+00 -1.02E+01 -1.07E+01 -1.12E+01 -1.17E+01 -1.21E+01 -1.25E+01 -1.28E+01 -1.31E+01 -1.35E+01 -1.40E+01 -1.47E+01 -1.54E+01 -1.63E+01 -1.71E+01 -1.76E+01 -1.80E+01 -1.82E+01 -1.85E+01 -1.88E+01 -1.91E+01 -1.95E+01 -1.98E+01 -2.01E+01 -2.04E+01 -2.05E+01 -2.07E+01 -2.11E+01 -2.15E+01 299 1.42E+10 1.43E+10 1.44E+10 1.45E+10 1.46E+10 1.47E+10 1.48E+10 1.49E+10 1.50E+10 1.51E+10 1.52E+10 1.53E+10 1.54E+10 1.55E+10 1.56E+10 1.57E+10 1.58E+10 1.59E+10 1.60E+10 1.61E+10 1.62E+10 1.63E+10 1.64E+10 1.65E+10 1.66E+10 1.67E+10 1.68E+10 1.69E+10 1.70E+10 1.71E+10 1.72E+10 1.73E+10 1.74E+10 1.75E+10 1.76E+10 1.77E+10 1.78E+10 1.79E+10 1.80E+10 1.81E+10 1.82E+10 1.83E+10 1.84E+10 1.85E+10 1.86E+10 1.87E+10 1.88E+10 -9.50E-02 -3.00E-02 3.30E-02 8.70E-02 1.27E-01 1.67E-01 2.05E-01 2.34E-01 2.65E-01 2.89E-01 3.13E-01 3.31E-01 3.57E-01 3.64E-01 3.73E-01 3.95E-01 4.17E-01 4.37E-01 4.45E-01 4.67E-01 4.81E-01 4.92E-01 5.10E-01 5.30E-01 5.42E-01 5.53E-01 5.64E-01 5.79E-01 5.99E-01 6.26E-01 6.62E-01 6.79E-01 7.01E-01 7.21E-01 7.40E-01 7.52E-01 7.70E-01 7.90E-01 8.02E-01 8.19E-01 8.37E-01 8.46E-01 8.62E-01 8.73E-01 8.89E-01 9.11E-01 9.61E-01 -2.20E+01 -2.26E+01 -2.33E+01 -2.39E+01 -2.45E+01 -2.52E+01 -2.59E+01 -2.65E+01 -2.71E+01 -2.78E+01 -2.84E+01 -2.90E+01 -2.97E+01 -3.02E+01 -3.07E+01 -3.13E+01 -3.19E+01 -3.24E+01 -3.30E+01 -3.35E+01 -3.41E+01 -3.46E+01 -3.51E+01 -3.56E+01 -3.62E+01 -3.67E+01 -3.72E+01 -3.76E+01 -3.81E+01 -3.86E+01 -3.90E+01 -3.95E+01 -4.01E+01 -4.07E+01 -4.12E+01 -4.18E+01 -4.23E+01 -4.29E+01 -4.34E+01 -4.40E+01 -4.46E+01 -4.51E+01 -4.56E+01 -4.61E+01 -4.67E+01 -4.71E+01 -4.75E+01 300 1.89E+10 1.90E+10 1.91E+10 1.92E+10 1.93E+10 1.94E+10 1.95E+10 1.96E+10 1.97E+10 1.98E+10 1.99E+10 2.00E+10 9.39E-01 9.56E-01 9.76E-01 9.92E-01 1.02E+00 1.04E+00 1.06E+00 1.10E+00 1.13E+00 1.16E+00 1.19E+00 1.21E+00 -4.83E+01 -4.89E+01 -4.94E+01 -5.00E+01 -5.05E+01 -5.10E+01 -5.16E+01 -5.22E+01 -5.28E+01 -5.35E+01 -5.42E+01 -5.49E+01 301 APPENDIX F CIRCUITS AND DATA FOR PLANAR SPIRAL INDUCTOR MODELING F.1. Introduction Input files and measured S-parameter data for test structure optimization for the planar spiral inductor modeling study described earlier in this thesis are presented in this appendix. In addition, the circuit file representing the complete model of the 4-turn spiral inductor is also show, with associated measured S-parameters. All circuit files are written for the Star-Hspice circuit simulator. It should be noted that in some cases, certain subcircuit (.subckt) calls are defined but are never used in the actual optimization runs. Additionally, only S11 and S21 results are shown for the measured data, since S22 and S11 are equal, and S12 and S21 are also equivalent for these structures. 302 F.2. Test Structure 1 F.2.1. Circuit Optimization Input File .option accurate dccap=1 node nopage ingold=2 post acct=2 probe * u circuit 10 mil in length .subckt mstl_sq 1 5 r1 1 2 rsq l1 2 3 lsq c1 3 0 csq r2 3 0 10g r1r 3 4 rsq l2r 4 5 lsq *k1 l1 l2r k=cou_l cc1 1 3 c_cou cc2 3 5 c_cou *cc3 1 5 c_cou2 .ends .subckt mstl_sq5 1 6 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq x4 4 5 mstl_sq x5 5 6 mstl_sq .ends x2 x3 x4 x5 ro 2 3 4 5 6 3 4 5 6 0 mstl_sq5 mstl_sq5 mstl_sq5 mstl_sq5 1g vpl 2 0 pulse(0 2.5 2n 0.1n 0.2n 10n 20n) ac 1 *.tran 1p 40n 1p *.ac dec 60 1mega 15giga .net v(6) vpl rin=50 rout=50 .param + c_cou = opt1(3.514e-14,1f,1n) + rsq = opt1(0.053,0.0001,1) + lsq = opt1(7.14e-11,1f,1u) + csq = opt1(3.1e-14,0.01f,1n) .ac data=measured optimize=opt1 + results=comp1,comp2,comp5,comp6 + model=converge .model converge opt relin=1e-8 relout=1e-8 close=100 itropt=1000 .measure .measure .measure .measure .measure .measure .measure ac ac ac ac ac ac ac comp1 comp2 comp3 comp4 comp5 comp6 comp7 err1 err1 err1 err1 err1 err1 err1 par(s11r) par(s11i) par(s12r) par(s12i) par(s21r) par(s21i) par(s22r) s11(r) s11(i) minval=10 s12(r) s12(i) minval=10 s21(r) s21(i) minval=10 s22(r) 303 .measure ac comp8 err1 par(s22i) s22(i) minval=10 .ac data=measured .print par(s11r) s11(r) par(s11i) s11(i) .print par(s12r) s12(r) par(s12i) s12(i) .print par(s21r) s21(r) par(s21i) s21(i) .print par(s22r) s22(r) par(s22i) s22(i) *.param s11db='20*log(sqrt((par(s11r))^2+(par(s11i))^2))' *.param s21db='20*log(sqrt((par(s21r))^2+(par(s21i))^2))' .print s11(db) .print s12(db) .print s21(db) .print s22(db) .data measured file = ‘line.txt’ freq=1 s11r=2 s11I=3 s21r=4 s21I=5 s12r=6 s12I=7 s22r=8 s22I=9 .enddata .param freq=100m, s11r=0, s11i=0, s21r=0, s21i=0, s12r=0, s12i=0, s22r=0, s22i=0 .end F.2.2. Method-of-Moments S-Parameter Data freq 1.00E+08 1.10E+08 1.19E+08 1.29E+08 1.39E+08 1.49E+08 1.58E+08 1.68E+08 1.78E+08 2.17E+08 2.56E+08 2.95E+08 3.34E+08 3.73E+08 4.11E+08 4.50E+08 4.89E+08 5.28E+08 5.67E+08 6.06E+08 6.45E+08 6.84E+08 7.23E+08 7.62E+08 8.01E+08 8.40E+08 8.79E+08 9.18E+08 MoM S11(R) 1.99E-02 2.02E-02 2.05E-02 2.06E-02 2.07E-02 2.07E-02 2.06E-02 2.05E-02 2.05E-02 2.06E-02 2.14E-02 2.21E-02 2.28E-02 2.34E-02 2.41E-02 2.48E-02 2.56E-02 2.65E-02 2.75E-02 2.86E-02 2.99E-02 3.13E-02 3.27E-02 3.41E-02 3.56E-02 3.71E-02 3.86E-02 4.02E-02 MoM S11(I) 7.95E-03 8.82E-03 9.66E-03 1.05E-02 1.12E-02 1.20E-02 1.27E-02 1.34E-02 1.42E-02 1.71E-02 2.03E-02 2.34E-02 2.64E-02 2.94E-02 3.23E-02 3.51E-02 3.80E-02 4.08E-02 4.37E-02 4.65E-02 4.92E-02 5.20E-02 5.47E-02 5.73E-02 5.99E-02 6.25E-02 6.50E-02 6.75E-02 304 MoM S21(R) 9.80E-01 9.80E-01 9.80E-01 9.79E-01 9.79E-01 9.79E-01 9.79E-01 9.79E-01 9.79E-01 9.78E-01 9.77E-01 9.76E-01 9.75E-01 9.74E-01 9.73E-01 9.71E-01 9.70E-01 9.68E-01 9.66E-01 9.64E-01 9.62E-01 9.60E-01 9.57E-01 9.55E-01 9.52E-01 9.50E-01 9.47E-01 9.44E-01 MoM S21(I) -2.74E-02 -3.00E-02 -3.27E-02 -3.53E-02 -3.80E-02 -4.07E-02 -4.34E-02 -4.61E-02 -4.88E-02 -5.95E-02 -7.01E-02 -8.07E-02 -9.13E-02 -1.02E-01 -1.12E-01 -1.23E-01 -1.33E-01 -1.44E-01 -1.54E-01 -1.64E-01 -1.75E-01 -1.85E-01 -1.95E-01 -2.05E-01 -2.16E-01 -2.26E-01 -2.36E-01 -2.46E-01 9.57E+08 9.95E+08 1.03E+09 1.07E+09 1.11E+09 1.15E+09 1.19E+09 1.23E+09 1.27E+09 1.31E+09 1.35E+09 1.38E+09 1.42E+09 1.46E+09 1.50E+09 1.54E+09 1.58E+09 1.62E+09 1.66E+09 1.70E+09 1.74E+09 1.77E+09 1.81E+09 1.85E+09 1.89E+09 1.93E+09 1.97E+09 2.01E+09 2.05E+09 2.09E+09 2.12E+09 2.16E+09 2.20E+09 2.24E+09 2.28E+09 2.32E+09 2.36E+09 2.40E+09 2.44E+09 2.47E+09 2.51E+09 2.55E+09 2.59E+09 2.65E+09 2.70E+09 2.75E+09 2.81E+09 4.18E-02 4.34E-02 4.50E-02 4.67E-02 4.85E-02 5.02E-02 5.20E-02 5.39E-02 5.58E-02 5.77E-02 5.96E-02 6.17E-02 6.37E-02 6.58E-02 6.79E-02 7.00E-02 7.22E-02 7.45E-02 7.67E-02 7.90E-02 8.14E-02 8.37E-02 8.61E-02 8.86E-02 9.10E-02 9.35E-02 9.60E-02 9.85E-02 1.01E-01 1.04E-01 1.06E-01 1.09E-01 1.11E-01 1.14E-01 1.17E-01 1.19E-01 1.22E-01 1.25E-01 1.27E-01 1.30E-01 1.33E-01 1.35E-01 1.38E-01 1.42E-01 1.45E-01 1.49E-01 1.53E-01 7.00E-02 7.23E-02 7.47E-02 7.70E-02 7.92E-02 8.14E-02 8.36E-02 8.57E-02 8.78E-02 8.98E-02 9.18E-02 9.37E-02 9.56E-02 9.74E-02 9.92E-02 1.01E-01 1.03E-01 1.04E-01 1.06E-01 1.08E-01 1.09E-01 1.11E-01 1.12E-01 1.13E-01 1.15E-01 1.16E-01 1.17E-01 1.18E-01 1.19E-01 1.20E-01 1.21E-01 1.22E-01 1.23E-01 1.24E-01 1.25E-01 1.26E-01 1.26E-01 1.27E-01 1.27E-01 1.28E-01 1.28E-01 1.29E-01 1.29E-01 1.29E-01 1.29E-01 1.30E-01 1.30E-01 305 9.41E-01 9.38E-01 9.35E-01 9.31E-01 9.28E-01 9.24E-01 9.21E-01 9.17E-01 9.13E-01 9.10E-01 9.06E-01 9.01E-01 8.97E-01 8.93E-01 8.89E-01 8.84E-01 8.80E-01 8.75E-01 8.70E-01 8.65E-01 8.60E-01 8.55E-01 8.50E-01 8.45E-01 8.40E-01 8.34E-01 8.29E-01 8.23E-01 8.18E-01 8.12E-01 8.06E-01 8.00E-01 7.94E-01 7.88E-01 7.82E-01 7.76E-01 7.70E-01 7.63E-01 7.57E-01 7.50E-01 7.44E-01 7.37E-01 7.31E-01 7.21E-01 7.12E-01 7.02E-01 6.93E-01 -2.56E-01 -2.66E-01 -2.75E-01 -2.85E-01 -2.95E-01 -3.05E-01 -3.14E-01 -3.24E-01 -3.34E-01 -3.43E-01 -3.53E-01 -3.62E-01 -3.71E-01 -3.81E-01 -3.90E-01 -3.99E-01 -4.08E-01 -4.17E-01 -4.26E-01 -4.35E-01 -4.44E-01 -4.53E-01 -4.62E-01 -4.70E-01 -4.79E-01 -4.87E-01 -4.96E-01 -5.04E-01 -5.13E-01 -5.21E-01 -5.29E-01 -5.37E-01 -5.45E-01 -5.53E-01 -5.61E-01 -5.69E-01 -5.77E-01 -5.85E-01 -5.92E-01 -6.00E-01 -6.07E-01 -6.15E-01 -6.22E-01 -6.32E-01 -6.42E-01 -6.52E-01 -6.61E-01 2.86E+09 2.91E+09 2.97E+09 3.02E+09 3.07E+09 3.13E+09 3.18E+09 3.23E+09 3.29E+09 3.34E+09 3.39E+09 3.45E+09 3.50E+09 3.56E+09 3.61E+09 3.66E+09 3.72E+09 3.77E+09 3.82E+09 3.88E+09 3.93E+09 3.98E+09 4.04E+09 4.09E+09 4.14E+09 4.20E+09 4.25E+09 4.30E+09 4.36E+09 4.41E+09 4.47E+09 4.52E+09 4.57E+09 4.63E+09 4.68E+09 4.73E+09 4.84E+09 4.95E+09 5.05E+09 5.16E+09 5.27E+09 5.38E+09 5.48E+09 5.59E+09 5.70E+09 5.80E+09 5.91E+09 1.57E-01 1.60E-01 1.64E-01 1.68E-01 1.71E-01 1.75E-01 1.79E-01 1.82E-01 1.86E-01 1.90E-01 1.93E-01 1.97E-01 2.00E-01 2.04E-01 2.07E-01 2.10E-01 2.14E-01 2.17E-01 2.20E-01 2.23E-01 2.26E-01 2.29E-01 2.32E-01 2.35E-01 2.38E-01 2.41E-01 2.44E-01 2.47E-01 2.49E-01 2.52E-01 2.54E-01 2.56E-01 2.59E-01 2.61E-01 2.63E-01 2.65E-01 2.69E-01 2.72E-01 2.75E-01 2.78E-01 2.80E-01 2.82E-01 2.84E-01 2.85E-01 2.86E-01 2.86E-01 2.86E-01 1.30E-01 1.29E-01 1.29E-01 1.29E-01 1.28E-01 1.28E-01 1.27E-01 1.26E-01 1.25E-01 1.24E-01 1.23E-01 1.22E-01 1.21E-01 1.19E-01 1.18E-01 1.16E-01 1.15E-01 1.13E-01 1.11E-01 1.09E-01 1.07E-01 1.05E-01 1.03E-01 1.00E-01 9.80E-02 9.55E-02 9.30E-02 9.04E-02 8.77E-02 8.49E-02 8.21E-02 7.92E-02 7.63E-02 7.32E-02 7.02E-02 6.70E-02 6.06E-02 5.39E-02 4.71E-02 4.01E-02 3.30E-02 2.58E-02 1.85E-02 1.10E-02 3.57E-03 -3.93E-03 -1.14E-02 306 6.83E-01 6.73E-01 6.63E-01 6.53E-01 6.43E-01 6.32E-01 6.22E-01 6.11E-01 6.01E-01 5.90E-01 5.79E-01 5.68E-01 5.57E-01 5.46E-01 5.35E-01 5.23E-01 5.12E-01 5.01E-01 4.89E-01 4.77E-01 4.66E-01 4.54E-01 4.42E-01 4.30E-01 4.19E-01 4.07E-01 3.95E-01 3.83E-01 3.70E-01 3.58E-01 3.46E-01 3.34E-01 3.21E-01 3.09E-01 2.97E-01 2.84E-01 2.59E-01 2.34E-01 2.09E-01 1.84E-01 1.58E-01 1.32E-01 1.07E-01 8.11E-02 5.54E-02 2.96E-02 3.73E-03 -6.71E-01 -6.80E-01 -6.89E-01 -6.98E-01 -7.07E-01 -7.15E-01 -7.24E-01 -7.32E-01 -7.40E-01 -7.48E-01 -7.56E-01 -7.63E-01 -7.71E-01 -7.78E-01 -7.85E-01 -7.92E-01 -7.99E-01 -8.06E-01 -8.12E-01 -8.19E-01 -8.25E-01 -8.31E-01 -8.37E-01 -8.42E-01 -8.48E-01 -8.53E-01 -8.59E-01 -8.64E-01 -8.68E-01 -8.73E-01 -8.78E-01 -8.82E-01 -8.86E-01 -8.90E-01 -8.94E-01 -8.98E-01 -9.05E-01 -9.11E-01 -9.17E-01 -9.22E-01 -9.26E-01 -9.30E-01 -9.33E-01 -9.36E-01 -9.38E-01 -9.39E-01 -9.39E-01 6.02E+09 6.12E+09 6.23E+09 6.34E+09 6.45E+09 6.55E+09 6.66E+09 6.77E+09 6.87E+09 6.93E+09 6.98E+09 7.03E+09 7.09E+09 7.14E+09 7.20E+09 7.25E+09 7.30E+09 7.36E+09 7.41E+09 7.46E+09 7.52E+09 7.57E+09 7.62E+09 7.68E+09 7.73E+09 7.78E+09 7.84E+09 7.89E+09 7.94E+09 8.00E+09 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2.41E-01 2.37E-01 2.34E-01 2.30E-01 2.27E-01 2.23E-01 2.19E-01 2.15E-01 2.11E-01 2.07E-01 2.03E-01 1.99E-01 1.95E-01 1.91E-01 1.87E-01 1.82E-01 1.78E-01 1.74E-01 1.70E-01 -1.20E-01 -1.22E-01 -1.25E-01 -1.27E-01 -1.30E-01 -1.32E-01 -1.35E-01 -1.37E-01 -1.39E-01 -1.41E-01 -1.43E-01 -1.44E-01 -1.46E-01 -1.47E-01 -1.49E-01 -1.50E-01 -1.51E-01 -1.52E-01 -1.53E-01 -1.54E-01 -1.54E-01 3.83E-01 3.96E-01 4.08E-01 4.21E-01 4.33E-01 4.46E-01 4.58E-01 4.70E-01 4.83E-01 4.95E-01 5.07E-01 5.19E-01 5.31E-01 5.43E-01 5.55E-01 5.66E-01 5.78E-01 5.90E-01 6.01E-01 6.12E-01 6.24E-01 F.3. Test Structure 2 F.3.1. Circuit Optimization Input File .option accurate dccap=1 node nopage ingold=2 post acct=2 probe * u circuit 10 mil in length .subckt mstlc1 1 6 5 10 r1l 1 2 rsq l1l 2 3 lsq c1 3 0 csq r1r 3 4 rsq l1r 4 5 lsq rg1 3 0 10mega ccs1 1 3 ccsq ccs2 3 5 ccsq ccouple2 3 8 c_cou r2l 6 7 rsq l2l 7 8 lsq c2 8 0 csq rg2 8 0 10mega l2r 8 9 lsq r2r 9 10 rsq ccs3 6 8 ccsq ccs4 8 10 ccsq k1 l1l l2l k=cou_l 312 8.64E-01 8.59E-01 8.54E-01 8.48E-01 8.42E-01 8.36E-01 8.30E-01 8.24E-01 8.18E-01 8.11E-01 8.04E-01 7.97E-01 7.90E-01 7.83E-01 7.75E-01 7.67E-01 7.59E-01 7.51E-01 7.43E-01 7.35E-01 7.26E-01 k2 l2r l1r k=cou_l .ends .subckt mstl_corner 1 5 r1 1 2 r2 l1 2 3 l2 c1 3 0 c2 r2 3 0 10g r1r 3 4 r2 l2r 4 5 l2 cc1 1 3 c_cou2cr cc2 3 5 c_cou2cr *cc3 1 5 c_cou3cr *k1 l1 l2r k=cou_lcr .ends .subckt mstl_sq 1 5 r1 1 2 rsq l1 2 3 lsq c1 3 0 csq r2 3 0 10g r1r 3 4 rsq l2r 4 5 lsq cc1 1 3 ccsq cc2 3 5 ccsq .ends .subckt mstlc5 1 2 11 12 x1 1 2 3 4 mstlc1 x2 3 4 5 6 mstlc1 x3 5 6 7 8 mstlc1 x4 7 8 9 10 mstlc1 x5 9 10 11 12 mstlc1 .ends .subckt mstlc4 1 2 9 10 x1 1 2 3 4 mstlc1 x2 3 4 5 6 mstlc1 x3 5 6 7 8 mstlc1 x4 7 8 9 10 mstlc1 .ends .subckt mstl_sq5 1 6 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq x4 4 5 mstl_sq x5 5 6 mstl_sq .ends x1 x2 x3 ro 1 3 5 2 2 4 6 0 3 4 mstlc4 5 6 mstlc5 mstl_corner 1g vpl 1 0 pulse(0 2.5 2n 0.1n 0.2n 10n 20n) ac 1 *.tran 1p 40n 1p *.ac dec 60 1mega 15giga .net v(2) vpl rin=50 rout=50 .param + c_cou = opt1(2.7e-15,.1f,1n) + cou_l = opt1(0.3,0.0001,0.5) + + + + r2 = opt1(1e-1,0.00001,10) l2 = opt1(4.7e-10,.01p,1u) c2 = opt1(3.8e-14,1f,1n) c_cou2cr = opt1(2.4e-15,.1f,1n) 313 + + + + csq = opt1(2.1e-14,0.01f,1n) rsq = 5.7e-2 lsq = opt1(7.14e-11,1e-11,1e-10) ccsq = 3.53e-15 .ac data=measured optimize=opt1 + results=comp1,comp2,comp5,comp6 + model=converge .model converge opt relin=1e-4 relout=1e-4 close=200 itropt=300 .measure .measure .measure .measure .measure .measure .measure .measure ac ac ac ac ac ac ac ac comp1 comp2 comp3 comp4 comp5 comp6 comp7 comp8 err1 err1 err1 err1 err1 err1 err1 err1 par(s11r) par(s11i) par(s12r) par(s12i) par(s21r) par(s21i) par(s22r) par(s22i) s11(r) s11(i) s12(r) s12(i) s21(r) s21(i) s22(r) s22(i) .ac data=measured .print par(s11r) s11(r) par(s11i) s11(i) .print par(s12r) s12(r) par(s12i) s12(i) .print par(s21r) s21(r) par(s21i) s21(i) .print par(s22r) s22(r) par(s22i) s22(i) *.param s11db='20*log(sqrt((par(s11r))^2+(par(s11i))^2))' *.param s21db='20*log(sqrt((par(s21r))^2+(par(s21i))^2))' .print s11(db) .print s12(db) .print s21(db) .print s22(db) .data measured file = ‘ubend.txt’ freq=1 s11r=2 s11I=3 s21r=4 s21I=5 s12r=6 s12I=7 s22r=8 s22I=9 .enddata .param freq=100m, s11r=0, s11i=0, s21r=0, s21i=0, s12r=0, s12i=0, s22r=0, s22i=0 .end F.3.2. Method-of-Moments S-Parameter Data freq 1.00E+08 1.39E+08 1.78E+08 2.17E+08 2.55E+08 2.94E+08 3.33E+08 3.72E+08 4.11E+08 4.50E+08 4.89E+08 5.28E+08 5.66E+08 6.05E+08 MoM S11(R) 2.22E-02 2.24E-02 2.26E-02 2.29E-02 2.33E-02 2.37E-02 2.41E-02 2.46E-02 2.51E-02 2.56E-02 2.62E-02 2.68E-02 2.74E-02 2.81E-02 MoM S11(I) 4.20E-03 5.81E-03 7.40E-03 8.97E-03 1.05E-02 1.20E-02 1.35E-02 1.50E-02 1.64E-02 1.78E-02 1.91E-02 2.05E-02 2.18E-02 2.30E-02 314 MoM S21(R) 9.78E-01 9.77E-01 9.77E-01 9.76E-01 9.76E-01 9.75E-01 9.74E-01 9.73E-01 9.72E-01 9.71E-01 9.69E-01 9.68E-01 9.67E-01 9.65E-01 MoM S21(I) -2.33E-02 -3.24E-02 -4.14E-02 -5.04E-02 -5.94E-02 -6.83E-02 -7.73E-02 -8.61E-02 -9.50E-02 -1.04E-01 -1.13E-01 -1.21E-01 -1.30E-01 -1.39E-01 6.44E+08 6.83E+08 7.22E+08 8.00E+08 8.77E+08 9.55E+08 1.03E+09 1.11E+09 1.19E+09 1.27E+09 1.34E+09 1.42E+09 1.50E+09 1.58E+09 1.65E+09 1.73E+09 1.81E+09 1.89E+09 1.97E+09 2.04E+09 2.12E+09 2.20E+09 2.28E+09 2.35E+09 2.43E+09 2.51E+09 2.59E+09 2.67E+09 2.74E+09 2.82E+09 2.90E+09 2.98E+09 3.05E+09 3.13E+09 3.21E+09 3.29E+09 3.36E+09 3.44E+09 3.52E+09 3.60E+09 3.68E+09 3.75E+09 3.83E+09 3.91E+09 3.99E+09 4.06E+09 2.87E-02 2.94E-02 3.01E-02 3.15E-02 3.30E-02 3.45E-02 3.61E-02 3.77E-02 3.93E-02 4.10E-02 4.27E-02 4.44E-02 4.62E-02 4.80E-02 4.98E-02 5.17E-02 5.35E-02 5.54E-02 5.73E-02 5.93E-02 6.12E-02 6.32E-02 6.52E-02 6.72E-02 6.92E-02 7.12E-02 7.32E-02 7.52E-02 7.72E-02 7.92E-02 8.12E-02 8.31E-02 8.50E-02 8.69E-02 8.88E-02 9.06E-02 9.24E-02 9.41E-02 9.57E-02 9.73E-02 9.88E-02 1.00E-01 1.02E-01 1.03E-01 1.04E-01 1.05E-01 2.42E-02 2.54E-02 2.66E-02 2.88E-02 3.10E-02 3.30E-02 3.50E-02 3.68E-02 3.86E-02 4.03E-02 4.18E-02 4.34E-02 4.48E-02 4.61E-02 4.74E-02 4.86E-02 4.97E-02 5.07E-02 5.16E-02 5.24E-02 5.32E-02 5.38E-02 5.44E-02 5.48E-02 5.52E-02 5.55E-02 5.56E-02 5.57E-02 5.57E-02 5.56E-02 5.53E-02 5.50E-02 5.45E-02 5.40E-02 5.33E-02 5.26E-02 5.17E-02 5.08E-02 4.97E-02 4.85E-02 4.73E-02 4.59E-02 4.44E-02 4.29E-02 4.12E-02 3.94E-02 315 9.64E-01 9.62E-01 9.60E-01 9.57E-01 9.53E-01 9.48E-01 9.44E-01 9.39E-01 9.34E-01 9.29E-01 9.23E-01 9.17E-01 9.11E-01 9.05E-01 8.98E-01 8.91E-01 8.84E-01 8.76E-01 8.68E-01 8.60E-01 8.52E-01 8.43E-01 8.34E-01 8.24E-01 8.15E-01 8.05E-01 7.94E-01 7.84E-01 7.73E-01 7.62E-01 7.50E-01 7.39E-01 7.26E-01 7.14E-01 7.01E-01 6.88E-01 6.75E-01 6.61E-01 6.47E-01 6.33E-01 6.19E-01 6.04E-01 5.89E-01 5.73E-01 5.57E-01 5.41E-01 -1.48E-01 -1.56E-01 -1.65E-01 -1.82E-01 -1.99E-01 -2.16E-01 -2.33E-01 -2.49E-01 -2.66E-01 -2.83E-01 -2.99E-01 -3.15E-01 -3.32E-01 -3.48E-01 -3.64E-01 -3.80E-01 -3.96E-01 -4.12E-01 -4.28E-01 -4.43E-01 -4.59E-01 -4.74E-01 -4.89E-01 -5.05E-01 -5.20E-01 -5.35E-01 -5.49E-01 -5.64E-01 -5.78E-01 -5.93E-01 -6.07E-01 -6.21E-01 -6.35E-01 -6.48E-01 -6.62E-01 -6.75E-01 -6.88E-01 -7.01E-01 -7.14E-01 -7.26E-01 -7.39E-01 -7.51E-01 -7.62E-01 -7.74E-01 -7.85E-01 -7.96E-01 4.14E+09 4.22E+09 4.30E+09 4.38E+09 4.45E+09 4.49E+09 4.53E+09 4.57E+09 4.61E+09 4.65E+09 4.69E+09 4.73E+09 4.76E+09 4.80E+09 4.84E+09 4.88E+09 4.92E+09 4.96E+09 5.00E+09 5.04E+09 5.08E+09 5.19E+09 5.31E+09 5.42E+09 5.54E+09 5.66E+09 5.77E+09 5.89E+09 6.01E+09 6.07E+09 6.12E+09 6.18E+09 6.24E+09 6.30E+09 6.36E+09 6.42E+09 6.47E+09 6.53E+09 6.59E+09 6.65E+09 6.71E+09 6.77E+09 6.82E+09 6.88E+09 6.94E+09 7.00E+09 1.06E-01 1.07E-01 1.07E-01 1.08E-01 1.08E-01 1.08E-01 1.09E-01 1.09E-01 1.09E-01 1.09E-01 1.09E-01 1.09E-01 1.08E-01 1.08E-01 1.08E-01 1.08E-01 1.07E-01 1.07E-01 1.07E-01 1.06E-01 1.06E-01 1.04E-01 1.02E-01 9.92E-02 9.60E-02 9.24E-02 8.82E-02 8.36E-02 7.84E-02 7.56E-02 7.28E-02 6.97E-02 6.66E-02 6.33E-02 6.00E-02 5.65E-02 5.28E-02 4.91E-02 4.53E-02 4.14E-02 3.73E-02 3.32E-02 2.90E-02 2.47E-02 2.03E-02 1.58E-02 3.76E-02 3.57E-02 3.37E-02 3.16E-02 2.94E-02 2.83E-02 2.72E-02 2.61E-02 2.49E-02 2.38E-02 2.26E-02 2.14E-02 2.03E-02 1.91E-02 1.79E-02 1.66E-02 1.54E-02 1.42E-02 1.30E-02 1.17E-02 1.05E-02 6.78E-03 3.08E-03 -5.72E-04 -4.12E-03 -7.53E-03 -1.08E-02 -1.38E-02 -1.65E-02 -1.77E-02 -1.89E-02 -1.99E-02 -2.08E-02 -2.17E-02 -2.24E-02 -2.30E-02 -2.34E-02 -2.38E-02 -2.39E-02 -2.40E-02 -2.38E-02 -2.35E-02 -2.30E-02 -2.23E-02 -2.14E-02 -2.04E-02 316 5.25E-01 5.08E-01 4.91E-01 4.74E-01 4.56E-01 4.47E-01 4.38E-01 4.29E-01 4.20E-01 4.11E-01 4.02E-01 3.92E-01 3.83E-01 3.73E-01 3.64E-01 3.54E-01 3.44E-01 3.35E-01 3.25E-01 3.15E-01 3.05E-01 2.74E-01 2.43E-01 2.11E-01 1.79E-01 1.46E-01 1.12E-01 7.79E-02 4.32E-02 2.56E-02 7.98E-03 -9.81E-03 -2.77E-02 -4.57E-02 -6.38E-02 -8.20E-02 -1.00E-01 -1.19E-01 -1.37E-01 -1.56E-01 -1.74E-01 -1.93E-01 -2.12E-01 -2.30E-01 -2.49E-01 -2.68E-01 -8.07E-01 -8.18E-01 -8.28E-01 -8.38E-01 -8.48E-01 -8.52E-01 -8.57E-01 -8.62E-01 -8.66E-01 -8.71E-01 -8.75E-01 -8.79E-01 -8.83E-01 -8.87E-01 -8.91E-01 -8.95E-01 -8.99E-01 -9.03E-01 -9.06E-01 -9.10E-01 -9.13E-01 -9.23E-01 -9.32E-01 -9.40E-01 -9.47E-01 -9.53E-01 -9.57E-01 -9.61E-01 -9.63E-01 -9.64E-01 -9.65E-01 -9.65E-01 -9.64E-01 -9.64E-01 -9.63E-01 -9.62E-01 -9.60E-01 -9.58E-01 -9.56E-01 -9.53E-01 -9.50E-01 -9.46E-01 -9.42E-01 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1.92E+10 1.93E+10 1.93E+10 1.94E+10 1.95E+10 1.95E+10 1.96E+10 1.96E+10 1.97E+10 1.97E+10 1.98E+10 1.98E+10 1.99E+10 1.99E+10 2.00E+10 2.00E+10 2.01E+10 2.93E-01 2.92E-01 2.91E-01 2.89E-01 2.88E-01 2.86E-01 2.85E-01 2.83E-01 2.81E-01 2.79E-01 2.77E-01 2.75E-01 2.72E-01 2.70E-01 2.67E-01 2.65E-01 2.62E-01 2.59E-01 2.56E-01 2.53E-01 2.50E-01 2.47E-01 2.44E-01 2.41E-01 2.37E-01 2.34E-01 2.30E-01 2.27E-01 2.23E-01 2.19E-01 2.15E-01 2.11E-01 2.07E-01 2.03E-01 1.99E-01 1.95E-01 1.91E-01 1.87E-01 1.82E-01 1.78E-01 1.74E-01 1.70E-01 -4.55E-02 -4.95E-02 -5.35E-02 -5.74E-02 -6.14E-02 -6.52E-02 -6.91E-02 -7.29E-02 -7.66E-02 -8.03E-02 -8.40E-02 -8.75E-02 -9.11E-02 -9.45E-02 -9.79E-02 -1.01E-01 -1.05E-01 -1.08E-01 -1.11E-01 -1.14E-01 -1.17E-01 -1.20E-01 -1.22E-01 -1.25E-01 -1.27E-01 -1.30E-01 -1.32E-01 -1.35E-01 -1.37E-01 -1.39E-01 -1.41E-01 -1.43E-01 -1.44E-01 -1.46E-01 -1.47E-01 -1.49E-01 -1.50E-01 -1.51E-01 -1.52E-01 -1.53E-01 -1.54E-01 -1.54E-01 322 1.10E-01 1.23E-01 1.36E-01 1.49E-01 1.63E-01 1.76E-01 1.89E-01 2.02E-01 2.15E-01 2.28E-01 2.41E-01 2.54E-01 2.67E-01 2.80E-01 2.93E-01 3.06E-01 3.19E-01 3.32E-01 3.45E-01 3.58E-01 3.70E-01 3.83E-01 3.96E-01 4.08E-01 4.21E-01 4.33E-01 4.46E-01 4.58E-01 4.70E-01 4.83E-01 4.95E-01 5.07E-01 5.19E-01 5.31E-01 5.43E-01 5.55E-01 5.66E-01 5.78E-01 5.90E-01 6.01E-01 6.12E-01 6.24E-01 9.32E-01 9.30E-01 9.28E-01 9.27E-01 9.25E-01 9.22E-01 9.20E-01 9.18E-01 9.15E-01 9.12E-01 9.09E-01 9.06E-01 9.02E-01 8.99E-01 8.95E-01 8.91E-01 8.87E-01 8.83E-01 8.78E-01 8.74E-01 8.69E-01 8.64E-01 8.59E-01 8.54E-01 8.48E-01 8.42E-01 8.36E-01 8.30E-01 8.24E-01 8.18E-01 8.11E-01 8.04E-01 7.97E-01 7.90E-01 7.83E-01 7.75E-01 7.67E-01 7.59E-01 7.51E-01 7.43E-01 7.35E-01 7.26E-01 F.4. Test Structure 3 F.4.1. Circuit Optimization Input File .option accurate node nopage ingold=2 post acct=2 probe *microstrip coupled 2 lines .subckt mstl_c2 1 6 5 10 r1l 1 2 rsq l1l 2 3 lsq c1 3 0 csq r1r 3 4 rsq l1r 4 5 lsq rg1 3 0 10mega ccs1 1 3 ccsq ccs2 3 5 ccsq *ccouple1 1 6 c_cou ccouple2 3 8 c_cou_line r2l 6 7 rsq l2l 7 8 lsq c2 8 0 csq rg2 8 0 10mega l2r 8 9 lsq r2r 9 10 rsq ccs3 6 8 ccsq ccs4 8 10 ccsq k1 l1l l2l k=cou_l_line k2 l2r l1r k=cou_l_line .param + rsq = 5.2e-2 + lsq = 5.74e-11 + csq = 2.11e-14 + ccsq = 3.53e-14 + c_cou = c_cou_line + cou_l = cou_l_line .ends *set of 5 microstrip coupled 2 lines .subckt mstl_c2_5 1 2 11 12 x1 1 2 3 4 mstl_c2 x2 3 4 5 6 mstl_c2 x3 5 6 7 8 mstl_c2 x4 7 8 9 10 mstl_c2 x5 9 10 11 12 mstl_c2 .ends *set of 6 microstrip coupled 2 lines .subckt mstl_c2_6 1 2 13 14 x1 1 2 3 4 mstl_c2 x2 3 4 5 6 mstl_c2 x3 5 6 7 8 mstl_c2 x4 7 8 9 10 mstl_c2 x5 9 10 11 12 mstl_c2 x6 11 12 13 14 mstl_c2 .ends *set of 4 microstrip coupled 2 lines 323 .subckt mstl_c2_4 1 2 9 10 x1 1 2 3 4 mstl_c2 x2 3 4 5 6 mstl_c2 x3 5 6 7 8 mstl_c2 x4 7 8 9 10 mstl_c2 .ends *set of 3 microstrip coupled 2 lines .subckt mstl_c2_3 1 2 7 8 x1 1 2 3 4 mstl_c2 x2 3 4 5 6 mstl_c2 x3 5 6 7 8 mstl_c2 .ends *set of 2 microstrip coupled 2 lines .subckt mstl_c2_2 1 2 5 6 x1 1 2 3 4 mstl_c2 x2 3 4 5 6 mstl_c2 .ends ************************************************************* *microstrip striaght line .subckt mstl_sq 1 5 r1 1 2 rsq l1 2 3 lsq c1 3 0 csq r2 3 0 10g r1r 3 4 rsq l2r 4 5 lsq cc1 1 3 ccsq cc2 3 5 ccsq .param + rsq = 5.2e-2 + lsq = 7.14e-11 + csq = 3.11e-14 + ccsq = 3.53e-17 .ends *set of 5 microstrip blocks .subckt mstl_sq_5 1 6 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq x4 4 5 mstl_sq x5 5 6 mstl_sq .ends *set of 4 microstrip blocks .subckt mstl_sq_4 1 5 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq x4 4 5 mstl_sq .ends *set of 3 microstrip blocks .subckt mstl_sq_3 1 4 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq .ends *set of 2 microstrip blocks .subckt mstl_sq_2 1 3 x1 1 2 mstl_sq x2 2 3 mstl_sq 324 .ends ********************************************************* *corner bend (shape of l with 3 blocks) .subckt mstl_corn_l 1 3 x1 1 2 mstl_corn_l_half x2 2 3 mstl_corn_l_half .ends *composite corner (u shaped made of 5 squares) .subckt mstl_corn_u 1 5 r1 1 2 r2 l1 2 3 l2 c1 3 0 c2 r2 3 0 10g r2r 3 4 r2 l2r 4 5 l2 cc1 1 3 c_cou2cr cc2 3 5 c_cou2cr .param + r2 = 4.4e-1 + l2 = 2.834e-10 + c2 = 1.8e-13 + c_cou2cr = 5.477e-14 .ends .subckt corn_st 1 6 15 10 r1l 1 2 rsq l1l 2 3 lsq c1 3 0 csq r1r 3 4 rsq l1r 4 5 lsq rg1 3 0 10mega ccs1 1 3 ccsq ccs2 3 5 ccsq ccouple2 3 8 c_cou *k1 l1l l2l k=cou_l *k2 l1r l2r k=cou_l k23a l2l l3l k=cou_l k23b l2r l3r k=cou_l r2l 6 7 r2 l2l 7 8 l2 c2 8 0 c2 rg2 8 0 10mega l2r 8 9 l2 r2r 9 10 r2 ccs3 6 8 c_cou2cr ccs4 8 10 c_cou2cr r3l 5 12 r2 l3l 12 13 l2 c3 13 0 c2 rg3 13 0 10mega l3r 13 14 l2 r3r 14 15 r2 ccs6 5 13 c_cou2cr ccs7 13 15 c_cou2cr *k3 l3l l2r k=cou_l_2 *lshunt1 5 15 ls1 *lshunt2 6 10 ls1 *kshunt lshunt1 lshunt2 k=cou_l_2 *ct 15 10 ctest .param *+ rsq = .107 325 *+ lsq = 1.428e-10 *+ csq = 6.206e-14 *+ ccsq = 4.47e-14 + rsq = 0.11 + lsq = 11.4e-11 *+ csq = 4.2e-14 +ccsq = 1e-14 *+ r2 = 1.1 + c_cou2cr = 2.332e-13 .ends .subckt mstl_corn_lcomp 1 2 8 7 x1 1 2 3 4 corn_st x4 8 7 3 4 corn_st .ends x1 x2 x3 x4 x5 x6 1 2 3 4 mstl_c2_5 3 4 5 6 mstl_c2_2 5 6 7 8 mstl_corn_lcomp 7 8 9 10 mstl_c2_5 9 10 11 12 mstl_c2 11 12 mstl_corn_u ro 2 0 1g vpl 1 0 pulse(0 2.5 2n 0.1n 0.2n 10n 20n) ac 1 *.tran 1p 40n 1p *.ac dec 60 1mega 15giga .net v(2) vpl rin=50 rout=50 .param + c_cou = opt1(1.0e-15,.01f,1n) + cou_l = opt1(0.20,0.0001,1) + l2 = opt1(2e-10,1e-11,1e-9) + c2 = opt1(2e-14,1e-15,1e-13) + c_cou_line = opt1(4e-15,1e-15,1e-14) + cou_l_line = opt1(0.32,0.1,1) + r2 = opt1(0.37,0.01,4) + csq = 3.7e-14 .ac data=measured optimize=opt1 + results=comp1,comp2,comp5,comp6 + model=converge .model converge opt relin=1e-2 relout=1e-2 close=200 itropt=300 .measure .measure .measure .measure .measure .measure .print .print .print .print .print .print .print .print ac ac ac ac ac ac comp1 comp2 comp5 comp6 comp7 comp8 par(s11r) par(s12r) par(s21r) par(s22r) s11(db) s12(db) s21(db) s22(db) err1 err1 err1 err1 err1 err1 s11(r) s12(r) s21(r) s22(r) par(s11r) par(s11i) par(s21r) par(s21i) par(s22r) par(s22i) par(s11i) par(s12i) par(s21i) par(s22i) s11(r) s11(i) s21(r) s21(i) s22(r) s22(i) s11(i) s12(i) s21(i) s22(i) .data measured file = ‘ts3.txt’ freq=1 s11r=2 s11I=3 s21r=4 s21I=5 s12r=6 s12I=7 s22r=8 s22I=9 .enddata 326 .param freq=100m, s11r=0, s11i=0, s21r=0, s21i=0, s12r=0, s12i=0, s22r=0, s22i=0 .end F.4.2. Method-o-Moments S-Parameter Data freq 1.00E+08 1.39E+08 1.78E+08 2.17E+08 2.55E+08 2.94E+08 3.33E+08 3.72E+08 4.11E+08 4.50E+08 4.89E+08 5.28E+08 5.66E+08 6.05E+08 6.44E+08 6.83E+08 7.22E+08 7.61E+08 8.00E+08 8.38E+08 8.77E+08 9.16E+08 9.55E+08 9.94E+08 1.03E+09 1.07E+09 1.11E+09 1.15E+09 1.19E+09 1.23E+09 1.27E+09 1.30E+09 1.34E+09 1.38E+09 1.42E+09 1.46E+09 1.50E+09 1.54E+09 1.58E+09 1.62E+09 MoM S11(R) 3.92E-02 3.96E-02 4.01E-02 4.07E-02 4.15E-02 4.24E-02 4.33E-02 4.44E-02 4.56E-02 4.68E-02 4.82E-02 4.96E-02 5.10E-02 5.26E-02 5.41E-02 5.58E-02 5.74E-02 5.92E-02 6.09E-02 6.27E-02 6.45E-02 6.64E-02 6.82E-02 7.02E-02 7.21E-02 7.41E-02 7.61E-02 7.82E-02 8.02E-02 8.21E-02 8.40E-02 8.58E-02 8.77E-02 8.95E-02 9.14E-02 9.32E-02 9.51E-02 9.69E-02 9.88E-02 1.01E-01 MoM S11(I) 6.97E-03 9.64E-03 1.23E-02 1.49E-02 1.74E-02 1.99E-02 2.23E-02 2.46E-02 2.69E-02 2.91E-02 3.12E-02 3.32E-02 3.51E-02 3.69E-02 3.87E-02 4.03E-02 4.19E-02 4.34E-02 4.47E-02 4.60E-02 4.72E-02 4.84E-02 4.94E-02 5.03E-02 5.12E-02 5.21E-02 5.29E-02 5.36E-02 5.44E-02 5.51E-02 5.57E-02 5.62E-02 5.66E-02 5.68E-02 5.70E-02 5.71E-02 5.72E-02 5.71E-02 5.70E-02 5.68E-02 327 MoM S21(R) 9.60E-01 9.59E-01 9.58E-01 9.57E-01 9.55E-01 9.53E-01 9.50E-01 9.47E-01 9.45E-01 9.41E-01 9.38E-01 9.34E-01 9.30E-01 9.26E-01 9.22E-01 9.17E-01 9.12E-01 9.07E-01 9.02E-01 8.96E-01 8.90E-01 8.84E-01 8.78E-01 8.72E-01 8.65E-01 8.58E-01 8.51E-01 8.44E-01 8.37E-01 8.29E-01 8.21E-01 8.13E-01 8.05E-01 7.96E-01 7.87E-01 7.78E-01 7.69E-01 7.60E-01 7.50E-01 7.40E-01 MoM S21(I) -4.03E-02 -5.59E-02 -7.15E-02 -8.70E-02 -1.03E-01 -1.18E-01 -1.33E-01 -1.49E-01 -1.64E-01 -1.79E-01 -1.94E-01 -2.09E-01 -2.24E-01 -2.38E-01 -2.53E-01 -2.67E-01 -2.82E-01 -2.96E-01 -3.11E-01 -3.25E-01 -3.39E-01 -3.53E-01 -3.67E-01 -3.80E-01 -3.94E-01 -4.08E-01 -4.21E-01 -4.35E-01 -4.48E-01 -4.61E-01 -4.74E-01 -4.87E-01 -5.00E-01 -5.13E-01 -5.26E-01 -5.38E-01 -5.51E-01 -5.63E-01 -5.75E-01 -5.87E-01 1.65E+09 1.69E+09 1.73E+09 1.77E+09 1.81E+09 1.85E+09 1.89E+09 1.93E+09 1.97E+09 2.00E+09 2.04E+09 2.08E+09 2.12E+09 2.16E+09 2.20E+09 2.24E+09 2.28E+09 2.32E+09 2.35E+09 2.39E+09 2.43E+09 2.47E+09 2.51E+09 2.55E+09 2.59E+09 2.63E+09 2.67E+09 2.70E+09 2.74E+09 2.78E+09 2.82E+09 2.86E+09 2.90E+09 2.94E+09 2.98E+09 3.02E+09 3.05E+09 3.09E+09 3.13E+09 3.17E+09 3.21E+09 3.25E+09 3.29E+09 3.33E+09 3.36E+09 3.40E+09 3.44E+09 1.02E-01 1.04E-01 1.06E-01 1.08E-01 1.10E-01 1.11E-01 1.13E-01 1.15E-01 1.16E-01 1.18E-01 1.19E-01 1.21E-01 1.22E-01 1.24E-01 1.25E-01 1.26E-01 1.28E-01 1.29E-01 1.30E-01 1.31E-01 1.32E-01 1.33E-01 1.34E-01 1.34E-01 1.35E-01 1.36E-01 1.36E-01 1.37E-01 1.37E-01 1.37E-01 1.37E-01 1.37E-01 1.37E-01 1.37E-01 1.36E-01 1.36E-01 1.35E-01 1.35E-01 1.34E-01 1.33E-01 1.32E-01 1.30E-01 1.29E-01 1.28E-01 1.26E-01 1.24E-01 1.23E-01 5.65E-02 5.62E-02 5.58E-02 5.53E-02 5.48E-02 5.41E-02 5.34E-02 5.27E-02 5.19E-02 5.10E-02 5.00E-02 4.90E-02 4.78E-02 4.67E-02 4.54E-02 4.41E-02 4.28E-02 4.14E-02 3.99E-02 3.83E-02 3.68E-02 3.51E-02 3.34E-02 3.17E-02 2.99E-02 2.80E-02 2.61E-02 2.42E-02 2.23E-02 2.03E-02 1.83E-02 1.63E-02 1.42E-02 1.21E-02 1.01E-02 7.99E-03 5.91E-03 3.84E-03 1.78E-03 -2.72E-04 -2.30E-03 -4.30E-03 -6.26E-03 -8.19E-03 -1.01E-02 -1.19E-02 -1.37E-02 328 7.30E-01 7.20E-01 7.09E-01 6.99E-01 6.88E-01 6.77E-01 6.65E-01 6.54E-01 6.42E-01 6.30E-01 6.18E-01 6.05E-01 5.93E-01 5.80E-01 5.67E-01 5.54E-01 5.40E-01 5.26E-01 5.13E-01 4.98E-01 4.84E-01 4.70E-01 4.55E-01 4.40E-01 4.25E-01 4.09E-01 3.94E-01 3.78E-01 3.62E-01 3.46E-01 3.30E-01 3.13E-01 2.96E-01 2.79E-01 2.62E-01 2.45E-01 2.28E-01 2.10E-01 1.92E-01 1.74E-01 1.56E-01 1.38E-01 1.19E-01 1.00E-01 8.16E-02 6.26E-02 4.36E-02 -5.99E-01 -6.11E-01 -6.23E-01 -6.34E-01 -6.46E-01 -6.57E-01 -6.68E-01 -6.79E-01 -6.90E-01 -7.00E-01 -7.11E-01 -7.21E-01 -7.31E-01 -7.41E-01 -7.51E-01 -7.61E-01 -7.70E-01 -7.79E-01 -7.88E-01 -7.97E-01 -8.06E-01 -8.14E-01 -8.22E-01 -8.30E-01 -8.38E-01 -8.45E-01 -8.53E-01 -8.60E-01 -8.66E-01 -8.73E-01 -8.79E-01 -8.85E-01 -8.91E-01 -8.96E-01 -9.01E-01 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3.25E-01 3.05E-01 2.83E-01 2.62E-01 2.41E-01 2.19E-01 1.97E-01 1.75E-01 1.52E-01 1.30E-01 1.07E-01 8.39E-02 6.08E-02 3.76E-02 1.43E-02 -9.18E-03 -3.27E-02 -5.63E-02 -8.00E-02 -1.04E-01 -1.28E-01 -1.51E-01 -1.75E-01 -1.99E-01 1.29E+10 1.29E+10 1.30E+10 1.31E+10 1.31E+10 1.32E+10 1.32E+10 1.33E+10 1.34E+10 1.34E+10 1.35E+10 1.35E+10 1.36E+10 1.36E+10 1.37E+10 1.38E+10 1.38E+10 1.39E+10 1.39E+10 1.40E+10 1.41E+10 1.41E+10 1.42E+10 1.42E+10 1.43E+10 1.43E+10 1.44E+10 1.45E+10 1.45E+10 1.46E+10 1.46E+10 1.47E+10 1.47E+10 1.47E+10 1.48E+10 1.48E+10 1.48E+10 1.48E+10 1.49E+10 1.49E+10 1.49E+10 1.50E+10 1.50E+10 1.50E+10 -1.62E-01 -1.67E-01 -1.71E-01 -1.76E-01 -1.81E-01 -1.85E-01 -1.89E-01 -1.93E-01 -1.97E-01 -2.01E-01 -2.04E-01 -2.08E-01 -2.11E-01 -2.14E-01 -2.17E-01 -2.20E-01 -2.22E-01 -2.25E-01 -2.27E-01 -2.29E-01 -2.31E-01 -2.33E-01 -2.34E-01 -2.36E-01 -2.37E-01 -2.38E-01 -2.39E-01 -2.39E-01 -2.40E-01 -2.40E-01 -2.39E-01 -2.39E-01 -2.39E-01 -2.38E-01 -2.38E-01 -2.37E-01 -2.36E-01 -2.35E-01 -2.34E-01 -2.33E-01 -2.32E-01 -2.30E-01 -2.29E-01 -2.27E-01 -1.28E-01 -1.20E-01 -1.11E-01 -1.03E-01 -9.41E-02 -8.52E-02 -7.63E-02 -6.72E-02 -5.81E-02 -4.88E-02 -3.94E-02 -2.98E-02 -2.02E-02 -1.03E-02 -3.97E-04 9.70E-03 1.99E-02 3.04E-02 4.09E-02 5.17E-02 6.27E-02 7.38E-02 8.52E-02 9.68E-02 1.09E-01 1.21E-01 1.33E-01 1.46E-01 1.59E-01 1.72E-01 1.86E-01 1.93E-01 2.00E-01 2.07E-01 2.14E-01 2.21E-01 2.29E-01 2.36E-01 2.44E-01 2.51E-01 2.59E-01 2.67E-01 2.75E-01 2.83E-01 334 9.01E-01 8.95E-01 8.87E-01 8.80E-01 8.71E-01 8.62E-01 8.51E-01 8.40E-01 8.29E-01 8.16E-01 8.03E-01 7.89E-01 7.74E-01 7.58E-01 7.42E-01 7.24E-01 7.06E-01 6.87E-01 6.67E-01 6.46E-01 6.25E-01 6.02E-01 5.79E-01 5.55E-01 5.30E-01 5.04E-01 4.78E-01 4.50E-01 4.22E-01 3.93E-01 3.63E-01 3.48E-01 3.33E-01 3.18E-01 3.02E-01 2.86E-01 2.70E-01 2.54E-01 2.38E-01 2.22E-01 2.05E-01 1.89E-01 1.72E-01 1.56E-01 -2.23E-01 -2.46E-01 -2.70E-01 -2.94E-01 -3.17E-01 -3.40E-01 -3.64E-01 -3.87E-01 -4.09E-01 -4.32E-01 -4.54E-01 -4.77E-01 -4.98E-01 -5.20E-01 -5.41E-01 -5.62E-01 -5.82E-01 -6.02E-01 -6.21E-01 -6.40E-01 -6.59E-01 -6.76E-01 -6.94E-01 -7.10E-01 -7.25E-01 -7.40E-01 -7.54E-01 -7.67E-01 -7.79E-01 -7.90E-01 -8.00E-01 -8.05E-01 -8.09E-01 -8.13E-01 -8.17E-01 -8.20E-01 -8.23E-01 -8.26E-01 -8.28E-01 -8.30E-01 -8.31E-01 -8.33E-01 -8.33E-01 -8.34E-01 F.5. 4-Turn Spiral Inductor F.5.1. Circuit File for 4-Turn Spiral Inductor .option accurate node nopage ingold=2 post acct=2 probe * 4 coupled lines .subckt mstl_c4 1 6 11 16 5 10 15 20 r1l 1 2 rsq l1l 2 3 lsq c1 3 0 csq r1r 3 4 rsq l1r 4 5 lsq rg1 3 0 10mega cca 1 3 ccsq ccb 3 5 ccsq cc12a 3 8 c_cou r2l 6 7 rsq l2l 7 8 lsq c2 8 0 csq rg2 8 0 10mega l2r 8 9 lsq r2r 9 10 rsq ccb1 6 8 ccsq ccb2 8 10 ccsq k1 l1l l2l k=cou_l k2 l2r l1r k=cou_l r3l 11 12 rsq l3l 12 13 lsq c3 13 0 csq rg3 13 0 10mega l3r 13 14 lsq r3r 14 15 rsq ccc1 11 13 ccsq ccc2 13 15 ccsq cc34b 13 18 c_cou r4l 16 17 rsq l4l 17 18 lsq c4 18 0 csq rg4 18 0 10mega l4r 18 19 lsq r4r 19 20 rsq ccd1 16 18 ccsq ccd2 18 20 ccsq k34a l3l l4l k=cou_l k34b l3r l4r k=cou_l cc23b 8 13 c_cou k23a l2l l3l k=cou_l k23b l2r l3r k=cou_l .param + rsq = 5.2e-2 + lsq = 5.74e-11 + csq = 2.11e-14 + ccsq = 3.53e-14 + cou_l = 0.32 + c_cou = 5e-15 .ends .subckt mstl_c4_5 1 2 3 4 21 22 23 24 x1 1 2 3 4 5 6 7 8 mstl_c4 x2 5 6 7 8 9 10 11 12 mstl_c4 335 x3 9 10 11 12 13 14 15 16 mstl_c4 x4 13 14 15 16 17 18 19 20 mstl_c4 x5 17 18 19 20 21 22 23 24 mstl_c4 .ends .subckt mstl_c4_4 1 2 3 4 17 18 19 20 x1 1 2 3 4 5 6 7 8 mstl_c4 x2 5 6 7 8 9 10 11 12 mstl_c4 x3 9 10 11 12 13 14 15 16 mstl_c4 x4 13 14 15 16 17 18 19 20 mstl_c4 .ends .subckt mstl_c4_3 1 2 3 4 13 14 15 16 x1 1 2 3 4 5 6 7 8 mstl_c4 x2 5 6 7 8 9 10 11 12 mstl_c4 x3 9 10 11 12 13 14 15 16 mstl_c4 .ends .subckt mstl_c4_2 1 2 3 4 9 10 11 12 x1 1 2 3 4 5 6 7 8 mstl_c4 x2 5 6 7 8 9 10 11 12 mstl_c4 .ends * 3 coupled lines .subckt mstl_c3 1 6 11 r1l 1 2 rsq l1l 2 3 lsq c1 3 0 csq r1r 3 4 rsq l1r 4 5 lsq rg1 3 0 10mega cca 1 3 ccsq ccb 3 5 ccsq cc12a 3 8 c_cou r2l 6 7 rsq l2l 7 8 lsq c2 8 0 csq rg2 8 0 10mega l2r 8 9 lsq r2r 9 10 rsq ccb1 6 8 ccsq ccb2 8 10 ccsq k1 l1l l2l k=cou_l k2 l2r l1r k=cou_l r3l 11 12 rsq l3l 12 13 lsq c3 13 0 csq rg3 13 0 10mega l3r 13 14 lsq r3r 14 15 rsq ccc1 11 13 ccsq ccc2 13 15 ccsq 5 10 15 cc23b 8 13 c_cou k23a l2l l3l k=cou_l k23b l2r l3r k=cou_l .param + rsq = 5.7e-2 + lsq = 5.77e-11 + csq = 2.11e-14 + ccsq = 3.53e-16 + cou_l = 0.32 + c_cou = 5e-15 .ends * microstrip coupled 3 lines set of 5 336 .subckt mstl_c3_5 1 2 3 16 17 18 x1 1 2 3 4 5 6 mstl_c3 x2 4 5 6 7 8 9 mstl_c3 x3 7 8 9 10 11 12 mstl_c3 x4 10 11 12 13 14 15 mstl_c3 x5 13 14 15 16 17 18 mstl_c3 .ends * microstrip coupled 3 lines set of 4 .subckt mstl_c3_4 1 2 3 13 14 15 x1 1 2 3 4 5 6 mstl_c3 x2 4 5 6 7 8 9 mstl_c3 x3 7 8 9 10 11 12 mstl_c3 x4 10 11 12 13 14 15 mstl_c3 .ends * microstrip coupled 3 lines set of 3 .subckt mstl_c3_3 1 2 3 10 11 12 x1 1 2 3 4 5 6 mstl_c3 x2 4 5 6 7 8 9 mstl_c3 x3 7 8 9 10 11 12 mstl_c3 .ends * microstrip coupled 3 lines set of 2 .subckt mstl_c3_2 1 2 3 7 8 9 x1 1 2 3 4 5 6 mstl_c3 x2 4 5 6 7 8 9 mstl_c3 .ends ************************************************************* *microstrip coupled 2 lines .subckt mstl_c2 1 6 5 10 r1l 1 2 rsq l1l 2 3 lsq c1 3 0 csq r1r 3 4 rsq l1r 4 5 lsq rg1 3 0 10mega ccs1 1 3 ccsq ccs2 3 5 ccsq *ccouple1 1 6 c_cou ccouple2 3 8 c_cou r2l 6 7 rsq l2l 7 8 lsq c2 8 0 csq rg2 8 0 10mega l2r 8 9 lsq r2r 9 10 rsq ccs3 6 8 ccsq ccs4 8 10 ccsq k1 l1l l2l k=cou_l k2 l2r l1r k=cou_l .param + rsq = 5.7e-2 + lsq = 5.74e-11 *+ 3.11e-14 + csq = 2.11e-14 + ccsq = 3.53e-17 + c_cou = 5e-15 + cou_l = 0.32 .ends *set of 5 microstrip coupled 2 lines .subckt mstl_c2_5 1 2 11 12 x1 1 2 3 4 mstl_c2 x2 3 4 5 6 mstl_c2 x3 5 6 7 8 mstl_c2 337 x4 7 8 9 10 mstl_c2 x5 9 10 11 12 mstl_c2 .ends *set of 6 microstrip coupled 2 lines .subckt mstl_c2_6 1 2 13 14 x1 1 2 3 4 mstl_c2 x2 3 4 5 6 mstl_c2 x3 5 6 7 8 mstl_c2 x4 7 8 9 10 mstl_c2 x5 9 10 11 12 mstl_c2 x6 11 12 13 14 mstl_c2 .ends *set of 4 microstrip coupled 2 lines .subckt mstl_c2_4 1 2 9 10 x1 1 2 3 4 mstl_c2 x2 3 4 5 6 mstl_c2 x3 5 6 7 8 mstl_c2 x4 7 8 9 10 mstl_c2 .ends *set of 3 microstrip coupled 2 lines .subckt mstl_c2_3 1 2 7 8 x1 1 2 3 4 mstl_c2 x2 3 4 5 6 mstl_c2 x3 5 6 7 8 mstl_c2 .ends *set of 2 microstrip coupled 2 lines .subckt mstl_c2_2 1 2 5 6 x1 1 2 3 4 mstl_c2 x2 3 4 5 6 mstl_c2 .ends ************************************************************* *microstrip striaght line .subckt mstl_sq 1 5 r1 1 2 rsq l1 2 3 lsq c1 3 0 csq r2 3 0 10g r1r 3 4 rsq l2r 4 5 lsq cc1 1 3 ccsq cc2 3 5 ccsq .param + rsq = 5.7e-2 + lsq = 7.14e-11 + csq = 3.11e-14 + ccsq = 3.53e-17 .ends *set of 5 microstrip blocks .subckt mstl_sq_5 1 6 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq x4 4 5 mstl_sq x5 5 6 mstl_sq .ends *set of 4 microstrip blocks .subckt mstl_sq_4 1 5 x1 1 2 mstl_sq 338 x2 2 3 mstl_sq x3 3 4 mstl_sq x4 4 5 mstl_sq .ends *set of 3 microstrip blocks .subckt mstl_sq_3 1 4 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq .ends *set of 2 microstrip blocks .subckt mstl_sq_2 1 3 x1 1 2 mstl_sq x2 2 3 mstl_sq .ends ********************************************************* *composite corner (u shaped made of 5 squares) .subckt mstl_corn_u 1 5 r1 1 2 r2 l1 2 3 l2 c1 3 0 c2 r2 3 0 10g r2r 3 4 r2 l2r 4 5 l2 cc1 1 3 c_cou2cr cc2 3 5 c_cou2cr .param + r2 = 1.71e-1 + l2 = 3.234e-10 + c2 = 1e-15 + c_cou2cr = 1.377e-15 .ends .subckt corn_st 1 6 5 10 r1l 1 2 rsq l1l 2 3 lsq c1 3 0 csq r1r 3 4 rsq l1r 4 5 lsq rg1 3 0 10mega ccs1 1 3 ccsq ccs2 3 5 ccsq ccouple2 3 8 c_cou k1 l1l l2l k=cou_l k2 l1r l2r k=cou_l r2l 6 7 r2 l2l 7 8 l2 c2 8 0 c2 rg2 8 0 10mega l2r 8 9 l2 r2r 9 10 r2 ccs3 6 8 c_cou2cr ccs4 8 10 c_cou2cr .param + rsq = .1111 + lsq = 1.448e-10 + csq = 6.154e-14 + ccsq = 9.786e-15 + r2 = 6.6e-2 + l2 = 1.143e-10 + c2 = 4.668e-14 + c_cou2cr = 1e-17 339 .ends .subckt corn4_corn2 1 6 15 10 r1l 1 2 rsq l1l 2 3 lsq c1 3 0 csq r1r 3 4 rsq l1r 4 5 lsq rg1 3 0 10mega ccs1 1 3 ccsq ccs2 3 5 ccsq ccouple2 3 8 c_cou k1 l1l l2l k=cou_l k2 l1r l2r k=cou_l r2l 6 7 r2 l2l 7 8 l2 c2 8 0 c2 rg2 8 0 10mega l2r 8 9 l2 r2r 9 10 r2 ccs3 6 8 c_cou2cr ccs4 8 10 c_cou2cr r3l 5 12 r2 l3l 12 13 l2 c3 13 0 c2 rg3 13 0 10mega l3r 13 14 l2 r3r 14 15 r2 ccs6 5 13 c_cou2cr ccs7 13 15 c_cou2cr *k3 l3r l2r k=cou_l_2 lshunt1 5 15 ls1 lshunt2 6 10 ls2 kshunt lshunt1 lshunt2 k=cou_l_2 .param + rsq = .107 + lsq = 1.428e-10 + csq = 6.206e-14 + ccsq = 4.47e-14 + r2 = 6.2e-2 + l2 = 1.443e-10 + c2 = 4.668e-14 + c_cou2cr = 1e-17 .ends .subckt corn6_corn3 41 1 6 70 15 10 r1l 1 2 rsq l1l 2 3 lsq c1 3 0 csq r1r 3 4 rsq l1r 4 5 lsq rg1 3 0 10mega ccs1 1 3 ccsq ccs2 3 5 ccsq ccouple2 13 8 c_cou k1 l1l l2l k=cou_l k2 l1r l2r k=cou_l k23a l2l l3l k=cou_l k23b l2r l3r k=cou_l r2l 6 7 r2 l2l 7 8 l2 c2 8 0 c2 rg2 8 0 10mega l2r 8 9 l2 r2r 9 10 r2 340 ccs3 6 8 c_cou2cr ccs4 8 10 c_cou2cr r3l 5 12 r2 l3l 12 13 l2 c3 13 0 c2 rg3 13 0 10mega l3r 13 14 l2 r3r 14 15 r2 ccs6 5 13 c_cou2cr ccs7 13 15 c_cou2cr *k3 l3r l2r k=cou_l_2 lshunt1 5 15 ls1 lshunt2 6 10 ls2 kshunt lshunt1 lshunt2 k=cou_l_2 r4l 41 42 rsq l4l 42 43 lsq c4 43 0 csq r4r 43 44 rsq l4r 44 45 lsq rg4 43 0 10mega ccs41 41 43 ccsq ccs42 43 45 ccsq ccouple4_1 43 3 c_cou_line r5l 45 52 rsq l5l 52 53 lsq c5 53 0 csq r5r 53 54 rsq l5r 54 55 lsq rg5 53 0 10mega ccs51 45 53 ccsq ccs52 53 55 ccsq r6l 55 67 r2 l6l 67 68 l2 c6 68 0 c2 rg6 68 0 10mega l6r 68 69 l2 r6r 69 70 r2 ccs61 55 68 c_cou2cr ccs62 68 70 c_cou2cr ccouple5_2 68 13 c_cou lshunt3 55 70 ls1 kshunt3 lshunt2 lshunt3 k=cou_l_2 k14a l1l k14b l1r k35a l3l k35b l3r k36a l3l k36b l3r .ends l4l l4r l5l l5r l6l l6r k=cou_line k=cou_line k=cou_l k=cou_l k=cou_l k=cou_l .subckt mstl_corn_lcomp 1 2 8 7 x1 1 2 3 4 corn4_corn2 x4 8 7 3 4 corn4_corn2 .ends .subckt mstl_corn_6 1 2 3 9 10 11 x1 1 2 3 4 5 6 corn6_corn3 x4 9 10 11 4 5 6 corn6_corn3 .ends .subckt corn8_corn4 121 41 1 6 100 70 15 10 r1l 1 2 rsq l1l 2 3 lsq 341 c1 3 0 csq r1r 3 4 rsq l1r 4 5 lsq rg1 3 0 10mega ccs1 1 3 ccsq ccs2 3 5 ccsq ccouple2 13 8 c_cou k1 l1l l2l k=cou_l k2 l1r l2r k=cou_l k23a l2l l3l k=cou_l k23b l2r l3r k=cou_l r2l 6 7 r2 l2l 7 8 l2 c2 8 0 c2 rg2 8 0 10mega l2r 8 9 l2 r2r 9 10 r2 ccs3 6 8 c_cou2cr ccs4 8 10 c_cou2cr r3l 5 12 r2 l3l 12 13 l2 c3 13 0 c2 rg3 13 0 10mega l3r 13 14 l2 r3r 14 15 r2 ccs6 5 13 c_cou2cr ccs7 13 15 c_cou2cr *k3 l3r l2r k=cou_l_2 lshunt1 5 15 ls1 lshunt2 6 10 ls2 kshunt lshunt1 lshunt2 k=cou_l_2 r4l 41 42 rsq l4l 42 43 lsq c4 43 0 csq r4r 43 44 rsq l4r 44 45 lsq rg4 43 0 10mega ccs41 41 43 ccsq ccs42 43 45 ccsq ccouple4_1 43 3 c_cou_line r5l 45 52 rsq l5l 52 53 lsq c5 53 0 csq r5r 53 54 rsq l5r 54 55 lsq rg5 53 0 10mega ccs51 45 53 ccsq ccs52 53 55 ccsq r6l 55 67 r2 l6l 67 68 l2 c6 68 0 c2 rg6 68 0 10mega l6r 68 69 l2 r6r 69 70 r2 ccs61 55 68 c_cou2cr ccs62 68 70 c_cou2cr lshunt3 55 70 ls1 kshunt3 lshunt2 lshunt3 k=cou_l_2 k14a k14b k35a k35b l1l l1r l3l l3r l4l l4r l5l l5r k=cou_line k=cou_line k=cou_l k=cou_l 342 k36a l3l l6l k=cou_l k36b l3r l6r k=cou_l ccouple36 13 68 c_cou r10l 121 122 rsq l10l 122 123 lsq c10 123 0 csq r10r 123 124 rsq l10r 124 71 lsq cc10a 121 123 ccsq cc10b 123 71 ccsq r7l 71 72 rsq l7l 72 73 lsq c7 73 0 csq r7r 73 74 rsq l7r 74 75 lsq rg7 73 0 10mega ccs71 71 73 ccsq ccs72 73 75 ccsq ccouple7_1 73 53 c_cou_line k17a l7l l5l k=cou_line k17b l7r l5r k=cou_line ccouple10_4 123 43 c_cou_line k104a l10l l4l k=cou_line k104b l10r l4r k=cou_line r8l 75 82 rsq l8l 82 83 lsq c8 83 0 csq r8r 83 84 rsq l8r 84 85 lsq rg8 83 0 10mega ccs81 75 83 ccsq ccs82 83 85 ccsq ccouple8_1 98 68 c_cou k28a l8l l6l k=cou_l k28b l8r l6r k=cou_l k96a l9l l6l k=cou_l k96b l9r l6r k=cou_l r9l 85 97 r2 l9l 97 98 l2 c9 98 0 c2 rg9 98 0 10mega l9r 98 99 l2 r9r 99 100 r2 ccs91 85 98 c_cou2cr ccs92 98 100 c_cou2cr lshunt5 85 100 ls1 kshunt5 lshunt5 lshunt3 k=cou_l_2 .ends .subckt mstl_corn_8 1 2 3 4 9 10 11 12 x1 1 2 3 4 5 6 7 8 corn8_corn4 x4 9 10 11 12 5 6 7 8 corn8_corn4 .ends x1 x2 x3 x4 x5 x6 1 2 3 4 5 6 7 8 mstl_c4_5 5 6 7 8 9 10 11 12 mstl_c4_4 9 10 11 12 13 14 15 16 mstl_corn_8 13 14 15 16 17 18 19 20 mstl_c4_3 17 18 19 20 21 22 23 24 mstl_corn_8 21 22 23 24 25 26 27 28 mstl_c4_5 343 x7 25 26 27 28 x8 29 30 31 33 x9 33 34 35 36 x10 36 37 38 2 29 30 31 32 mstl_c4_4 34 35 mstl_corn_6 37 38 mstl_c3_5 3 4 mstl_corn_6 ro 32 0 1g vpl 1 0 pulse(0 2.5 2n 0.1n 0.2n 10n 20n) ac 1 *.tran 1p 40n 1p *.ac dec 60 1mega 15giga .net v(32) vpl rin=50 rout=50 .param + ls1 = 1e-11 + ls2 = ls1 + + + + + + + + + + + + + c_cou = 1e-15 cou_l = 0.2 l2 = 2e-10 c2 = 2e-14 r2 = 0.37 rsq = .1 lsq = 11.4e-11 csq = 3.7e-14 ccsq = 1.0e-14 c_cou2cr = 1e-14 cou_line = 0.32 cou_l_2 = 0.01 c_cou_line = 4.0e-15 .measure .measure .measure .measure .measure .measure .measure .measure ac ac ac ac ac ac ac ac comp1 comp2 comp3 comp4 comp5 comp6 comp7 comp8 err1 err1 err1 err1 err1 err1 err1 err1 .ac data=measured .print par(s11r) s11(r) .print par(s12r) s12(r) .print par(s21r) s21(r) .print par(s22r) s22(r) .print s11(db) .print s12(db) .print s21(db) .print s22(db) par(s11r) par(s11i) par(s12r) par(s12i) par(s21r) par(s21i) par(s22r) par(s22i) par(s11i) par(s12i) par(s21i) par(s22i) s11(r) s11(i) s12(r) s12(i) s21(r) s21(i) s22(r) s22(i) minval=10 minval=10 minval=10 minval=10 s11(i) s12(i) s21(i) s22(i) .data measured file = ‘line.txt’ freq=1 s11r=2 s11I=3 s21r=4 s21I=5 s12r=6 s12I=7 s22r=8 s22I=9 .enddata .param freq=100m, s11r=0, s11i=0, s21r=0, s21i=0, s12r=0, s12i=0, s22r=0, s22i=0 .end F.5.2. Method-of-Moments S-Parameter Data Frequency MoM S11R MoM S11I MoM S21R MoM S21I 1.00E+08 1.92E+08 2.06E-01 2.82E-01 8.79E-02 1.35E-01 7.82E-01 6.76E-01 -2.42E-01 -4.30E-01 344 1.97E+08 2.03E+08 2.09E+08 2.15E+08 2.83E+08 2.89E+08 2.95E+08 3.01E+08 3.06E+08 3.12E+08 3.18E+08 3.23E+08 3.29E+08 3.35E+08 3.41E+08 3.46E+08 3.52E+08 3.58E+08 3.64E+08 3.69E+08 3.92E+08 3.98E+08 4.04E+08 4.09E+08 4.15E+08 4.93E+08 5.07E+08 5.20E+08 5.87E+08 6.00E+08 6.14E+08 6.94E+08 7.07E+08 7.21E+08 7.34E+08 7.47E+08 7.61E+08 7.74E+08 7.88E+08 8.01E+08 8.14E+08 8.28E+08 8.41E+08 8.54E+08 8.68E+08 8.81E+08 8.94E+08 9.08E+08 9.21E+08 9.35E+08 9.48E+08 2.87E-01 2.93E-01 2.99E-01 3.04E-01 3.74E-01 3.80E-01 3.85E-01 3.91E-01 3.97E-01 4.02E-01 4.08E-01 4.13E-01 4.18E-01 4.24E-01 4.29E-01 4.34E-01 4.39E-01 4.44E-01 4.49E-01 4.54E-01 4.71E-01 4.76E-01 4.80E-01 4.84E-01 4.88E-01 5.28E-01 5.33E-01 5.36E-01 5.43E-01 5.42E-01 5.41E-01 5.15E-01 5.09E-01 5.02E-01 4.94E-01 4.85E-01 4.76E-01 4.66E-01 4.56E-01 4.45E-01 4.34E-01 4.22E-01 4.10E-01 3.97E-01 3.84E-01 3.70E-01 3.56E-01 3.42E-01 3.28E-01 3.13E-01 2.98E-01 1.36E-01 1.38E-01 1.39E-01 1.40E-01 1.35E-01 1.33E-01 1.31E-01 1.29E-01 1.27E-01 1.25E-01 1.22E-01 1.20E-01 1.17E-01 1.14E-01 1.11E-01 1.07E-01 1.04E-01 1.00E-01 9.66E-02 9.27E-02 7.60E-02 7.16E-02 6.70E-02 6.22E-02 5.74E-02 -1.74E-02 -3.14E-02 -4.57E-02 -1.20E-01 -1.35E-01 -1.49E-01 -2.36E-01 -2.50E-01 -2.63E-01 -2.76E-01 -2.89E-01 -3.01E-01 -3.13E-01 -3.24E-01 -3.35E-01 -3.45E-01 -3.55E-01 -3.64E-01 -3.73E-01 -3.81E-01 -3.88E-01 -3.94E-01 -4.00E-01 -4.06E-01 -4.10E-01 -4.14E-01 345 6.68E-01 6.59E-01 6.51E-01 6.42E-01 5.30E-01 5.20E-01 5.10E-01 5.00E-01 4.90E-01 4.80E-01 4.70E-01 4.60E-01 4.50E-01 4.39E-01 4.29E-01 4.19E-01 4.09E-01 3.98E-01 3.88E-01 3.78E-01 3.36E-01 3.26E-01 3.16E-01 3.05E-01 2.95E-01 1.56E-01 1.32E-01 1.09E-01 -5.25E-03 -2.76E-02 -4.99E-02 -1.80E-01 -2.01E-01 -2.22E-01 -2.43E-01 -2.64E-01 -2.84E-01 -3.05E-01 -3.25E-01 -3.45E-01 -3.65E-01 -3.85E-01 -4.04E-01 -4.24E-01 -4.43E-01 -4.61E-01 -4.80E-01 -4.98E-01 -5.16E-01 -5.34E-01 -5.51E-01 -4.40E-01 -4.50E-01 -4.60E-01 -4.69E-01 -5.69E-01 -5.76E-01 -5.83E-01 -5.89E-01 -5.95E-01 -6.02E-01 -6.08E-01 -6.14E-01 -6.19E-01 -6.25E-01 -6.30E-01 -6.35E-01 -6.40E-01 -6.45E-01 -6.49E-01 -6.54E-01 -6.70E-01 -6.73E-01 -6.77E-01 -6.80E-01 -6.83E-01 -7.11E-01 -7.14E-01 -7.15E-01 -7.15E-01 -7.13E-01 -7.11E-01 -6.86E-01 -6.80E-01 -6.73E-01 -6.66E-01 -6.59E-01 -6.51E-01 -6.42E-01 -6.33E-01 -6.23E-01 -6.13E-01 -6.02E-01 -5.90E-01 -5.78E-01 -5.66E-01 -5.53E-01 -5.39E-01 -5.24E-01 -5.09E-01 -4.94E-01 -4.78E-01 9.61E+08 9.75E+08 9.88E+08 1.00E+09 1.01E+09 1.23E+09 1.24E+09 1.26E+09 1.27E+09 1.28E+09 1.38E+09 1.39E+09 1.40E+09 1.42E+09 1.75E+09 1.76E+09 1.78E+09 1.79E+09 1.80E+09 1.82E+09 1.83E+09 1.84E+09 1.86E+09 1.87E+09 1.88E+09 1.90E+09 1.91E+09 2.00E+09 2.02E+09 2.30E+09 2.32E+09 2.79E+09 2.80E+09 2.97E+09 2.99E+09 3.00E+09 3.22E+09 3.23E+09 3.25E+09 3.45E+09 3.45E+09 3.46E+09 3.47E+09 3.48E+09 3.48E+09 3.64E+09 3.66E+09 3.68E+09 3.90E+09 3.92E+09 4.04E+09 2.83E-01 2.68E-01 2.53E-01 2.38E-01 2.23E-01 2.38E-02 1.65E-02 9.93E-03 4.24E-03 -5.79E-04 -9.80E-03 -7.70E-03 -4.82E-03 -1.17E-03 1.86E-01 1.91E-01 1.95E-01 1.99E-01 2.03E-01 2.05E-01 2.07E-01 2.09E-01 2.10E-01 2.10E-01 2.09E-01 2.08E-01 2.06E-01 1.77E-01 1.71E-01 -3.50E-02 -4.74E-02 -1.44E-01 -1.38E-01 -6.00E-02 -5.24E-02 -4.48E-02 4.44E-02 4.77E-02 5.02E-02 -1.72E-02 -2.31E-02 -2.91E-02 -3.51E-02 -4.11E-02 -4.71E-02 -1.35E-01 -1.38E-01 -1.40E-01 -1.10E-01 -1.05E-01 -6.72E-02 -4.17E-01 -4.19E-01 -4.20E-01 -4.21E-01 -4.21E-01 -3.17E-01 -3.05E-01 -2.94E-01 -2.82E-01 -2.70E-01 -1.84E-01 -1.73E-01 -1.62E-01 -1.51E-01 -1.08E-01 -1.14E-01 -1.22E-01 -1.29E-01 -1.37E-01 -1.46E-01 -1.54E-01 -1.63E-01 -1.71E-01 -1.80E-01 -1.89E-01 -1.98E-01 -2.07E-01 -2.66E-01 -2.74E-01 -3.21E-01 -3.16E-01 4.91E-03 1.24E-02 6.27E-02 6.43E-02 6.55E-02 2.37E-02 1.64E-02 8.58E-03 -8.59E-02 -8.66E-02 -8.70E-02 -8.70E-02 -8.66E-02 -8.59E-02 -7.46E-03 2.84E-03 1.31E-02 1.37E-01 1.43E-01 1.82E-01 346 -5.68E-01 -5.85E-01 -6.01E-01 -6.17E-01 -6.32E-01 -7.80E-01 -7.81E-01 -7.81E-01 -7.80E-01 -7.78E-01 -7.31E-01 -7.19E-01 -7.07E-01 -6.93E-01 -1.07E-01 -8.02E-02 -5.32E-02 -2.63E-02 3.65E-04 2.68E-02 5.31E-02 7.90E-02 1.05E-01 1.30E-01 1.55E-01 1.80E-01 2.04E-01 3.61E-01 3.82E-01 6.85E-01 6.94E-01 4.61E-01 4.39E-01 1.51E-01 1.22E-01 9.29E-02 -3.11E-01 -3.38E-01 -3.64E-01 -6.28E-01 -6.35E-01 -6.42E-01 -6.48E-01 -6.54E-01 -6.60E-01 -7.16E-01 -7.15E-01 -7.12E-01 -5.52E-01 -5.34E-01 -3.64E-01 -4.61E-01 -4.43E-01 -4.25E-01 -4.06E-01 -3.86E-01 5.85E-03 3.41E-02 6.26E-02 9.12E-02 1.20E-01 3.18E-01 3.46E-01 3.73E-01 3.99E-01 7.69E-01 7.71E-01 7.71E-01 7.71E-01 7.70E-01 7.67E-01 7.64E-01 7.60E-01 7.55E-01 7.49E-01 7.43E-01 7.36E-01 7.27E-01 6.51E-01 6.38E-01 2.42E-01 2.16E-01 -5.68E-01 -5.86E-01 -7.21E-01 -7.27E-01 -7.31E-01 -6.68E-01 -6.54E-01 -6.40E-01 -3.68E-01 -3.55E-01 -3.42E-01 -3.29E-01 -3.16E-01 -3.02E-01 -5.87E-03 2.27E-02 5.12E-02 4.35E-01 4.56E-01 5.96E-01 4.06E+09 4.07E+09 4.09E+09 4.23E+09 4.24E+09 4.26E+09 4.47E+09 4.48E+09 4.70E+09 4.71E+09 4.73E+09 4.90E+09 4.91E+09 5.14E+09 5.16E+09 5.45E+09 5.46E+09 5.48E+09 5.49E+09 6.03E+09 6.04E+09 6.52E+09 6.53E+09 6.55E+09 6.56E+09 6.58E+09 7.05E+09 7.07E+09 7.08E+09 7.34E+09 7.36E+09 7.62E+09 7.63E+09 7.97E+09 7.98E+09 8.23E+09 8.24E+09 8.26E+09 8.49E+09 8.50E+09 8.72E+09 8.73E+09 8.75E+09 8.76E+09 8.98E+09 9.11E+09 9.12E+09 9.13E+09 9.32E+09 9.33E+09 9.34E+09 -6.20E-02 -5.66E-02 -5.12E-02 3.39E-03 1.02E-02 1.71E-02 1.19E-01 1.25E-01 1.17E-01 1.12E-01 1.07E-01 6.48E-02 6.27E-02 5.76E-02 5.85E-02 9.14E-02 9.38E-02 9.62E-02 9.88E-02 2.33E-01 2.38E-01 4.40E-01 4.46E-01 4.52E-01 4.58E-01 4.63E-01 4.00E-01 3.91E-01 3.82E-01 2.08E-01 1.98E-01 5.31E-02 4.59E-02 -8.57E-02 -9.07E-02 -1.62E-01 -1.66E-01 -1.70E-01 -2.20E-01 -2.22E-01 -2.50E-01 -2.51E-01 -2.52E-01 -2.53E-01 -2.56E-01 -2.49E-01 -2.48E-01 -2.48E-01 -2.28E-01 -2.27E-01 -2.25E-01 1.87E-01 1.91E-01 1.94E-01 2.20E-01 2.22E-01 2.23E-01 1.97E-01 1.90E-01 9.83E-02 9.56E-02 9.36E-02 1.05E-01 1.08E-01 1.52E-01 1.54E-01 1.98E-01 2.00E-01 2.02E-01 2.04E-01 2.28E-01 2.26E-01 7.83E-02 6.89E-02 5.92E-02 4.92E-02 3.90E-02 -2.90E-01 -2.96E-01 -3.02E-01 -3.42E-01 -3.42E-01 -3.07E-01 -3.04E-01 -2.43E-01 -2.40E-01 -2.15E-01 -2.14E-01 -2.13E-01 -2.13E-01 -2.14E-01 -2.31E-01 -2.33E-01 -2.35E-01 -2.37E-01 -2.63E-01 -2.77E-01 -2.78E-01 -2.79E-01 -2.93E-01 -2.93E-01 -2.94E-01 347 -3.40E-01 -3.16E-01 -2.91E-01 -5.30E-02 -2.56E-02 1.95E-03 3.73E-01 3.97E-01 6.55E-01 6.67E-01 6.77E-01 7.04E-01 6.98E-01 4.41E-01 4.14E-01 -2.13E-01 -2.45E-01 -2.77E-01 -3.08E-01 -5.60E-01 -5.42E-01 2.49E-01 2.73E-01 2.98E-01 3.21E-01 3.44E-01 6.27E-01 6.19E-01 6.10E-01 3.04E-01 2.78E-01 -2.58E-01 -2.91E-01 -8.58E-01 -8.69E-01 -7.89E-01 -7.69E-01 -7.47E-01 -3.07E-01 -2.74E-01 1.36E-01 1.60E-01 1.83E-01 2.05E-01 4.18E-01 4.60E-01 4.60E-01 4.61E-01 4.25E-01 4.22E-01 4.16E-01 6.10E-01 6.22E-01 6.34E-01 6.96E-01 6.97E-01 6.98E-01 5.96E-01 5.81E-01 2.74E-01 2.46E-01 2.18E-01 -1.17E-01 -1.48E-01 -5.56E-01 -5.76E-01 -6.60E-01 -6.48E-01 -6.34E-01 -6.18E-01 3.72E-01 3.97E-01 6.09E-01 5.98E-01 5.86E-01 5.72E-01 5.58E-01 -1.84E-01 -2.11E-01 -2.38E-01 -6.26E-01 -6.42E-01 -7.34E-01 -7.27E-01 -2.13E-01 -1.77E-01 4.05E-01 4.36E-01 4.66E-01 7.31E-01 7.35E-01 6.48E-01 6.33E-01 6.17E-01 6.01E-01 3.13E-01 1.21E-01 1.11E-01 9.98E-02 -1.50E-01 -1.59E-01 -1.77E-01 9.88E+09 9.91E+09 9.94E+09 9.97E+09 1.00E+10 -1.60E-01 -1.56E-01 -1.53E-01 -1.50E-01 -1.47E-01 -2.94E-01 -2.93E-01 -2.92E-01 -2.90E-01 -2.89E-01 2.77E-02 6.93E-04 -2.64E-02 -5.37E-02 -8.10E-02 -6.07E-01 -6.20E-01 -6.33E-01 -6.45E-01 -6.56E-01 APPENDIX G CIRCUITS AND DATA FOR LOW TEMPERATURE COFIRED CERAMIC (LTCC) STRUCTURES MODELING G.1. Introduction Input files and measured S-parameter data for test structure optimization for the LTCC inductors and capacitor study described earlier in this thesis are presented in this appendix. In addition, the circuit files representing the complete models of the 4 coil benchmark inductors and large gridded plate capacitor are also shown, with associated measured S-parameters. All circuit files are written for the Star-Hspice circuit simulator. It should be noted that in some cases, certain subcircuit (.subckt) calls are defined but are never used in the actual optimization runs. Additionally, only S11 and S21 results are shown for the measured data, since S22 and S11 are equal, and S12 and S21 are also equivalent for these structures. 348 G.2. Test structure 1 G.2.1. Circuit Optimization Input File Hspice input circuit for optimization and parameter extraction of test structire 1 building blocks is shown below. .option accurate dccap=1 node nopage ingold=2 post acct=2 probe .subckt mstl_pad 1 5 r1 1 2 r2 l1 2 3 l2 c1 3 0 c2 r2 3 0 10g r1r 3 4 r2 l2r 4 5 l2 .ends .subckt mstl_sq 1 5 r1 1 2 rsq l1 2 3 lsq c1 3 0 csq *r2 3 0 rl r1r 3 4 rsq l2r 4 5 lsq *k1 l1 l2r k=cou_l cc1 1 3 c_cou cc2 3 5 c_cou *cc3 1 5 c_cou2 .ends .subckt mstl_sq5 1 6 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq x4 4 5 mstl_sq x5 5 6 mstl_sq .ends .subckt mstl_sq10 1 3 x1 1 2 mstl_sq5 x2 2 3 mstl_sq5 .ends .subckt line3 1 8 x1 1 2 mstl_pad x2 2 3 mstl_sq10 x3 3 4 mstl_sq10 x4 4 5 mstl_sq10 x5 5 6 mstl_sq10 x7 6 8 mstl_pad r0 8 0 1g .ends .subckt line2 1 4 x1 1 2 mstl_pad x2 2 3 mstl_sq10 x3 3 4 mstl_pad 349 ro 4 0 1g .ends *vpl 1 0 pulse(0 2.5 2n 0.1n 0.2n 10n 20n) ac 1 *.tran 1p 40n 1p *.ac dec 60 1mega 15giga *.net v(8) vpl rin=50 rout=50 .param + c_cou = opt1(6.4e-12,1f,1n) + rl = opt1(1e4,1,1e8) + r2 = opt1(4.7e-1,0.00001,10) + l2 = opt1(1.2e-11,.01p,1u) + c2 = opt1(9.2e-15,0.1f,1n) + rsq = opt1(0.30,0.01,10) + lsq = opt1(0.4e-11,1f,1u) + csq = opt1(2.1e-15,0.01f,1n) ****************************** * circuit for 1st subcircuit ****************************** v1i 1 0 dc 0 ac 1 r1i 1 2 50 x1 2 3 line2 r1o 3 4 50 v1o 4 0 dc 6 ac 0 e11 5 0 (2,0) 2 v11 5 11 ac 1 r11 11 0 1g e21 21 0 (3,0) 2 r21 21 0 1g .ac data=measured optimize=opt1 + results=comp1,comp2,comp5,comp6 + model=converge .model converge opt relin=1e-3 relout=1e-3 close=200 itropt=500 *.model converge opt .measure .measure .measure .measure ac ac ac ac comp1 comp2 comp5 comp6 err1 err1 err1 err1 par(s11r) par(s11i) par(s21r) par(s21i) vr(11) vi(11) vr(21) vi(21) .print par(s11r) vr(11) par(s11i) vi(11) .print par(s21r) vr(21) par(s21i) vi(21) .data measured mer file= 'dev23' freq=1 s11r=2 s11i=3 s21r=4 s21i=5 s12r=6 s12i=7 s22r=8 s22i=9 out = 'dev23_data.txt' .enddata .param freq=500m, s11r=0, s11i=0, s21r=0, s21i=0, s12r=0, s12i=0, s22r=0, s22i=0 .param sa11r=0, sa11i=0, sa21r=0, sa21i=0, sa12r=0, sa12i=0, sa22r=0, + sa22i=0 .end 350 G.2.2. S-Parameter Measured Data freq 4.50E+07 1.45E+08 2.45E+08 3.44E+08 4.44E+08 5.44E+08 6.44E+08 7.43E+08 8.43E+08 9.43E+08 1.04E+09 1.14E+09 1.24E+09 1.34E+09 1.44E+09 1.54E+09 1.64E+09 1.74E+09 1.84E+09 1.94E+09 2.04E+09 2.14E+09 2.24E+09 2.34E+09 2.44E+09 2.54E+09 2.64E+09 2.74E+09 2.84E+09 2.94E+09 3.04E+09 3.14E+09 3.24E+09 3.34E+09 3.44E+09 3.54E+09 3.64E+09 3.74E+09 3.84E+09 3.94E+09 4.04E+09 S11(R) Meas. 2.22E-03 3.13E-03 4.52E-03 4.74E-03 4.17E-02 2.63E-03 -2.51E-03 4.69E-03 7.77E-03 1.05E-02 1.25E-02 1.32E-02 1.41E-02 1.59E-02 1.73E-02 1.89E-02 2.06E-02 2.25E-02 2.45E-02 2.69E-02 2.92E-02 3.13E-02 3.35E-02 3.63E-02 3.86E-02 4.20E-02 4.46E-02 4.73E-02 4.99E-02 5.28E-02 5.52E-02 5.80E-02 6.10E-02 6.45E-02 6.77E-02 7.15E-02 7.44E-02 7.84E-02 8.30E-02 8.72E-02 9.06E-02 S11(I) Meas. 2.26E-03 4.54E-03 5.48E-03 6.60E-03 6.77E-03 7.23E-03 1.80E-02 2.07E-02 2.25E-02 2.38E-02 2.41E-02 2.48E-02 2.65E-02 2.82E-02 2.95E-02 3.10E-02 3.25E-02 3.40E-02 3.54E-02 3.67E-02 3.77E-02 3.84E-02 3.96E-02 4.06E-02 4.16E-02 4.18E-02 4.20E-02 4.21E-02 4.23E-02 4.22E-02 4.28E-02 4.26E-02 4.26E-02 4.27E-02 4.24E-02 4.21E-02 4.08E-02 4.08E-02 4.00E-02 3.73E-02 3.69E-02 351 S21(R) Meas. 9.98E-01 9.95E-01 9.93E-01 9.89E-01 9.84E-01 9.78E-01 9.67E-01 9.68E-01 9.63E-01 9.58E-01 9.51E-01 9.42E-01 9.32E-01 9.22E-01 9.11E-01 8.99E-01 8.86E-01 8.73E-01 8.59E-01 8.43E-01 8.27E-01 8.11E-01 7.94E-01 7.76E-01 7.57E-01 7.39E-01 7.22E-01 7.04E-01 6.86E-01 6.67E-01 6.47E-01 6.26E-01 6.05E-01 5.82E-01 5.59E-01 5.34E-01 5.10E-01 4.85E-01 4.59E-01 4.33E-01 4.07E-01 S21(I) Meas. -1.48E-02 -4.33E-02 -7.22E-02 -1.01E-01 -1.29E-01 -1.58E-01 -1.76E-01 -2.03E-01 -2.30E-01 -2.57E-01 -2.85E-01 -3.13E-01 -3.39E-01 -3.64E-01 -3.90E-01 -4.16E-01 -4.41E-01 -4.66E-01 -4.90E-01 -5.14E-01 -5.37E-01 -5.59E-01 -5.81E-01 -6.02E-01 -6.21E-01 -6.40E-01 -6.59E-01 -6.78E-01 -6.97E-01 -7.16E-01 -7.35E-01 -7.53E-01 -7.71E-01 -7.88E-01 -8.05E-01 -8.21E-01 -8.35E-01 -8.50E-01 -8.63E-01 -8.74E-01 -8.87E-01 4.14E+09 4.24E+09 4.34E+09 4.44E+09 4.53E+09 4.63E+09 4.73E+09 4.83E+09 4.93E+09 5.03E+09 9.52E-02 9.97E-02 1.04E-01 1.09E-01 1.14E-01 1.19E-01 1.24E-01 1.30E-01 1.35E-01 1.40E-01 3.53E-02 3.33E-02 3.10E-02 2.83E-02 2.56E-02 2.18E-02 1.80E-02 1.34E-02 8.28E-03 2.82E-03 3.80E-01 3.52E-01 3.24E-01 2.96E-01 2.66E-01 2.37E-01 2.08E-01 1.77E-01 1.47E-01 1.17E-01 G.3. Test Structure 2 G.3.1. Circuit Optimization Input File .option accurate dccap=1 node nopage ingold=2 post acct=2 probe .subckt mstl_pad 1 5 r1 1 2 r2 l1 2 3 l2 c1 3 0 c2 r2 3 0 10g r1r 3 4 r2 l2r 4 5 l2 .ends .subckt mstl_sq 1 5 r1 1 2 rsq l1 2 3 lsq c1 3 0 csq r1r 3 4 rsq l2r 4 5 lsq cc1 1 3 c_cou cc2 3 5 c_cou *cc3 1 5 c_cou2 .ends .subckt mstlc5 1 2 11 12 x1 1 2 3 4 mstlc1 x2 3 4 5 6 mstlc1 x3 5 6 7 8 mstlc1 x4 7 8 9 10 mstlc1 x5 9 10 11 12 mstlc1 .ends .subckt mstl_sq5 1 6 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq 352 -8.98E-01 -9.08E-01 -9.17E-01 -9.25E-01 -9.32E-01 -9.38E-01 -9.43E-01 -9.46E-01 -9.49E-01 -9.50E-01 x4 4 5 mstl_sq x5 5 6 mstl_sq .ends .subckt mstl_sq4 1 5 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq x4 4 5 mstl_sq .ends .subckt ind_blk 1 15 rl1 1 2 ri ll1 2 3 li c1 3 0 ci rg1 3 0 rg1 rr1 3 4 ri lr1 4 5 li lv1 5 10 lvia cv1 5 10 cvia rl2 10 11 ri2 ll2 11 12 li2 c2 12 0 ci2 rg2 12 0 rg2 rr2 12 13 ri2 lr2 13 14 li2 cc12 3 12 csi lv2 14 15 lvia cv2 14 15 cvia .ends .subckt ind8 1 9 x1 1 2 ind_blk x2 2 3 ind_blk x3 3 4 ind_blk x7 4 5 ind_blk x8 5 6 ind_blk x9 6 7 ind_blk x10 7 8 ind_blk x11 8 9 ind_blk .ends x1 1 2 mstl_pad x2 2 4 ind8 *x3 3 4 ind3 x7 4 8 mstl_pad r0 8 0 1g vpl 1 0 pulse(0 2.5 2n 0.1n 0.2n 10n 20n) ac 1 *.tran 1p 40n 1p *.ac dec 60 1mega 15giga .net v(8) vpl rin=50 rout=50 .param + c_cou = opt1(6.4e-14,1f,1n) + rl = opt1(1e2,1,1e8) + ri = opt1(0.01,0.001,10) + li = opt1(0.1e-9,1f,1u) + ci = opt1(4.5e-14,0.01f,1n) + csi = opt1(1.4e-15,0.01f,1n) 353 + + + + + + ri2 = opt1(1.01,0.001,10) li2 = opt1(0.4e-9,1f,1u) ci2 = opt1(4.5e-14,0.01f,1n) csi = opt1(1.4e-15,0.01f,1n) lvia = opt1(0.4e-9,1f,1u) cvia = opt1(4.4e-14,0.01f,1n) + rg1 = opt1(1e7,1e2,1e8) + rg2 = opt1(1e4, 1e2, 1e8) + cou_l = opt1(0.4,0.01,1) + + + + rv = opt1(0.10,0.01,10) lv = opt1(1.0e-9,1f,1u) cv = opt1(1.4e-13,0.01f,1n) csv = opt1(8.4e-15,0.01f,1n) .param r2 .param l2 .param c2 = 1.000e-05 = 3.228e-10 = 1.863e-13 $ $ $ 2.131e+01 1.886e+01 1.432e+01 -4.957e-06 1.186e-06 -1.809e-05 .ac data=measured optimize=opt1 + results=comp1,comp2,comp5,comp6 + model=converge .model converge opt relin=1e-4 relout=1e-3 close=100 itropt=1000 .measure .measure .measure .measure .measure .measure .measure .measure ac ac ac ac ac ac ac ac comp1 comp2 comp3 comp4 comp5 comp6 comp7 comp8 err1 err1 err1 err1 err1 err1 err1 err1 par(s11r) par(s11i) par(s12r) par(s12i) par(s21r) par(s21i) par(s22r) par(s22i) s11(r) s11(i) s12(r) s12(i) s21(r) s21(i) s22(r) s22(i) .ac data=measured .print par(s11r) s11(r) par(s11i) s11(i) .print par(s12r) s12(r) par(s12i) s12(i) .print par(s21r) s21(r) par(s21i) s21(i) .print par(s22r) s22(r) par(s22i) s22(i) *.param s11db='20*log(sqrt((par(s11r))^2+(par(s11i))^2))' *.param s21db='20*log(sqrt((par(s21r))^2+(par(s21i))^2))' .print s11(db) .print s12(db) .print s21(db) .print s22(db) .data measured file = ‘dev21.txt’ freq=1 s11r=2 s11I=3 s21r=4 s21I=5 s12r=6 s12I=7 s22r=8 s22I=9 .enddata .param freq=100m, s11r=0, s11i=0, s21r=0, s21i=0, s12r=0, s12i=0, s22r=0, s22i=0 .end G.3.2. Measured S-Parameter Data freq 4.50E+07 S11(R) Meas. 2.00E-03 S11(I) Meas. 2.70E-02 354 S21(R) Meas. 9.98E-01 S21(I) Meas. -7.30E-02 6.98E+07 9.46E+07 1.19E+08 1.44E+08 1.69E+08 1.94E+08 2.18E+08 2.43E+08 2.68E+08 2.93E+08 3.18E+08 3.42E+08 3.67E+08 3.92E+08 4.17E+08 4.41E+08 4.66E+08 4.91E+08 5.16E+08 5.40E+08 5.65E+08 5.90E+08 6.15E+08 6.40E+08 6.64E+08 6.89E+08 7.14E+08 7.39E+08 7.63E+08 7.88E+08 8.13E+08 8.38E+08 8.63E+08 8.87E+08 9.12E+08 9.37E+08 9.62E+08 9.86E+08 1.01E+09 1.04E+09 1.06E+09 1.09E+09 1.11E+09 1.14E+09 1.16E+09 1.18E+09 8.00E-03 1.40E-02 1.60E-02 1.50E-02 3.00E-02 3.90E-02 4.90E-02 5.70E-02 6.60E-02 7.60E-02 8.40E-02 9.40E-02 1.07E-01 1.16E-01 1.26E-01 1.34E-01 1.42E-01 1.57E-01 1.60E-01 1.63E-01 1.76E-01 1.60E-01 1.93E-01 2.45E-01 2.75E-01 3.03E-01 3.10E-01 3.38E-01 3.50E-01 3.53E-01 3.64E-01 3.80E-01 3.96E-01 4.09E-01 4.18E-01 4.26E-01 4.35E-01 4.42E-01 4.49E-01 4.53E-01 4.59E-01 4.63E-01 4.64E-01 4.66E-01 4.68E-01 4.68E-01 3.90E-02 5.20E-02 6.40E-02 8.10E-02 9.70E-02 1.06E-01 1.14E-01 1.25E-01 1.33E-01 1.40E-01 1.47E-01 1.58E-01 1.61E-01 1.65E-01 1.70E-01 1.74E-01 1.75E-01 1.82E-01 1.79E-01 1.93E-01 1.97E-01 2.12E-01 2.54E-01 2.62E-01 2.47E-01 2.36E-01 2.23E-01 2.08E-01 1.89E-01 1.78E-01 1.76E-01 1.69E-01 1.55E-01 1.40E-01 1.26E-01 1.12E-01 9.80E-02 8.30E-02 6.80E-02 5.30E-02 3.80E-02 2.20E-02 7.00E-03 -9.00E-03 -2.30E-02 -3.80E-02 355 9.86E-01 9.78E-01 9.69E-01 9.53E-01 9.49E-01 9.38E-01 9.25E-01 9.08E-01 8.91E-01 8.71E-01 8.49E-01 8.22E-01 8.04E-01 7.76E-01 7.46E-01 7.19E-01 6.94E-01 6.74E-01 6.40E-01 6.01E-01 5.67E-01 5.10E-01 5.08E-01 5.18E-01 5.10E-01 4.93E-01 4.54E-01 4.48E-01 4.19E-01 3.84E-01 3.57E-01 3.35E-01 3.12E-01 2.85E-01 2.58E-01 2.28E-01 2.01E-01 1.72E-01 1.45E-01 1.17E-01 9.00E-02 6.20E-02 3.30E-02 4.00E-03 -2.30E-02 -5.10E-02 -1.13E-01 -1.50E-01 -1.92E-01 -2.24E-01 -2.55E-01 -2.93E-01 -3.31E-01 -3.66E-01 -4.01E-01 -4.37E-01 -4.67E-01 -4.92E-01 -5.24E-01 -5.52E-01 -5.76E-01 -5.96E-01 -6.17E-01 -6.40E-01 -6.70E-01 -6.81E-01 -6.94E-01 -6.90E-01 -6.65E-01 -6.68E-01 -6.97E-01 -7.18E-01 -7.37E-01 -7.54E-01 -7.77E-01 -7.90E-01 -7.93E-01 -8.01E-01 -8.13E-01 -8.24E-01 -8.33E-01 -8.40E-01 -8.46E-01 -8.50E-01 -8.55E-01 -8.59E-01 -8.62E-01 -8.63E-01 -8.63E-01 -8.63E-01 -8.62E-01 -8.60E-01 1.21E+09 1.23E+09 1.26E+09 1.28E+09 1.31E+09 1.33E+09 1.36E+09 1.38E+09 1.41E+09 1.43E+09 1.46E+09 1.48E+09 1.51E+09 1.53E+09 1.56E+09 1.58E+09 1.61E+09 1.63E+09 1.66E+09 1.68E+09 1.70E+09 1.73E+09 1.75E+09 1.78E+09 1.80E+09 1.83E+09 1.85E+09 1.88E+09 1.90E+09 1.93E+09 1.95E+09 1.98E+09 2.00E+09 2.03E+09 2.05E+09 2.08E+09 2.10E+09 2.13E+09 2.15E+09 2.18E+09 2.20E+09 2.23E+09 2.25E+09 2.27E+09 2.30E+09 2.32E+09 4.68E-01 4.66E-01 4.63E-01 4.60E-01 4.69E-01 4.65E-01 4.60E-01 4.55E-01 4.49E-01 4.42E-01 4.35E-01 4.27E-01 4.18E-01 4.08E-01 3.99E-01 3.89E-01 3.78E-01 3.67E-01 3.53E-01 3.41E-01 3.28E-01 3.14E-01 3.01E-01 2.87E-01 2.73E-01 2.58E-01 2.43E-01 2.27E-01 2.11E-01 1.96E-01 1.82E-01 1.66E-01 1.50E-01 1.35E-01 1.20E-01 1.05E-01 9.00E-02 7.60E-02 6.20E-02 4.90E-02 4.00E-02 4.50E-02 5.00E-02 2.00E-02 4.00E-03 -7.00E-03 -5.30E-02 -6.80E-02 -8.00E-02 -8.50E-02 -1.06E-01 -1.24E-01 -1.40E-01 -1.56E-01 -1.71E-01 -1.85E-01 -1.98E-01 -2.11E-01 -2.25E-01 -2.37E-01 -2.49E-01 -2.60E-01 -2.70E-01 -2.81E-01 -2.91E-01 -2.99E-01 -3.08E-01 -3.11E-01 -3.21E-01 -3.27E-01 -3.33E-01 -3.38E-01 -3.41E-01 -3.44E-01 -3.46E-01 -3.46E-01 -3.46E-01 -3.45E-01 -3.44E-01 -3.41E-01 -3.37E-01 -3.32E-01 -3.26E-01 -3.18E-01 -3.09E-01 -2.98E-01 -2.83E-01 -2.63E-01 -2.87E-01 -2.83E-01 -2.70E-01 -2.55E-01 356 -7.80E-02 -1.07E-01 -1.35E-01 -1.66E-01 -1.79E-01 -2.05E-01 -2.32E-01 -2.57E-01 -2.84E-01 -3.09E-01 -3.36E-01 -3.61E-01 -3.86E-01 -4.12E-01 -4.37E-01 -4.63E-01 -4.88E-01 -5.10E-01 -5.35E-01 -5.57E-01 -5.81E-01 -6.02E-01 -6.25E-01 -6.47E-01 -6.68E-01 -6.90E-01 -7.09E-01 -7.29E-01 -7.48E-01 -7.67E-01 -7.84E-01 -8.02E-01 -8.19E-01 -8.36E-01 -8.51E-01 -8.64E-01 -8.79E-01 -8.91E-01 -9.01E-01 -9.11E-01 -9.16E-01 -9.07E-01 -9.01E-01 -9.23E-01 -9.33E-01 -9.36E-01 -8.57E-01 -8.52E-01 -8.47E-01 -8.32E-01 -8.29E-01 -8.27E-01 -8.22E-01 -8.15E-01 -8.07E-01 -7.99E-01 -7.89E-01 -7.80E-01 -7.68E-01 -7.56E-01 -7.44E-01 -7.30E-01 -7.15E-01 -7.01E-01 -6.84E-01 -6.67E-01 -6.50E-01 -6.37E-01 -6.13E-01 -5.93E-01 -5.72E-01 -5.50E-01 -5.28E-01 -5.05E-01 -4.81E-01 -4.57E-01 -4.30E-01 -4.04E-01 -3.76E-01 -3.47E-01 -3.18E-01 -2.88E-01 -2.56E-01 -2.24E-01 -1.87E-01 -1.52E-01 -1.12E-01 -6.80E-02 -6.50E-02 -3.30E-02 6.00E-03 4.60E-02 2.35E+09 2.37E+09 2.40E+09 2.42E+09 2.45E+09 2.47E+09 2.50E+09 2.52E+09 2.55E+09 2.57E+09 2.60E+09 2.62E+09 2.65E+09 2.67E+09 2.70E+09 2.72E+09 2.75E+09 2.77E+09 2.80E+09 2.82E+09 2.84E+09 2.87E+09 2.89E+09 2.92E+09 2.94E+09 2.97E+09 2.99E+09 3.02E+09 3.04E+09 3.07E+09 3.09E+09 3.12E+09 3.14E+09 3.17E+09 3.19E+09 3.22E+09 3.24E+09 3.27E+09 3.29E+09 3.32E+09 3.34E+09 3.36E+09 3.39E+09 3.41E+09 3.44E+09 3.46E+09 -1.60E-02 -2.40E-02 -3.00E-02 -3.50E-02 -3.90E-02 -4.20E-02 -4.40E-02 -4.50E-02 -4.50E-02 -4.30E-02 -3.90E-02 -2.70E-02 -2.80E-02 -2.60E-02 -2.30E-02 -2.10E-02 -1.80E-02 -1.70E-02 -1.70E-02 -2.00E-02 -2.70E-02 -3.20E-02 -1.00E-03 3.20E-02 1.00E-02 -2.50E-02 -6.90E-02 -1.15E-01 -1.58E-01 -1.77E-01 -1.50E-01 -7.90E-02 1.20E-02 9.10E-02 1.50E-01 1.87E-01 2.14E-01 2.50E-01 2.85E-01 3.15E-01 3.36E-01 3.45E-01 3.62E-01 3.87E-01 4.03E-01 4.16E-01 -2.42E-01 -2.28E-01 -2.14E-01 -2.00E-01 -1.87E-01 -1.73E-01 -1.61E-01 -1.47E-01 -1.35E-01 -1.20E-01 -1.04E-01 -1.01E-01 -9.60E-02 -8.70E-02 -7.90E-02 -7.30E-02 -6.80E-02 -6.50E-02 -6.10E-02 -5.60E-02 -4.70E-02 -2.50E-02 2.00E-03 -3.20E-02 -6.20E-02 -7.40E-02 -6.60E-02 -3.70E-02 1.80E-02 1.05E-01 2.02E-01 2.79E-01 3.12E-01 3.09E-01 2.85E-01 2.61E-01 2.50E-01 2.39E-01 2.21E-01 1.95E-01 1.67E-01 1.42E-01 1.36E-01 1.12E-01 8.50E-02 6.00E-02 357 -9.37E-01 -9.36E-01 -9.32E-01 -9.28E-01 -9.20E-01 -9.11E-01 -8.98E-01 -8.86E-01 -8.70E-01 -8.50E-01 -8.26E-01 -8.12E-01 -7.90E-01 -7.64E-01 -7.36E-01 -7.03E-01 -6.69E-01 -6.33E-01 -5.92E-01 -5.49E-01 -5.05E-01 -4.64E-01 -4.50E-01 -4.13E-01 -3.43E-01 -2.75E-01 -2.10E-01 -1.55E-01 -1.17E-01 -1.10E-01 -1.35E-01 -1.79E-01 -2.08E-01 -2.13E-01 -1.93E-01 -1.61E-01 -1.32E-01 -1.08E-01 -8.40E-02 -5.40E-02 -1.90E-02 2.60E-02 5.30E-02 7.50E-02 1.04E-01 1.34E-01 8.50E-02 1.24E-01 1.62E-01 2.02E-01 2.41E-01 2.80E-01 3.20E-01 3.59E-01 3.98E-01 4.36E-01 4.71E-01 4.99E-01 5.39E-01 5.75E-01 6.10E-01 6.42E-01 6.73E-01 7.03E-01 7.29E-01 7.49E-01 7.63E-01 7.66E-01 7.76E-01 8.20E-01 8.40E-01 8.38E-01 8.18E-01 7.83E-01 7.29E-01 6.70E-01 6.25E-01 6.18E-01 6.43E-01 6.83E-01 7.21E-01 7.45E-01 7.53E-01 7.65E-01 7.79E-01 7.91E-01 8.03E-01 8.05E-01 7.89E-01 7.93E-01 7.94E-01 7.93E-01 3.49E+09 3.51E+09 3.54E+09 3.56E+09 3.59E+09 3.61E+09 3.64E+09 3.66E+09 3.69E+09 3.71E+09 3.74E+09 3.76E+09 3.79E+09 3.81E+09 3.84E+09 3.86E+09 3.89E+09 3.91E+09 3.93E+09 3.96E+09 3.98E+09 4.01E+09 4.03E+09 4.06E+09 4.08E+09 4.11E+09 4.13E+09 4.16E+09 4.18E+09 4.21E+09 4.23E+09 4.26E+09 4.28E+09 4.31E+09 4.33E+09 4.36E+09 4.38E+09 4.41E+09 4.43E+09 4.45E+09 4.48E+09 4.50E+09 4.53E+09 4.55E+09 4.58E+09 4.60E+09 4.26E-01 4.33E-01 4.38E-01 4.41E-01 4.42E-01 4.43E-01 4.41E-01 4.38E-01 4.33E-01 4.28E-01 4.20E-01 4.12E-01 4.02E-01 3.91E-01 3.79E-01 3.66E-01 3.58E-01 3.44E-01 3.27E-01 3.10E-01 2.93E-01 2.75E-01 2.57E-01 2.38E-01 2.19E-01 2.00E-01 1.80E-01 1.59E-01 1.40E-01 1.19E-01 9.80E-02 7.80E-02 5.90E-02 3.90E-02 2.00E-02 3.00E-03 -1.20E-02 -1.20E-02 1.00E-02 7.80E-02 7.20E-02 -2.00E-03 -5.40E-02 -8.40E-02 -1.05E-01 -1.21E-01 3.40E-02 8.00E-03 -1.70E-02 -4.30E-02 -6.50E-02 -9.00E-02 -1.12E-01 -1.35E-01 -1.57E-01 -1.78E-01 -1.99E-01 -2.20E-01 -2.39E-01 -2.57E-01 -2.74E-01 -2.88E-01 -3.04E-01 -3.21E-01 -3.36E-01 -3.48E-01 -3.61E-01 -3.71E-01 -3.80E-01 -3.88E-01 -3.94E-01 -3.99E-01 -4.03E-01 -4.06E-01 -4.06E-01 -4.06E-01 -4.03E-01 -3.99E-01 -3.92E-01 -3.84E-01 -3.72E-01 -3.55E-01 -3.32E-01 -3.01E-01 -2.71E-01 -2.80E-01 -3.74E-01 -4.01E-01 -3.82E-01 -3.57E-01 -3.33E-01 -3.10E-01 358 1.66E-01 1.97E-01 2.28E-01 2.59E-01 2.89E-01 3.19E-01 3.47E-01 3.76E-01 4.03E-01 4.30E-01 4.57E-01 4.82E-01 5.07E-01 5.33E-01 5.58E-01 5.80E-01 5.99E-01 6.22E-01 6.44E-01 6.66E-01 6.86E-01 7.05E-01 7.25E-01 7.42E-01 7.58E-01 7.73E-01 7.89E-01 8.02E-01 8.16E-01 8.27E-01 8.39E-01 8.49E-01 8.57E-01 8.64E-01 8.70E-01 8.72E-01 8.68E-01 8.57E-01 8.26E-01 7.45E-01 7.38E-01 7.93E-01 8.23E-01 8.31E-01 8.32E-01 8.25E-01 7.91E-01 7.87E-01 7.81E-01 7.73E-01 7.62E-01 7.51E-01 7.39E-01 7.25E-01 7.12E-01 6.97E-01 6.80E-01 6.63E-01 6.44E-01 6.26E-01 6.05E-01 5.83E-01 5.62E-01 5.42E-01 5.20E-01 4.95E-01 4.70E-01 4.44E-01 4.17E-01 3.90E-01 3.62E-01 3.32E-01 3.03E-01 2.71E-01 2.40E-01 2.07E-01 1.74E-01 1.38E-01 1.02E-01 6.40E-02 2.30E-02 -2.00E-02 -6.80E-02 -1.18E-01 -1.75E-01 -1.91E-01 -1.27E-01 -1.28E-01 -1.72E-01 -2.19E-01 -2.64E-01 -3.08E-01 4.63E+09 4.65E+09 4.68E+09 4.70E+09 4.73E+09 4.75E+09 4.78E+09 4.80E+09 4.83E+09 4.85E+09 4.88E+09 4.90E+09 4.93E+09 4.95E+09 4.98E+09 5.00E+09 -1.33E-01 -1.43E-01 -1.50E-01 -1.56E-01 -1.61E-01 -1.63E-01 -1.64E-01 -1.62E-01 -1.59E-01 -1.56E-01 -1.49E-01 -1.43E-01 -1.34E-01 -1.25E-01 -1.13E-01 -1.03E-01 -2.89E-01 -2.67E-01 -2.47E-01 -2.24E-01 -2.03E-01 -1.82E-01 -1.61E-01 -1.42E-01 -1.22E-01 -1.03E-01 -8.40E-02 -6.70E-02 -5.00E-02 -3.40E-02 -1.90E-02 -5.00E-03 8.17E-01 8.06E-01 7.91E-01 7.74E-01 7.55E-01 7.34E-01 7.11E-01 6.86E-01 6.59E-01 6.30E-01 6.00E-01 5.68E-01 5.33E-01 4.98E-01 4.61E-01 4.23E-01 G.4. Test Structure 3 G.4.1. Circuit Optimization Input File .option accurate dccap=1 node nopage ingold=2 post acct=2 probe .subckt mstl_pad 1 5 r1 1 2 r2 l1 2 3 l2 c1 3 0 c2 r2 3 0 10g r1r 3 4 r2 l2r 4 5 l2 .ends .subckt mstl_sq 1 5 r1 1 2 rsq l1 2 3 lsq c1 3 0 csq *r2 3 0 rl r1r 3 4 rsq l2r 4 5 lsq cc1 1 3 c_cou cc2 3 5 c_cou .ends .subckt mstl_sq5 1 6 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq 359 -3.50E-01 -3.91E-01 -4.29E-01 -4.69E-01 -5.07E-01 -5.43E-01 -5.78E-01 -6.13E-01 -6.46E-01 -6.78E-01 -7.08E-01 -7.37E-01 -7.64E-01 -7.91E-01 -8.13E-01 -8.36E-01 x4 4 5 mstl_sq x5 5 6 mstl_sq .ends .subckt mstl_sq4 1 5 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq x4 4 5 mstl_sq .ends .subckt ind_blk 1 3 r1 1 2 ri l1 2 3 li c1 1 0 ci cs 1 3 csi .ends .subckt ind_nc 1 5 10 14 rl1 1 2 ri ll1 2 3 li c1 3 0 ci rr1 3 4 ri lr1 4 5 li rl2 10 11 ri2 ll2 11 12 li2 c2 12 0 ci2 rr2 12 13 ri2 lr2 13 14 li2 cc12 3 12 csi .ends .subckt vind 1 2 l1 1 2 lvia c1 1 2 cvia .ends .subckt ind_blk_3 x1 1 2 3 4 ind_nc lv1 2 3 lvia cv1 2 3 cvia lv2 4 5 lvia cv2 4 5 cvia rt1 5 0 1g 1 10 20 5 14 24 x2 10 11 12 13 ind_nc lv3 11 12 lvia cv3 11 12 cvia lv4 13 14 lvia cv4 13 14 cvia rt2 14 0 1g x3 20 21 22 23 ind_nc lv5 21 22 lvia cv5 21 22 cvia lv6 23 24 lvia cv6 23 24 cvia rt3 24 0 1g k1 lv1 lv4 k=cou_l k2 lv3 lv6 k=cou_l cc1 2 13 c_cou cc2 11 23 c_cou 360 rg1 1 0 1g rg2 5 0 1g rg3 10 0 1g rg4 2 13 1g rg5 11 23 1g .ends .subckt inductor 4 24 x2 4 5 6 7 8 9 ind_blk_3 x3 7 8 9 10 11 12 ind_blk_3 x4 10 11 12 13 14 15 ind_blk_3 x5 13 14 15 16 17 18 ind_blk_3 x6 16 17 18 19 20 21 ind_blk_3 x7 19 20 21 22 23 24 ind_blk_3 ls1 5 6 1e-10 ls2 22 23 1e-10 .ends x1 x2 x7 r0 2 3 4 8 3 4 8 0 mstl_pad inductor mstl_pad 1g vpl 2 0 pulse(0 2.5 2n 0.1n 0.2n 10n 20n) ac 1 *.tran 1p 40n 1p *.ac lin 100 45mega 4giga .net v(8) vpl rin=50 rout=50 .param .param c_cou = opt1(1.550e-13,1e-14,1e-12) .param rl = 1.000e+02 $ 0. .param cou_l = opt1(3.000e-01,-1,1) .param .param .param .param .param .param .param .param .param ri li ci csi ri2 li2 ci2 lvia cvia = = = = = = = = = .param r2 .param l2 .param c2 0. 1.000e-02 7.418e-11 1.385e-13 8.234e-15 1.703e-02 4.432e-10 1.017e-13 4.920e-10 8.663e-13 $ $ $ $ $ $ $ $ $ 8.159e-03 2.021e+00 2.098e+01 1.820e-01 1.393e-01 1.187e+01 1.532e+01 3.555e+01 1.391e+01 4.661e-02 -3.904e-05 7.933e-06 3.444e-04 -8.590e-05 1.300e-05 -1.432e-06 1.982e-06 3.154e-07 = 1.000e-05 = 3.228e-10 = 1.863e-13 $ $ $ 2.131e+01 1.886e+01 1.432e+01 -4.957e-06 1.186e-06 -1.809e-05 .ac data=measured optimize=opt1 + results=comp1,comp2,comp5,comp6 + model=converge .model converge opt relin=1e-5 relout=1e-4 close=500 itropt=500 .measure .measure .measure .measure .measure .measure .measure .measure ac ac ac ac ac ac ac ac comp1 comp2 comp3 comp4 comp5 comp6 comp7 comp8 err1 err1 err1 err1 err1 err1 err1 err1 par(s11r) par(s11i) par(s12r) par(s12i) par(s21r) par(s21i) par(s22r) par(s22i) s11(r) s11(i) s12(r) s12(i) s21(r) s21(i) s22(r) s22(i) .ac data=measured .print par(s11r) s11(r) par(s11i) s11(i) .print par(s12r) s12(r) par(s12i) s12(i) .data measured 361 file = ‘dev23.txt’ freq=1 s11r=2 s11I=3 s21r=4 s21I=5 s12r=6 s12I=7 s22r=8 s22I=9 .enddata .param freq=100m, s11r=0, s11i=0, s21r=0, s21i=0, s12r=0, s12i=0, s22r=0, s22i=0 .end G.4.2. Measured S-Parameter Data freq 4.50E+07 6.98E+07 9.46E+07 1.19E+08 1.44E+08 1.69E+08 1.94E+08 2.18E+08 2.43E+08 2.68E+08 2.93E+08 3.18E+08 3.42E+08 3.67E+08 3.92E+08 4.17E+08 4.41E+08 4.66E+08 4.91E+08 5.16E+08 5.40E+08 5.65E+08 5.90E+08 6.15E+08 6.40E+08 6.64E+08 6.89E+08 7.14E+08 7.39E+08 7.63E+08 7.88E+08 8.13E+08 8.38E+08 8.63E+08 S11(R) Meas. 2.40E-02 3.00E-02 4.90E-02 5.60E-02 6.10E-02 1.16E-01 1.54E-01 1.80E-01 2.10E-01 2.37E-01 2.60E-01 2.80E-01 2.97E-01 3.26E-01 3.24E-01 3.29E-01 3.37E-01 3.57E-01 4.00E-01 3.68E-01 3.91E-01 3.80E-01 4.07E-01 4.81E-01 5.83E-01 6.43E-01 7.06E-01 7.06E-01 6.99E-01 6.70E-01 6.51E-01 6.80E-01 6.98E-01 6.87E-01 S11(I) Meas. 6.00E-02 8.50E-02 1.08E-01 1.37E-01 1.72E-01 2.05E-01 2.10E-01 2.11E-01 2.10E-01 2.08E-01 1.98E-01 1.91E-01 1.99E-01 1.71E-01 1.56E-01 1.54E-01 1.61E-01 1.69E-01 1.38E-01 1.14E-01 1.53E-01 1.32E-01 2.08E-01 2.12E-01 2.11E-01 1.41E-01 8.80E-02 -2.90E-02 -8.70E-02 -1.41E-01 -1.61E-01 -1.83E-01 -2.43E-01 -3.22E-01 362 S21(R) Meas. 9.79E-01 9.57E-01 9.29E-01 8.93E-01 8.31E-01 8.20E-01 7.83E-01 7.34E-01 6.81E-01 6.23E-01 5.58E-01 4.88E-01 4.14E-01 3.58E-01 2.69E-01 1.91E-01 1.25E-01 7.80E-02 3.90E-02 -6.70E-02 -1.20E-01 -1.97E-01 -2.32E-01 -2.06E-01 -1.57E-01 -1.38E-01 -1.17E-01 -1.28E-01 -1.72E-01 -2.29E-01 -2.71E-01 -2.60E-01 -2.52E-01 -2.67E-01 S21(I) Meas. -1.48E-01 -2.21E-01 -2.94E-01 -3.63E-01 -4.15E-01 -4.56E-01 -5.21E-01 -5.80E-01 -6.31E-01 -6.75E-01 -7.20E-01 -7.50E-01 -7.57E-01 -7.91E-01 -8.03E-01 -7.90E-01 -7.66E-01 -7.46E-01 -7.57E-01 -7.57E-01 -6.76E-01 -6.53E-01 -5.22E-01 -4.70E-01 -4.11E-01 -4.24E-01 -4.11E-01 -4.69E-01 -4.75E-01 -4.68E-01 -4.24E-01 -3.78E-01 -3.68E-01 -3.79E-01 8.87E+08 9.12E+08 9.37E+08 9.62E+08 9.86E+08 1.01E+09 1.04E+09 1.06E+09 1.09E+09 1.11E+09 1.14E+09 1.16E+09 1.18E+09 1.21E+09 1.23E+09 1.26E+09 1.28E+09 1.31E+09 1.33E+09 1.36E+09 1.38E+09 1.41E+09 1.43E+09 1.46E+09 1.48E+09 1.51E+09 1.53E+09 1.56E+09 1.58E+09 1.61E+09 1.63E+09 1.66E+09 1.68E+09 1.70E+09 1.73E+09 1.75E+09 1.78E+09 1.80E+09 1.83E+09 1.85E+09 1.88E+09 1.90E+09 1.93E+09 1.95E+09 1.98E+09 2.00E+09 6.59E-01 6.17E-01 5.72E-01 5.22E-01 4.64E-01 4.07E-01 3.38E-01 2.67E-01 1.89E-01 1.07E-01 2.90E-02 -4.30E-02 -1.08E-01 -1.55E-01 -1.85E-01 -1.99E-01 -1.98E-01 -1.88E-01 -1.62E-01 -1.20E-01 -7.20E-02 -2.70E-02 7.00E-03 3.10E-02 4.20E-02 4.10E-02 3.30E-02 1.80E-02 1.00E-03 -1.60E-02 -3.30E-02 -4.60E-02 -5.60E-02 -6.00E-02 -6.70E-02 -5.60E-02 -4.60E-02 -3.10E-02 -1.40E-02 4.00E-03 2.50E-02 4.40E-02 6.10E-02 7.70E-02 9.20E-02 1.07E-01 -3.92E-01 -4.48E-01 -4.98E-01 -5.45E-01 -5.80E-01 -6.15E-01 -6.40E-01 -6.57E-01 -6.64E-01 -6.54E-01 -6.29E-01 -5.91E-01 -5.38E-01 -4.70E-01 -4.01E-01 -3.27E-01 -2.60E-01 -2.01E-01 -1.41E-01 -9.60E-02 -7.20E-02 -6.60E-02 -7.60E-02 -9.20E-02 -1.13E-01 -1.31E-01 -1.45E-01 -1.53E-01 -1.53E-01 -1.47E-01 -1.36E-01 -1.18E-01 -9.80E-02 -7.60E-02 -5.20E-02 -2.80E-02 -5.00E-03 1.40E-02 3.00E-02 4.30E-02 4.90E-02 5.20E-02 5.30E-02 5.20E-02 4.80E-02 4.30E-02 363 -2.95E-01 -3.32E-01 -3.68E-01 -4.05E-01 -4.44E-01 -4.77E-01 -5.16E-01 -5.55E-01 -5.96E-01 -6.35E-01 -6.64E-01 -6.87E-01 -6.97E-01 -6.83E-01 -6.46E-01 -5.86E-01 -5.09E-01 -4.23E-01 -3.09E-01 -1.72E-01 -2.40E-02 1.27E-01 2.70E-01 4.04E-01 5.22E-01 6.24E-01 7.10E-01 7.81E-01 8.36E-01 8.77E-01 9.03E-01 9.17E-01 9.19E-01 9.10E-01 8.94E-01 8.63E-01 8.24E-01 7.78E-01 7.27E-01 6.70E-01 6.09E-01 5.46E-01 4.82E-01 4.15E-01 3.48E-01 2.79E-01 -3.83E-01 -3.72E-01 -3.55E-01 -3.37E-01 -3.05E-01 -2.76E-01 -2.39E-01 -1.95E-01 -1.43E-01 -7.70E-02 3.00E-03 9.20E-02 1.95E-01 3.11E-01 4.23E-01 5.34E-01 6.26E-01 7.21E-01 8.06E-01 8.68E-01 8.99E-01 8.99E-01 8.72E-01 8.21E-01 7.52E-01 6.69E-01 5.77E-01 4.78E-01 3.74E-01 2.68E-01 1.63E-01 5.70E-02 -4.60E-02 -1.46E-01 -2.40E-01 -3.34E-01 -4.21E-01 -5.00E-01 -5.71E-01 -6.34E-01 -6.91E-01 -7.40E-01 -7.81E-01 -8.19E-01 -8.50E-01 -8.75E-01 2.03E+09 2.05E+09 2.08E+09 2.10E+09 2.13E+09 2.15E+09 2.18E+09 2.20E+09 2.23E+09 2.25E+09 2.27E+09 2.30E+09 2.32E+09 2.35E+09 2.37E+09 2.40E+09 2.42E+09 2.45E+09 2.47E+09 2.50E+09 2.52E+09 2.55E+09 2.57E+09 2.60E+09 2.62E+09 2.65E+09 2.67E+09 2.70E+09 2.72E+09 2.75E+09 2.77E+09 2.80E+09 2.82E+09 2.84E+09 2.87E+09 2.89E+09 2.92E+09 2.94E+09 2.97E+09 2.99E+09 3.02E+09 3.04E+09 3.07E+09 3.09E+09 3.12E+09 3.14E+09 1.20E-01 1.33E-01 1.41E-01 1.44E-01 1.45E-01 1.45E-01 1.43E-01 1.35E-01 1.26E-01 1.08E-01 9.40E-02 8.10E-02 6.40E-02 4.70E-02 3.10E-02 1.50E-02 0.00E+00 -1.30E-02 -2.40E-02 -3.10E-02 -3.50E-02 -3.40E-02 -2.90E-02 -1.50E-02 1.00E-03 2.00E-02 4.60E-02 7.80E-02 1.14E-01 1.58E-01 2.04E-01 2.54E-01 3.05E-01 3.59E-01 4.12E-01 4.63E-01 5.15E-01 5.58E-01 6.01E-01 6.36E-01 6.66E-01 6.91E-01 7.08E-01 7.16E-01 7.18E-01 7.10E-01 3.50E-02 2.50E-02 1.20E-02 -2.00E-03 -1.40E-02 -2.60E-02 -4.00E-02 -5.10E-02 -6.40E-02 -7.10E-02 -7.10E-02 -7.20E-02 -7.00E-02 -6.40E-02 -5.50E-02 -4.40E-02 -2.60E-02 -7.00E-03 1.70E-02 4.30E-02 7.20E-02 1.04E-01 1.38E-01 1.73E-01 2.02E-01 2.34E-01 2.66E-01 2.96E-01 3.21E-01 3.42E-01 3.57E-01 3.67E-01 3.70E-01 3.64E-01 3.54E-01 3.34E-01 3.09E-01 2.77E-01 2.38E-01 1.96E-01 1.48E-01 9.70E-02 4.40E-02 -1.10E-02 -6.50E-02 -1.19E-01 364 2.08E-01 1.37E-01 6.70E-02 -1.00E-03 -7.00E-02 -1.38E-01 -2.01E-01 -2.64E-01 -3.22E-01 -3.76E-01 -4.41E-01 -5.01E-01 -5.56E-01 -6.09E-01 -6.59E-01 -7.05E-01 -7.48E-01 -7.87E-01 -8.20E-01 -8.49E-01 -8.74E-01 -8.91E-01 -9.02E-01 -9.03E-01 -9.04E-01 -8.97E-01 -8.79E-01 -8.53E-01 -8.21E-01 -7.83E-01 -7.38E-01 -6.87E-01 -6.32E-01 -5.74E-01 -5.15E-01 -4.55E-01 -3.93E-01 -3.35E-01 -2.76E-01 -2.18E-01 -1.62E-01 -1.07E-01 -5.40E-02 -1.00E-03 5.10E-02 1.04E-01 -8.96E-01 -9.08E-01 -9.15E-01 -9.18E-01 -9.14E-01 -9.05E-01 -8.92E-01 -8.74E-01 -8.49E-01 -8.34E-01 -8.09E-01 -7.76E-01 -7.38E-01 -6.95E-01 -6.50E-01 -6.01E-01 -5.47E-01 -4.90E-01 -4.29E-01 -3.64E-01 -2.98E-01 -2.27E-01 -1.53E-01 -8.20E-02 -1.30E-02 6.20E-02 1.36E-01 2.08E-01 2.74E-01 3.37E-01 3.97E-01 4.48E-01 4.93E-01 5.31E-01 5.64E-01 5.87E-01 6.06E-01 6.18E-01 6.25E-01 6.28E-01 6.25E-01 6.20E-01 6.11E-01 5.98E-01 5.82E-01 5.63E-01 3.17E+09 3.19E+09 3.22E+09 3.24E+09 3.27E+09 3.29E+09 3.32E+09 3.34E+09 3.36E+09 3.39E+09 3.41E+09 3.44E+09 3.46E+09 3.49E+09 3.51E+09 3.54E+09 3.56E+09 3.59E+09 3.61E+09 3.64E+09 3.66E+09 3.69E+09 3.71E+09 3.74E+09 3.76E+09 3.79E+09 3.81E+09 3.84E+09 3.86E+09 3.89E+09 3.91E+09 3.93E+09 3.96E+09 3.98E+09 4.01E+09 4.03E+09 4.06E+09 4.08E+09 4.11E+09 4.13E+09 4.16E+09 4.18E+09 4.21E+09 4.23E+09 4.26E+09 4.28E+09 6.93E-01 6.73E-01 6.54E-01 6.36E-01 6.12E-01 5.91E-01 5.81E-01 5.89E-01 5.92E-01 5.71E-01 5.69E-01 5.60E-01 5.60E-01 5.58E-01 5.48E-01 5.30E-01 5.13E-01 4.86E-01 4.55E-01 4.23E-01 3.83E-01 3.43E-01 2.98E-01 2.50E-01 2.00E-01 1.45E-01 8.90E-02 3.00E-02 -3.20E-02 -9.50E-02 -1.61E-01 -2.26E-01 -2.92E-01 -3.56E-01 -4.18E-01 -4.71E-01 -5.12E-01 -5.30E-01 -5.15E-01 -4.60E-01 -3.63E-01 -2.39E-01 -1.14E-01 -1.50E-02 4.50E-02 6.60E-02 -1.65E-01 -2.03E-01 -2.33E-01 -2.62E-01 -2.85E-01 -2.97E-01 -3.01E-01 -3.14E-01 -3.50E-01 -3.72E-01 -3.91E-01 -4.15E-01 -4.38E-01 -4.71E-01 -5.08E-01 -5.47E-01 -5.82E-01 -6.20E-01 -6.55E-01 -6.86E-01 -7.17E-01 -7.46E-01 -7.68E-01 -7.89E-01 -8.06E-01 -8.18E-01 -8.26E-01 -8.28E-01 -8.26E-01 -8.16E-01 -7.99E-01 -7.72E-01 -7.35E-01 -6.90E-01 -6.28E-01 -5.52E-01 -4.58E-01 -3.46E-01 -2.26E-01 -1.11E-01 -2.00E-02 2.40E-02 1.50E-02 -4.10E-02 -1.21E-01 -2.06E-01 365 1.59E-01 2.11E-01 2.56E-01 2.91E-01 3.27E-01 3.56E-01 3.65E-01 3.50E-01 3.17E-01 3.12E-01 3.05E-01 2.93E-01 2.70E-01 2.42E-01 2.17E-01 1.96E-01 1.74E-01 1.56E-01 1.40E-01 1.26E-01 1.15E-01 1.06E-01 9.80E-02 9.00E-02 8.60E-02 8.30E-02 8.30E-02 8.50E-02 8.80E-02 9.30E-02 1.01E-01 1.09E-01 1.17E-01 1.27E-01 1.35E-01 1.39E-01 1.35E-01 1.15E-01 6.80E-02 -1.20E-02 -1.32E-01 -2.79E-01 -4.31E-01 -5.66E-01 -6.70E-01 -7.38E-01 5.35E-01 4.99E-01 4.53E-01 4.07E-01 3.55E-01 2.93E-01 2.20E-01 1.54E-01 1.12E-01 9.30E-02 5.60E-02 1.80E-02 -1.80E-02 -4.10E-02 -5.70E-02 -7.00E-02 -8.00E-02 -8.50E-02 -8.90E-02 -9.20E-02 -9.40E-02 -9.60E-02 -9.80E-02 -9.80E-02 -1.00E-01 -1.01E-01 -1.02E-01 -1.07E-01 -1.12E-01 -1.20E-01 -1.32E-01 -1.48E-01 -1.70E-01 -1.96E-01 -2.33E-01 -2.80E-01 -3.41E-01 -4.15E-01 -4.98E-01 -5.81E-01 -6.43E-01 -6.67E-01 -6.40E-01 -5.68E-01 -4.66E-01 -3.50E-01 4.31E+09 4.33E+09 4.36E+09 4.38E+09 4.41E+09 4.43E+09 4.45E+09 4.48E+09 4.50E+09 4.53E+09 4.55E+09 4.58E+09 4.60E+09 4.63E+09 4.65E+09 4.68E+09 4.70E+09 4.73E+09 4.75E+09 4.78E+09 4.80E+09 4.83E+09 4.85E+09 4.88E+09 4.90E+09 4.93E+09 4.95E+09 4.98E+09 5.00E+09 5.80E-02 2.80E-02 -1.40E-02 -6.20E-02 -1.10E-01 -1.61E-01 -2.19E-01 -2.74E-01 -3.18E-01 -3.53E-01 -3.84E-01 -4.13E-01 -4.38E-01 -4.60E-01 -4.77E-01 -4.91E-01 -4.99E-01 -5.06E-01 -5.08E-01 -5.07E-01 -5.04E-01 -4.96E-01 -4.83E-01 -4.64E-01 -4.47E-01 -4.23E-01 -3.98E-01 -3.66E-01 -3.30E-01 -2.83E-01 -3.44E-01 -3.88E-01 -4.18E-01 -4.36E-01 -4.46E-01 -4.46E-01 -4.30E-01 -3.99E-01 -3.69E-01 -3.37E-01 -3.03E-01 -2.64E-01 -2.26E-01 -1.84E-01 -1.43E-01 -9.60E-02 -5.20E-02 -6.00E-03 4.10E-02 8.70E-02 1.35E-01 1.86E-01 2.31E-01 2.77E-01 3.24E-01 3.73E-01 4.17E-01 4.65E-01 -7.77E-01 -7.93E-01 -7.91E-01 -7.77E-01 -7.53E-01 -7.18E-01 -6.80E-01 -6.53E-01 -6.36E-01 -6.03E-01 -5.62E-01 -5.17E-01 -4.68E-01 -4.17E-01 -3.63E-01 -3.05E-01 -2.45E-01 -1.83E-01 -1.18E-01 -5.30E-02 1.30E-02 7.90E-02 1.38E-01 2.02E-01 2.67E-01 3.27E-01 3.83E-01 4.31E-01 4.74E-01 G.5. Test Structure 4 G.5.1. Circuit Optimization Input File .option accurate dccap=1 node nopage ingold=2 post acct=2 probe .subckt mstl_pad 1 5 r1 1 2 r2 l1 2 3 l2 c1 3 0 c2 r2 3 0 10g 366 -2.34E-01 -1.26E-01 -2.60E-02 6.50E-02 1.46E-01 2.21E-01 2.73E-01 3.16E-01 3.71E-01 4.31E-01 4.84E-01 5.32E-01 5.74E-01 6.12E-01 6.46E-01 6.73E-01 6.97E-01 7.15E-01 7.26E-01 7.30E-01 7.29E-01 7.17E-01 7.03E-01 6.87E-01 6.57E-01 6.20E-01 5.76E-01 5.23E-01 4.67E-01 r1r 3 4 r2 l2r 4 5 l2 .ends .subckt line_seg 1 5 6 10 r1 1 2 rsq l1 2 3 lsq c1 3 0 cmid rgm1 3 0 10g r1r 3 4 rsq l1r 4 5 lsq ce2 5 10 csq rg1 5 10 rg ce1 1 6 csq rg2 1 6 rg r2 6 7 rsq l2 7 8 lsq c2 8 0 cmid rgm2 8 0 10g r2r 8 9 rsq l2r 9 10 lsq .ends .subckt line1x3 1 2 3 4 x1 1 2 line_seg x2 2 3 line_seg x3 3 4 line_seg .ends .subckt line3x3 4 15 x1 1 2 10 11 line_seg x2 2 3 11 12 line_seg x3 4 5 13 14 line_seg x4 5 6 14 15 line_seg x5 7 8 16 17 line_seg x6 8 9 17 18 line_seg x7 1 4 10 13 line_seg x8 2 5 11 14 line_seg x9 3 6 12 15 line_seg x10 4 7 13 16 line_seg x11 5 8 14 17 line_seg x12 6 9 15 18 line_seg rs 10 0 0.1 .ends .subckt mstl_sq5 1 6 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq x4 4 5 mstl_sq x5 5 6 mstl_sq .ends .subckt mstl_sq4 1 5 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq x4 4 5 mstl_sq .ends x1 x2 x7 r0 1 2 4 8 2 4 8 0 mstl_pad line3x3 mstl_pad 1g 367 vpl 1 0 pulse(0 2.5 2n 0.1n 0.2n 10n 20n) ac 1 *.tran 1p 40n 1p *.ac dec 60 1mega 15giga .net v(8) vpl rin=50 rout=50 .param + c_cou = opt1(6.4e-14,1f,1n) + rl = opt1(1e2,1,1e8) + ri = opt1(0.05,0.01,10) + li = opt1(2.0e-9,1f,1u) + ci = opt1(4.5e-13,0.01f,1n) + csi = opt1(1.4e-15,0.01f,1n) + cou_l = opt1(0.1,0.01,1) + + + + + + rsq = opt1(0.50,0.01,10) rg = opt1(1e7,1e6,1e11) lsq = opt1(0.1e-9,1f,1u) csq = opt1(3.4e-14,0.01f,1n) cmid = opt1(1.4e-14,0.01f,1n) csv = opt1(8.4e-15,0.01f,1n) .param r2 .param l2 .param c2 = 1.000e-05 = 3.228e-10 = 1.863e-13 $ $ $ 2.131e+01 1.886e+01 1.432e+01 -4.957e-06 1.186e-06 -1.809e-05 .ac data=measured optimize=opt1 + results=comp1,comp2,comp5,comp6 + model=converge .model converge opt relin=1e-4 relout=1e-4 close=100 itropt=900 .measure .measure .measure .measure ac ac ac ac comp1 comp2 comp3 comp4 err1 err1 err1 err1 par(s11r) par(s11i) par(s12r) par(s12i) s11(r) s11(i) s12(r) s12(i) .ac data=measured .print par(s11r) s11(r) par(s11i) s11(i) .print par(s12r) s12(r) par(s12i) s12(i) .data measured file = ‘dev26.txt’ freq=1 s11r=2 s11I=3 s21r=4 s21I=5 s12r=6 s12I=7 s22r=8 s22I=9 .enddata .param freq=100m, s11r=0, s11i=0, s21r=0, s21i=0, s12r=0, s12i=0, s22r=0, s22i=0 .end G.5.2. Measured S-Parameter Data freq 4.50E+07 6.98E+07 9.46E+07 1.19E+08 1.44E+08 1.69E+08 1.94E+08 S11(R) Meas. 9.69E-01 9.16E-01 8.56E-01 7.78E-01 6.87E-01 6.24E-01 5.64E-01 S11(I) Meas. -2.17E-01 -3.20E-01 -4.16E-01 -4.92E-01 -5.43E-01 -5.74E-01 -6.14E-01 368 S21(R) Meas. 3.00E-02 7.20E-02 1.23E-01 1.83E-01 2.36E-01 3.02E-01 3.74E-01 S21(I) Meas. 1.47E-01 2.19E-01 2.77E-01 3.22E-01 3.51E-01 3.78E-01 3.84E-01 2.18E+08 2.43E+08 2.68E+08 2.93E+08 3.18E+08 3.42E+08 3.67E+08 3.92E+08 4.17E+08 4.41E+08 4.66E+08 4.91E+08 5.16E+08 5.40E+08 5.65E+08 5.90E+08 6.15E+08 6.40E+08 6.64E+08 6.89E+08 7.14E+08 7.39E+08 7.63E+08 7.88E+08 8.13E+08 8.38E+08 8.63E+08 8.87E+08 9.12E+08 9.37E+08 9.62E+08 9.86E+08 1.01E+09 1.04E+09 1.06E+09 1.09E+09 1.11E+09 1.14E+09 1.16E+09 1.18E+09 1.21E+09 1.23E+09 1.26E+09 1.28E+09 1.31E+09 1.33E+09 4.91E-01 4.23E-01 3.58E-01 2.97E-01 2.39E-01 1.89E-01 1.45E-01 1.00E-01 6.40E-02 3.30E-02 1.00E-03 -2.20E-02 -6.10E-02 -1.03E-01 -1.36E-01 -1.95E-01 -2.29E-01 -3.00E-01 -2.94E-01 -3.00E-01 -1.56E-01 -1.16E-01 -9.00E-02 -1.22E-01 -1.42E-01 -1.49E-01 -1.38E-01 -1.33E-01 -1.37E-01 -1.49E-01 -1.48E-01 -1.53E-01 -1.60E-01 -1.69E-01 -1.78E-01 -1.86E-01 -1.97E-01 -2.09E-01 -2.18E-01 -2.27E-01 -2.36E-01 -2.45E-01 -2.58E-01 -2.67E-01 -2.71E-01 -2.72E-01 -6.41E-01 -6.59E-01 -6.68E-01 -6.71E-01 -6.67E-01 -6.51E-01 -6.52E-01 -6.37E-01 -6.22E-01 -6.07E-01 -6.00E-01 -5.91E-01 -5.93E-01 -5.78E-01 -5.68E-01 -5.42E-01 -5.09E-01 -4.48E-01 -3.58E-01 -2.66E-01 -2.22E-01 -2.49E-01 -3.08E-01 -3.36E-01 -3.21E-01 -3.11E-01 -3.08E-01 -3.13E-01 -3.22E-01 -3.24E-01 -3.22E-01 -3.28E-01 -3.32E-01 -3.34E-01 -3.35E-01 -3.38E-01 -3.39E-01 -3.38E-01 -3.34E-01 -3.32E-01 -3.29E-01 -3.28E-01 -3.22E-01 -3.13E-01 -3.04E-01 -3.00E-01 369 4.33E-01 4.88E-01 5.33E-01 5.74E-01 6.08E-01 6.33E-01 6.65E-01 6.81E-01 6.95E-01 7.08E-01 7.20E-01 7.35E-01 7.40E-01 7.28E-01 7.23E-01 6.86E-01 6.67E-01 6.11E-01 6.13E-01 6.02E-01 7.24E-01 7.86E-01 8.31E-01 8.21E-01 8.12E-01 8.16E-01 8.36E-01 8.52E-01 8.57E-01 8.51E-01 8.57E-01 8.56E-01 8.55E-01 8.49E-01 8.43E-01 8.37E-01 8.28E-01 8.17E-01 8.08E-01 7.99E-01 7.90E-01 7.80E-01 7.66E-01 7.55E-01 7.48E-01 7.44E-01 3.78E-01 3.65E-01 3.45E-01 3.20E-01 2.95E-01 2.73E-01 2.41E-01 2.10E-01 1.85E-01 1.62E-01 1.32E-01 1.06E-01 6.40E-02 3.60E-02 6.00E-03 -1.90E-02 -2.20E-02 -1.20E-02 4.30E-02 9.30E-02 1.44E-01 1.11E-01 3.70E-02 -1.30E-02 -2.70E-02 -4.00E-02 -6.00E-02 -8.70E-02 -1.21E-01 -1.49E-01 -1.69E-01 -1.97E-01 -2.23E-01 -2.48E-01 -2.70E-01 -2.93E-01 -3.15E-01 -3.36E-01 -3.51E-01 -3.70E-01 -3.86E-01 -4.03E-01 -4.18E-01 -4.29E-01 -4.37E-01 -4.49E-01 1.36E+09 1.38E+09 1.41E+09 1.43E+09 1.46E+09 1.48E+09 1.51E+09 1.53E+09 1.56E+09 1.58E+09 1.61E+09 1.63E+09 1.66E+09 1.68E+09 1.70E+09 1.73E+09 1.75E+09 1.78E+09 1.80E+09 1.83E+09 1.85E+09 1.88E+09 1.90E+09 1.93E+09 1.95E+09 1.98E+09 2.00E+09 2.03E+09 2.05E+09 2.08E+09 2.10E+09 2.13E+09 2.15E+09 2.18E+09 2.20E+09 2.23E+09 2.25E+09 2.27E+09 2.30E+09 2.32E+09 2.35E+09 2.37E+09 2.40E+09 2.42E+09 2.45E+09 2.47E+09 -2.76E-01 -2.81E-01 -2.87E-01 -2.93E-01 -3.01E-01 -3.09E-01 -3.18E-01 -3.26E-01 -3.33E-01 -3.39E-01 -3.43E-01 -3.49E-01 -3.55E-01 -3.61E-01 -3.68E-01 -3.75E-01 -3.80E-01 -3.83E-01 -3.85E-01 -3.89E-01 -3.91E-01 -3.94E-01 -4.00E-01 -4.05E-01 -4.10E-01 -4.16E-01 -4.23E-01 -4.29E-01 -4.31E-01 -4.34E-01 -4.37E-01 -4.42E-01 -4.46E-01 -4.48E-01 -4.51E-01 -4.55E-01 -4.60E-01 -4.65E-01 -4.67E-01 -4.70E-01 -4.73E-01 -4.75E-01 -4.77E-01 -4.79E-01 -4.83E-01 -4.85E-01 -2.97E-01 -2.96E-01 -2.94E-01 -2.94E-01 -2.91E-01 -2.88E-01 -2.84E-01 -2.78E-01 -2.72E-01 -2.67E-01 -2.62E-01 -2.59E-01 -2.54E-01 -2.49E-01 -2.44E-01 -2.36E-01 -2.30E-01 -2.22E-01 -2.17E-01 -2.12E-01 -2.07E-01 -2.04E-01 -2.02E-01 -1.97E-01 -1.93E-01 -1.88E-01 -1.81E-01 -1.74E-01 -1.66E-01 -1.59E-01 -1.55E-01 -1.49E-01 -1.41E-01 -1.36E-01 -1.30E-01 -1.26E-01 -1.19E-01 -1.10E-01 -1.04E-01 -9.80E-02 -9.10E-02 -8.40E-02 -7.60E-02 -6.70E-02 -6.30E-02 -5.60E-02 370 7.38E-01 7.29E-01 7.20E-01 7.11E-01 7.01E-01 6.91E-01 6.78E-01 6.65E-01 6.54E-01 6.42E-01 6.31E-01 6.19E-01 6.07E-01 5.96E-01 5.84E-01 5.71E-01 5.59E-01 5.47E-01 5.36E-01 5.24E-01 5.13E-01 5.02E-01 4.92E-01 4.79E-01 4.67E-01 4.55E-01 4.43E-01 4.29E-01 4.18E-01 4.05E-01 3.93E-01 3.79E-01 3.66E-01 3.56E-01 3.45E-01 3.33E-01 3.22E-01 3.09E-01 2.98E-01 2.85E-01 2.71E-01 2.58E-01 2.46E-01 2.33E-01 2.19E-01 2.06E-01 -4.66E-01 -4.81E-01 -4.94E-01 -5.09E-01 -5.24E-01 -5.38E-01 -5.51E-01 -5.63E-01 -5.74E-01 -5.85E-01 -5.95E-01 -6.05E-01 -6.15E-01 -6.23E-01 -6.34E-01 -6.40E-01 -6.49E-01 -6.56E-01 -6.64E-01 -6.71E-01 -6.78E-01 -6.86E-01 -6.93E-01 -7.00E-01 -7.07E-01 -7.15E-01 -7.21E-01 -7.27E-01 -7.32E-01 -7.38E-01 -7.44E-01 -7.48E-01 -7.51E-01 -7.55E-01 -7.60E-01 -7.64E-01 -7.69E-01 -7.72E-01 -7.78E-01 -7.83E-01 -7.87E-01 -7.90E-01 -7.94E-01 -7.96E-01 -7.98E-01 -8.00E-01 2.50E+09 2.52E+09 2.55E+09 2.57E+09 2.60E+09 2.62E+09 2.65E+09 2.67E+09 2.70E+09 2.72E+09 2.75E+09 2.77E+09 2.80E+09 2.82E+09 2.84E+09 2.87E+09 2.89E+09 2.92E+09 2.94E+09 2.97E+09 2.99E+09 3.02E+09 3.04E+09 3.07E+09 3.09E+09 3.12E+09 3.14E+09 3.17E+09 3.19E+09 3.22E+09 3.24E+09 3.27E+09 3.29E+09 3.32E+09 3.34E+09 3.36E+09 3.39E+09 3.41E+09 3.44E+09 3.46E+09 3.49E+09 3.51E+09 3.54E+09 3.56E+09 3.59E+09 3.61E+09 -4.88E-01 -4.90E-01 -4.93E-01 -4.94E-01 -4.96E-01 -4.98E-01 -4.98E-01 -4.97E-01 -4.98E-01 -4.96E-01 -4.94E-01 -4.93E-01 -4.90E-01 -4.89E-01 -4.88E-01 -4.88E-01 -4.86E-01 -4.83E-01 -4.80E-01 -4.75E-01 -4.71E-01 -4.69E-01 -4.66E-01 -4.62E-01 -4.58E-01 -4.53E-01 -4.43E-01 -4.34E-01 -4.25E-01 -4.22E-01 -4.27E-01 -4.29E-01 -4.26E-01 -4.20E-01 -4.17E-01 -4.12E-01 -4.08E-01 -4.05E-01 -4.04E-01 -4.01E-01 -3.98E-01 -3.93E-01 -3.86E-01 -3.80E-01 -3.74E-01 -3.67E-01 -4.80E-02 -4.00E-02 -3.20E-02 -2.50E-02 -1.60E-02 -5.00E-03 3.00E-03 1.20E-02 2.20E-02 3.00E-02 3.90E-02 4.70E-02 5.50E-02 6.20E-02 7.00E-02 7.80E-02 8.70E-02 9.40E-02 1.04E-01 1.10E-01 1.16E-01 1.23E-01 1.30E-01 1.36E-01 1.46E-01 1.55E-01 1.63E-01 1.67E-01 1.65E-01 1.61E-01 1.60E-01 1.71E-01 1.78E-01 1.84E-01 1.90E-01 1.96E-01 1.98E-01 2.03E-01 2.07E-01 2.15E-01 2.22E-01 2.30E-01 2.37E-01 2.42E-01 2.50E-01 2.54E-01 371 1.93E-01 1.81E-01 1.68E-01 1.56E-01 1.44E-01 1.33E-01 1.20E-01 1.09E-01 9.60E-02 8.30E-02 7.10E-02 5.70E-02 4.30E-02 3.10E-02 2.00E-02 6.00E-03 -6.00E-03 -1.90E-02 -3.30E-02 -4.40E-02 -5.70E-02 -6.90E-02 -8.00E-02 -9.20E-02 -1.06E-01 -1.20E-01 -1.35E-01 -1.48E-01 -1.62E-01 -1.68E-01 -1.74E-01 -1.86E-01 -1.97E-01 -2.09E-01 -2.22E-01 -2.34E-01 -2.45E-01 -2.56E-01 -2.67E-01 -2.76E-01 -2.89E-01 -3.02E-01 -3.15E-01 -3.30E-01 -3.43E-01 -3.54E-01 -8.02E-01 -8.04E-01 -8.04E-01 -8.05E-01 -8.06E-01 -8.08E-01 -8.08E-01 -8.10E-01 -8.12E-01 -8.12E-01 -8.14E-01 -8.14E-01 -8.15E-01 -8.13E-01 -8.13E-01 -8.13E-01 -8.12E-01 -8.13E-01 -8.10E-01 -8.08E-01 -8.05E-01 -8.03E-01 -8.02E-01 -8.01E-01 -8.01E-01 -7.98E-01 -7.95E-01 -7.89E-01 -7.82E-01 -7.75E-01 -7.73E-01 -7.73E-01 -7.72E-01 -7.69E-01 -7.67E-01 -7.63E-01 -7.58E-01 -7.54E-01 -7.51E-01 -7.49E-01 -7.47E-01 -7.44E-01 -7.40E-01 -7.37E-01 -7.31E-01 -7.28E-01 3.64E+09 3.66E+09 3.69E+09 3.71E+09 3.74E+09 3.76E+09 3.79E+09 3.81E+09 3.84E+09 3.86E+09 3.89E+09 3.91E+09 3.93E+09 3.96E+09 3.98E+09 4.01E+09 4.03E+09 4.06E+09 4.08E+09 4.11E+09 4.13E+09 4.16E+09 4.18E+09 4.21E+09 4.23E+09 4.26E+09 4.28E+09 4.31E+09 4.33E+09 4.36E+09 4.38E+09 4.41E+09 4.43E+09 4.45E+09 4.48E+09 4.50E+09 4.53E+09 4.55E+09 4.58E+09 4.60E+09 4.63E+09 4.65E+09 4.68E+09 4.70E+09 4.73E+09 4.75E+09 -3.60E-01 -3.51E-01 -3.45E-01 -3.37E-01 -3.27E-01 -3.20E-01 -3.10E-01 -3.00E-01 -2.93E-01 -2.84E-01 -2.75E-01 -2.68E-01 -2.60E-01 -2.51E-01 -2.42E-01 -2.33E-01 -2.23E-01 -2.13E-01 -2.01E-01 -1.91E-01 -1.82E-01 -1.70E-01 -1.59E-01 -1.46E-01 -1.37E-01 -1.25E-01 -1.14E-01 -1.03E-01 -9.50E-02 -8.70E-02 -7.90E-02 -6.80E-02 -5.70E-02 -4.50E-02 -3.30E-02 -2.30E-02 -1.40E-02 -5.00E-03 4.00E-03 1.30E-02 1.90E-02 2.40E-02 2.60E-02 3.10E-02 3.10E-02 3.10E-02 2.59E-01 2.64E-01 2.70E-01 2.73E-01 2.78E-01 2.81E-01 2.83E-01 2.86E-01 2.87E-01 2.89E-01 2.90E-01 2.91E-01 2.92E-01 2.95E-01 2.96E-01 2.96E-01 2.98E-01 2.98E-01 2.97E-01 2.96E-01 2.94E-01 2.92E-01 2.90E-01 2.86E-01 2.81E-01 2.77E-01 2.71E-01 2.63E-01 2.55E-01 2.49E-01 2.44E-01 2.39E-01 2.32E-01 2.22E-01 2.10E-01 1.99E-01 1.86E-01 1.73E-01 1.58E-01 1.42E-01 1.25E-01 1.07E-01 8.90E-02 7.20E-02 5.20E-02 3.30E-02 372 -3.69E-01 -3.84E-01 -3.96E-01 -4.10E-01 -4.25E-01 -4.38E-01 -4.51E-01 -4.66E-01 -4.78E-01 -4.91E-01 -5.03E-01 -5.15E-01 -5.25E-01 -5.36E-01 -5.47E-01 -5.58E-01 -5.70E-01 -5.81E-01 -5.93E-01 -6.04E-01 -6.16E-01 -6.28E-01 -6.39E-01 -6.52E-01 -6.64E-01 -6.75E-01 -6.87E-01 -6.97E-01 -7.06E-01 -7.13E-01 -7.20E-01 -7.30E-01 -7.40E-01 -7.50E-01 -7.60E-01 -7.67E-01 -7.75E-01 -7.84E-01 -7.91E-01 -7.97E-01 -8.03E-01 -8.07E-01 -8.07E-01 -8.11E-01 -8.11E-01 -8.11E-01 -7.22E-01 -7.15E-01 -7.09E-01 -7.03E-01 -6.95E-01 -6.88E-01 -6.79E-01 -6.71E-01 -6.61E-01 -6.52E-01 -6.41E-01 -6.31E-01 -6.21E-01 -6.12E-01 -6.02E-01 -5.91E-01 -5.82E-01 -5.71E-01 -5.61E-01 -5.50E-01 -5.38E-01 -5.26E-01 -5.15E-01 -5.01E-01 -4.87E-01 -4.73E-01 -4.58E-01 -4.41E-01 -4.23E-01 -4.07E-01 -3.92E-01 -3.78E-01 -3.63E-01 -3.45E-01 -3.27E-01 -3.07E-01 -2.88E-01 -2.69E-01 -2.47E-01 -2.24E-01 -2.02E-01 -1.78E-01 -1.55E-01 -1.31E-01 -1.07E-01 -8.20E-02 4.78E+09 4.80E+09 4.83E+09 4.85E+09 4.88E+09 4.90E+09 4.93E+09 4.95E+09 4.98E+09 5.00E+09 2.80E-02 2.70E-02 2.60E-02 2.50E-02 2.10E-02 1.60E-02 8.00E-03 -2.00E-03 -1.40E-02 -2.90E-02 1.50E-02 -2.00E-03 -2.00E-02 -3.90E-02 -6.10E-02 -8.40E-02 -1.06E-01 -1.28E-01 -1.50E-01 -1.71E-01 -8.07E-01 -8.05E-01 -8.04E-01 -8.02E-01 -7.97E-01 -7.92E-01 -7.83E-01 -7.75E-01 -7.62E-01 -7.50E-01 -5.90E-02 -3.60E-02 -1.30E-02 1.00E-02 3.50E-02 6.20E-02 8.80E-02 1.12E-01 1.39E-01 1.62E-01 G.6. Solenoid Inductors - 4 Coils, with 6 and 8 Turns per Coil G.6.1. Inductor Equivalent Circuit The circuits for the 6 turn and 8 turn per coil inductors are similar, except for the use of some additional subcircuits in the latter case. The substitutions that need to be made are highlighted in bold in the circuit below. .option accurate dccap=1 node nopage ingold=2 post acct=2 probe .subckt mstl_pad 1 5 r1 1 2 r2 l1 2 3 l2 c1 3 0 c2 r2 3 0 10g r1r 3 4 r2 l2r 4 5 l2 .ends .subckt mstl_sq 1 5 r1 1 2 rsq l1 2 3 lsq c1 3 0 csq r1r 3 4 rsq l2r 4 5 lsq cc1 1 3 c_cou cc2 3 5 c_cou .ends .subckt mstl_sq5 1 6 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq x4 4 5 mstl_sq 373 x5 5 6 mstl_sq .ends .subckt mstl_sq4 1 5 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq x4 4 5 mstl_sq .ends .subckt ind_blk 1 3 r1 1 2 ri l1 2 3 li c1 1 0 ci cs 1 3 csi .ends .subckt ind_nc 1 5 10 14 rl1 1 2 ri ll1 2 3 li c1 3 0 ci *rg1 3 0 rg rr1 3 4 ri lr1 4 5 li rl2 10 11 ri2 ll2 11 12 li2 c2 12 0 ci2 rg2 12 0 rg rr2 12 13 ri2 lr2 13 14 li2 cc12 3 12 csi .ends .subckt vind 1 2 l1 1 2 lvia c1 1 2 cvia .ends .subckt ind_blk_3 1 5 10 14 20 24 x1 1 2 3 4 ind_nc lv1 2 3 lvia cv1 2 3 cvia lv2 4 5 lvia cv2 4 5 cvia rt1 5 0 1g x2 10 11 12 13 ind_nc lv3 11 12 lvia cv3 11 12 cvia lv4 13 14 lvia cv4 13 14 cvia rt2 14 0 1g x3 20 21 22 23 ind_nc lv5 21 22 lvia cv5 21 22 cvia lv6 23 24 lvia cv6 23 24 cvia rt3 24 0 1g k1 lv1 lv4 k=cou_l k2 lv3 lv6 k=cou_l cc1 2 13 c_cou cc2 11 23 c_cou 374 rg1 1 0 1g rg2 5 0 1g rg3 10 0 1g rg4 2 13 1g rg5 11 23 1g .ends .subckt ind_blk_4 1 10 20 30 5 14 24 34 x1 1 2 3 4 ind_nc lv1 2 3 lvia cv1 2 3 cvia lv2 4 5 lvia cv2 4 5 cvia rt1 5 0 1g x2 10 11 12 13 ind_nc lv3 11 12 lvia cv3 11 12 cvia lv4 13 14 lvia cv4 13 14 cvia rt2 14 0 1g x3 20 21 22 23 ind_nc lv5 21 22 lvia cv5 21 22 cvia lv6 23 24 lvia cv6 23 24 cvia rt3 24 0 1g x4 30 31 32 33 ind_nc lv7 31 32 lvia cv7 31 32 cvia lv8 33 34 lvia cv8 33 34 cvia rt4 34 0 1g k1 lv1 lv4 k=cou_l k2 lv3 lv6 k=cou_l cc1 2 13 c_cou cc2 11 23 c_cou k3 lv5 lv8 k=cou_l cc3 21 33 cvia rg1 1 0 1g rg2 5 0 1g rg3 10 0 1g rg4 2 13 1g rg5 11 23 1g rg6 21 33 1g rg7 30 0 1g .ends .subckt inductor1 4 30 x2 4 5 6 7 8 9 ind_blk_3 x3 7 8 9 10 11 12 ind_blk_3 x4 10 11 12 13 14 15 ind_blk_3 x5 13 14 15 16 17 18 ind_blk_3 x6 16 17 18 19 20 21 ind_blk_3 x7 19 20 21 22 23 24 ind_blk_3 ls1 5 6 1e-9 ls2 28 29 1e-9 .ends 375 ******* Inductor is the circuit for the 4 coil, 6 turn per coil ******* inductor .subckt inductor 1 4 x1 1 2 3 4 5 6 7 8 ind_blk_4 x2 5 6 7 8 9 10 11 12 ind_blk_4 x3 9 10 11 12 13 14 15 16 ind_blk_4 x4 13 14 15 16 17 18 19 20 ind_blk_4 x5 17 18 19 20 21 22 23 24 ind_blk_4 x6 21 22 23 24 25 26 27 28 ind_blk_4 ls1 2 3 ls ls2 25 26 ls ls3 27 28 ls .ends ******* Inductor2 is the circuit for the 4 coil, 8 turn per coil ******* inductor .subckt inductor2 1 4 x1 1 2 3 4 5 6 7 8 ind_blk_4 x2 5 6 7 8 9 10 11 12 ind_blk_4 x3 9 10 11 12 13 14 15 16 ind_blk_4 x4 13 14 15 16 17 18 19 20 ind_blk_4 x5 17 18 19 20 21 22 23 24 ind_blk_4 x6 21 22 23 24 25 26 27 28 ind_blk_4 x7 25 26 27 28 29 30 31 32 ind_blk_4 x8 29 30 31 32 33 34 35 36 ind_blk_4 ls1 2 3 1e-10 ls2 33 34 1e-10 ls3 35 36 1e-10 .ends r1 1 2 1e-5 x1 2 3 mstl_pad ******* Use inductor2 for the 4 coil, 8 turn per coil inductor ******* in the next line in place of inductor x2 3 4 inductor x7 4 8 mstl_pad r0 8 0 1e-5 vpl 1 0 pulse(0 2.5 2n 0.1n 0.2n 10n 20n) ac 1 *.tran 1p 40n 1p *.ac lin 100 45mega 4giga .net v(2) vpl rin=50 rout=50 .param .param .param .param .param .param .param .param .param .param rl ri li ci csi ri2 li2 ci2 lvia cvia .param r2 .param l2 .param c2 = = = = = = = = = = 1.000e+02 1.000e-02 7.418e-11 1.385e-13 8.234e-15 1.703e-02 4.432e-10 1.017e-13 4.920e-10 8.663e-13 $ $ $ $ $ $ $ $ $ $ 0. 8.159e-03 2.021e+00 2.098e+01 1.820e-01 1.393e-01 1.187e+01 1.532e+01 3.555e+01 1.391e+01 0. 4.661e-02 -3.904e-05 7.933e-06 3.444e-04 -8.590e-05 1.300e-05 -1.432e-06 1.982e-06 3.154e-07 = 1.000e-05 = 3.228e-10 = 1.863e-13 $ $ $ 2.131e+01 1.886e+01 1.432e+01 -4.957e-06 1.186e-06 -1.809e-05 .param ls=1e-10 .param rg=9e3, ci=1.0e-13 cou_l = .4 ci2 = 1.8e-13 c_cou=1.4e-13 376 .ac data=measured .print par(s11r) s11(r) par(s11i) s11(i) .print par(s12r) s12(r) par(s12i) s12(i) .print z11(db) z11(p) z21(db) z21(p) .print z11(r) z11(i) .data measured file= 'dev19' freq=1 s11r=2 s11i=3 s21r=4 s21i=5 s12r=6 s12i=7 s22r=8 s22i=9 out = 'dev19_data.txt' .enddata .param freq=500m, s11r=0, s11i=0, s21r=0, s21i=0, s12r=0, s12i=0, s22r=0, s22i=0 .param sa11r=0, sa11i=0, sa21r=0, sa21i=0, sa12r=0, sa12i=0, sa22r=0, + sa22i=0 .end G.6.2. Measured S-Parameter Data The data for the 4 coil, 6 turn per coil inductor is given in colums 2-5, and the data for the 4 coil, 8 turn per coil inductor is given in columns 6-9. freq 4.50E+07 S11(R) Meas. (4C, 6T) 2.50E-02 S11(I) Meas. (4C, 6T) 7.90E-02 S21(R) Meas. (4C, 6T) 9.69E-01 S21(I) Meas. (4C, 6T) -1.78E-01 S11(R) Meas. (4C, 8T) 3.50E-02 S11(I) Meas. (4C, 8T) 1.06E-01 S21(R) Meas. (4C, 8T) 9.53E-01 S21(I) Meas. (4C, 8T) -2.26E-01 6.98E+07 4.40E-02 1.12E-01 9.38E-01 -2.72E-01 7.10E-02 1.43E-01 9.03E-01 -3.41E-01 9.46E+07 6.70E-02 1.37E-01 8.96E-01 -3.59E-01 1.07E-01 1.65E-01 8.38E-01 -4.50E-01 1.19E+08 8.40E-02 1.65E-01 8.40E-01 -4.42E-01 1.29E-01 1.89E-01 7.47E-01 -5.39E-01 1.44E+08 7.70E-02 2.08E-01 7.38E-01 -4.91E-01 1.15E-01 2.58E-01 5.99E-01 -5.59E-01 1.69E+08 1.63E-01 2.60E-01 7.30E-01 -5.17E-01 2.56E-01 2.89E-01 6.04E-01 -5.93E-01 1.94E+08 2.26E-01 2.56E-01 6.90E-01 -5.86E-01 3.41E-01 2.60E-01 5.46E-01 -6.65E-01 2.18E+08 2.64E-01 2.42E-01 6.21E-01 -6.50E-01 3.85E-01 2.18E-01 4.49E-01 -7.24E-01 2.43E+08 2.99E-01 2.28E-01 5.48E-01 -7.02E-01 4.20E-01 1.74E-01 3.47E-01 -7.64E-01 2.68E+08 3.32E-01 2.09E-01 4.71E-01 -7.43E-01 4.45E-01 1.26E-01 2.43E-01 -7.90E-01 2.93E+08 3.55E-01 1.79E-01 3.85E-01 -7.83E-01 4.50E-01 6.40E-02 1.27E-01 -8.14E-01 3.18E+08 3.70E-01 1.58E-01 2.97E-01 -8.03E-01 4.40E-01 2.50E-02 1.00E-02 -8.02E-01 3.42E+08 3.72E-01 1.52E-01 1.95E-01 -7.91E-01 4.29E-01 2.10E-02 -9.70E-02 -7.44E-01 3.67E+08 4.00E-01 1.11E-01 1.30E-01 -8.10E-01 4.08E-01 -4.80E-02 -2.02E-01 -7.41E-01 3.92E+08 3.66E-01 7.90E-02 7.00E-03 -8.10E-01 3.27E-01 -4.70E-02 -3.50E-01 -6.65E-01 4.17E+08 3.50E-01 9.60E-02 -9.20E-02 -7.52E-01 3.06E-01 3.40E-02 -4.31E-01 -5.10E-01 4.41E+08 3.38E-01 1.07E-01 -1.78E-01 -6.99E-01 3.24E-01 7.30E-02 -4.66E-01 -3.96E-01 4.66E+08 3.69E-01 1.55E-01 -2.16E-01 -6.07E-01 4.17E-01 9.80E-02 -4.27E-01 -2.92E-01 4.91E+08 4.27E-01 8.10E-02 -2.30E-01 -6.28E-01 4.27E-01 -2.20E-02 -4.58E-01 -3.14E-01 5.16E+08 3.90E-01 4.00E-02 -3.30E-01 -6.12E-01 3.52E-01 -3.00E-02 -5.55E-01 -2.29E-01 5.40E+08 3.70E-01 6.30E-02 -4.07E-01 -5.23E-01 3.88E-01 1.10E-02 -5.39E-01 -9.70E-02 5.65E+08 3.19E-01 3.90E-02 -5.03E-01 -4.78E-01 3.23E-01 9.00E-03 -6.06E-01 -6.00E-03 5.90E+08 2.63E-01 1.56E-01 -6.03E-01 -2.93E-01 3.79E-01 1.38E-01 -5.55E-01 2.08E-01 377 6.15E+08 3.63E-01 2.16E-01 -5.37E-01 -1.61E-01 4.87E-01 9.50E-02 -4.43E-01 2.64E-01 6.40E+08 4.09E-01 2.83E-01 -5.21E-01 -1.80E-02 5.60E-01 9.90E-02 -3.54E-01 3.61E-01 6.64E+08 5.53E-01 2.81E-01 -3.99E-01 5.50E-02 6.45E-01 1.00E-03 -2.42E-01 3.64E-01 6.89E+08 6.00E-01 2.14E-01 -3.67E-01 6.30E-02 6.21E-01 -7.00E-02 -2.22E-01 3.84E-01 7.14E+08 6.89E-01 1.91E-01 -2.89E-01 1.18E-01 6.64E-01 -1.37E-01 -1.30E-01 4.08E-01 7.39E+08 7.60E-01 8.80E-02 -2.18E-01 9.30E-02 6.44E-01 -2.52E-01 -8.20E-02 3.75E-01 7.63E+08 7.30E-01 -1.90E-02 -2.46E-01 6.10E-02 5.61E-01 -2.84E-01 -9.40E-02 4.22E-01 7.88E+08 7.17E-01 -3.80E-02 -2.50E-01 1.19E-01 5.53E-01 -3.01E-01 -1.80E-02 4.78E-01 8.13E+08 7.45E-01 -6.90E-02 -2.06E-01 1.65E-01 5.41E-01 -3.50E-01 6.90E-02 4.93E-01 8.38E+08 7.58E-01 -1.26E-01 -1.73E-01 1.82E-01 5.06E-01 -4.07E-01 1.43E-01 4.92E-01 8.63E+08 7.67E-01 -1.92E-01 -1.36E-01 1.90E-01 4.45E-01 -4.60E-01 2.06E-01 4.78E-01 8.87E+08 7.69E-01 -2.65E-01 -9.90E-02 1.91E-01 3.58E-01 -4.99E-01 2.57E-01 4.60E-01 9.12E+08 7.43E-01 -3.39E-01 -8.10E-02 1.87E-01 2.52E-01 -4.78E-01 2.97E-01 4.70E-01 9.37E+08 7.13E-01 -4.07E-01 -6.20E-02 1.87E-01 1.78E-01 -3.93E-01 3.61E-01 5.09E-01 9.62E+08 6.74E-01 -4.72E-01 -4.50E-02 1.86E-01 2.26E-01 -3.20E-01 5.17E-01 5.29E-01 9.86E+08 6.19E-01 -5.39E-01 -3.40E-02 1.81E-01 2.57E-01 -3.68E-01 6.68E-01 4.36E-01 1.01E+09 5.50E-01 -6.03E-01 -3.10E-02 1.72E-01 2.02E-01 -4.09E-01 7.57E-01 3.17E-01 1.04E+09 4.56E-01 -6.52E-01 -4.00E-02 1.72E-01 1.28E-01 -4.04E-01 8.16E-01 2.00E-01 1.06E+09 3.42E-01 -6.94E-01 -5.90E-02 1.71E-01 6.20E-02 -3.65E-01 8.59E-01 7.70E-02 1.09E+09 1.82E-01 -7.02E-01 -1.12E-01 1.86E-01 1.40E-02 -3.03E-01 8.82E-01 -5.50E-02 1.11E+09 -2.00E-03 -6.23E-01 -1.82E-01 2.61E-01 -1.10E-02 -2.25E-01 8.80E-01 -1.96E-01 1.14E+09 -1.27E-01 -4.21E-01 -2.09E-01 4.28E-01 -1.00E-02 -1.40E-01 8.48E-01 -3.37E-01 1.16E+09 -8.60E-02 -1.72E-01 -1.08E-01 6.31E-01 2.30E-02 -5.80E-02 7.86E-01 -4.72E-01 1.18E+09 9.70E-02 -4.60E-02 1.15E-01 7.41E-01 8.20E-02 9.00E-03 6.93E-01 -5.91E-01 1.21E+09 2.63E-01 -8.00E-02 3.39E-01 7.17E-01 1.62E-01 5.20E-02 5.72E-01 -6.85E-01 1.23E+09 3.34E-01 -1.68E-01 4.98E-01 6.32E-01 2.48E-01 6.90E-02 4.36E-01 -7.45E-01 1.26E+09 3.44E-01 -2.48E-01 6.10E-01 5.34E-01 3.36E-01 5.80E-02 2.95E-01 -7.70E-01 1.28E+09 3.20E-01 -3.06E-01 6.91E-01 4.34E-01 4.13E-01 2.90E-02 1.58E-01 -7.57E-01 1.31E+09 2.81E-01 -3.44E-01 7.53E-01 3.32E-01 4.90E-01 -1.30E-02 4.40E-02 -7.16E-01 1.33E+09 2.33E-01 -3.63E-01 8.00E-01 2.33E-01 5.63E-01 -8.20E-02 -4.80E-02 -6.65E-01 1.36E+09 1.79E-01 -3.62E-01 8.37E-01 1.33E-01 6.11E-01 -1.77E-01 -1.27E-01 -6.06E-01 1.38E+09 1.30E-01 -3.41E-01 8.65E-01 2.80E-02 6.32E-01 -2.78E-01 -1.89E-01 -5.37E-01 1.41E+09 9.10E-02 -3.03E-01 8.77E-01 -8.40E-02 6.29E-01 -3.82E-01 -2.36E-01 -4.65E-01 1.43E+09 6.70E-02 -2.56E-01 8.73E-01 -1.98E-01 6.03E-01 -4.81E-01 -2.69E-01 -3.91E-01 1.46E+09 5.50E-02 -2.02E-01 8.52E-01 -3.14E-01 5.61E-01 -5.72E-01 -2.85E-01 -3.19E-01 1.48E+09 6.30E-02 -1.45E-01 8.11E-01 -4.27E-01 5.04E-01 -6.54E-01 -2.90E-01 -2.52E-01 1.51E+09 8.60E-02 -9.20E-02 7.49E-01 -5.32E-01 4.30E-01 -7.27E-01 -2.85E-01 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-7.63E-01 9.00E-02 -1.46E-01 2.82E-01 -7.25E-01 1.24E-01 7.90E-02 4.73E+09 -5.11E-01 -7.24E-01 1.00E-01 -1.79E-01 9.90E-02 -7.08E-01 1.43E-01 5.70E-02 4.75E+09 -5.52E-01 -6.82E-01 1.02E-01 -2.13E-01 -1.04E-01 -5.96E-01 1.60E-01 2.40E-02 4.78E+09 -5.91E-01 -6.36E-01 9.60E-02 -2.46E-01 -2.55E-01 -3.32E-01 1.61E-01 -2.30E-02 4.80E+09 -6.25E-01 -5.89E-01 8.40E-02 -2.78E-01 -1.87E-01 3.70E-02 1.30E-01 -7.00E-02 4.83E+09 -6.56E-01 -5.39E-01 6.70E-02 -3.06E-01 1.47E-01 2.54E-01 7.60E-02 -8.10E-02 4.85E+09 -6.82E-01 -4.88E-01 4.40E-02 -3.31E-01 4.89E-01 1.85E-01 3.80E-02 -5.60E-02 4.88E+09 -7.06E-01 -4.35E-01 1.80E-02 -3.53E-01 6.72E-01 -1.30E-02 2.80E-02 -2.50E-02 4.90E+09 -7.25E-01 -3.83E-01 -1.20E-02 -3.71E-01 7.30E-01 -2.11E-01 3.50E-02 -1.00E-03 4.93E+09 -7.40E-01 -3.28E-01 -4.60E-02 -3.83E-01 7.21E-01 -3.72E-01 4.70E-02 1.40E-02 4.95E+09 -7.55E-01 -2.74E-01 -8.20E-02 -3.91E-01 6.81E-01 -4.98E-01 6.30E-02 2.20E-02 4.98E+09 -7.63E-01 -2.18E-01 -1.19E-01 -3.92E-01 6.25E-01 -5.94E-01 7.90E-02 2.60E-02 5.00E+09 -7.70E-01 -1.59E-01 -1.55E-01 -3.90E-01 5.62E-01 -6.70E-01 9.40E-02 2.60E-02 381 G.7. Capacitor Benchmark Structure G.7.1. Equivalent Circuit .option accurate dccap=1 node nopage ingold=2 post acct=2 probe .subckt mstl_pad 1 5 r1 1 2 r2 l1 2 3 l2 c1 3 0 cpad r2 3 0 10g r1r 3 4 r2 l2r 4 5 l2 .ends .subckt line_seg 1 5 6 10 r1 1 2 rsq l1 2 3 lsq c1 3 0 cmid rgm1 3 0 rg r1r 3 4 rsq l1r 4 5 lsq rt1 1 6 rg ce1 1 6 csq rt2 5 10 rg ce2 5 10 csq k1 l1 l2 k=cou_l k2 l1r l2r k=cou_l *ce2 3 8 csq *rg1 3 8 rg r2 6 7 rsq l2 7 8 lsq c2 8 0 cmid2 rgm2 8 0 rg r2r 8 9 rsq l2r 9 10 lsq rtm1 1 0 1g rtm2 5 0 1g rtm3 6 0 1g rtm4 10 0 1g .ends .subckt sq1 1 2 3 4 5 6 7 8 x1 1 2 5 6 line_seg x2 2 4 6 8 line_seg x3 3 4 7 8 line_seg x4 1 3 5 7 line_seg .ends .subckt sq1x3 x1 1 2 3 4 10 x2 2 5 4 6 11 x3 5 7 6 8 14 .ends 1 7 3 11 12 14 13 16 15 8 10 18 12 17 13 sq1 15 sq1 17 sq1 .subckt sq1x4 1 20 3 21 10 22 12 23 382 x1 1 2 3 4 10 11 12 13 sq1 x2 2 5 4 6 11 14 13 15 sq1 x3 5 7 6 8 14 16 15 17 sq1 x4 7 20 8 21 16 22 17 23 sq1 .ends .subckt sq1x6 1 13 3 14 20 32 22 33 x1 1 2 3 4 20 21 22 23 sq1 x2 2 5 4 6 21 24 23 25 sq1 x3 5 7 6 8 24 26 25 27 sq1 x4 7 9 8 10 26 28 27 29 sq1 x5 9 11 10 12 28 30 29 31 sq1 x6 11 13 12 14 30 32 31 33 sq1 .ends .subckt sq3x6 x1 1 2 3 4 10 x2 3 4 5 6 12 x3 5 6 7 8 14 .ends 1 2 7 11 12 13 14 15 16 8 10 11 16 17 13 sq1x6 15 sq1x6 17 sq1x6 .subckt sq9x6 x1 1 2 3 4 10 x2 3 4 5 6 12 x3 5 6 7 8 14 .ends 3 15 11 12 13 sq3x6 13 14 15 sq3x6 15 16 17 sq3x6 .subckt sq5x4 5 25 x1 1 2 3 4 20 21 22 23 sq1x4 x2 3 4 5 6 22 23 24 25 sq1x4 x3 5 6 7 8 24 25 26 27 sq1x4 x4 7 8 9 10 26 27 28 29 sq1x4 x5 9 10 11 12 28 29 30 31 sq1x4 .ends .subckt sq3x3 x1 1 2 3 4 10 x2 3 4 5 6 12 x3 5 6 7 8 14 .ends 3 15 11 12 13 sq1x3 13 14 15 sq1x3 15 16 17 sq1x3 .subckt line3x3 4 15 x1 1 2 10 11 line_seg x2 2 3 11 12 line_seg x3 4 5 13 14 line_seg x4 5 6 14 15 line_seg x5 7 8 16 17 line_seg x6 8 9 17 18 line_seg x7 1 4 10 13 line_seg x8 2 5 11 14 line_seg x9 3 6 12 15 line_seg x10 4 7 13 16 line_seg x11 5 8 14 17 line_seg x12 6 9 15 18 line_seg rs 10 0 0.1 ca1 2 11 c3 ca2 ca3 ca4 cb1 cb2 cb3 6 4 8 1 3 7 12 13 17 10 12 16 c3 c3 c3 c2 c2 c2 383 cb4 9 18 c2 cc1 5 14 c4 .ends .subckt mstl_sq5 1 6 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq x4 4 5 mstl_sq x5 5 6 mstl_sq .ends .subckt mstl_sq4 1 5 x1 1 2 mstl_sq x2 2 3 mstl_sq x3 3 4 mstl_sq x4 4 5 mstl_sq .ends r1 x1 x2 x7 r0 1 2 3 4 8 2 3 4 8 0 1e-5 mstl_pad sq9x6 mstl_pad 1e-5 vpl 1 0 pulse(0 2.5 2n 0.1n 0.2n 10n 20n) ac 1 *.tran 1p 40n 1p *.ac dec 60 1mega 15giga .net v(2) vpl rin=50 rout=50 .param r2 .param l2 .param cpad = 1.000e-05 = 3.228e-10 = 1.863e-13 $ $ .param .param .param .param .param .param .param = = = = = = = $ $ $ $ $ $ $ cou_l rsq rg lsq csq cmid cmid2 9.796e-01 4.709e-01 1.000e+07 9.100e-10 0.891e-13 1.315e-14 1.000e-14 2.131e+01 1.886e+01 $ 1.432e+01 6.578e+01 1.399e+00 0. 1.524e+01 6.057e+00 1.146e+01 5.546e-02 -4.957e-06 1.186e-06 -1.809e-05 2.840e-04 7.011e-02 0. -3.006e-03 8.651e-04 -1.923e-03 -1.905e+00 .ac data=measured .print par(s11r) s11(r) par(s11i) s11(i) .print par(s12r) s12(r) par(s12i) s12(i) .data measured file = ‘dev28.txt’ freq=1 s11r=2 s11I=3 s21r=4 s21I=5 s12r=6 s12I=7 s22r=8 s22I=9 .enddata .param freq=100m, s11r=0, s11i=0, s21r=0, s21i=0, s12r=0, s12i=0, s22r=0, s22i=0 .end G.7.2. Measured S-Parameter Data freq 4.50E+07 6.98E+07 S11(R) Meas. 5.22E-01 2.67E-01 S11(I) Meas. -6.07E-01 -6.20E-01 384 S21(R) Meas. 4.31E-01 6.35E-01 S21(I) Meas. 4.15E-01 3.47E-01 9.46E+07 1.19E+08 1.44E+08 1.69E+08 1.94E+08 2.18E+08 2.43E+08 2.68E+08 2.93E+08 3.18E+08 3.42E+08 3.67E+08 3.92E+08 4.17E+08 4.41E+08 4.66E+08 4.91E+08 5.16E+08 5.40E+08 5.65E+08 5.90E+08 6.15E+08 6.40E+08 6.64E+08 6.89E+08 7.14E+08 7.39E+08 7.63E+08 7.88E+08 8.13E+08 8.38E+08 8.63E+08 8.87E+08 9.12E+08 9.37E+08 9.62E+08 9.86E+08 1.01E+09 1.04E+09 1.06E+09 1.09E+09 1.11E+09 1.14E+09 1.16E+09 1.18E+09 1.21E+09 1.01E-01 -1.70E-02 -3.80E-02 -7.80E-02 -1.16E-01 -1.61E-01 -2.02E-01 -2.49E-01 -3.00E-01 -3.18E-01 -3.43E-01 -3.87E-01 -3.72E-01 -3.79E-01 -4.03E-01 -4.40E-01 -4.59E-01 -5.02E-01 -4.92E-01 -5.01E-01 -5.22E-01 -5.56E-01 -5.79E-01 -6.06E-01 -6.22E-01 -6.39E-01 -6.61E-01 -6.78E-01 -6.91E-01 -7.03E-01 -7.15E-01 -7.25E-01 -7.31E-01 -7.34E-01 -7.33E-01 -7.30E-01 -7.24E-01 -7.13E-01 -6.95E-01 -6.75E-01 -6.55E-01 -6.22E-01 -5.82E-01 -5.41E-01 -4.85E-01 -4.10E-01 -5.91E-01 -5.31E-01 -4.54E-01 -4.52E-01 -4.56E-01 -4.55E-01 -4.46E-01 -4.39E-01 -4.20E-01 -3.90E-01 -3.83E-01 -3.42E-01 -3.21E-01 -3.28E-01 -3.29E-01 -3.22E-01 -3.14E-01 -2.73E-01 -2.53E-01 -2.53E-01 -2.51E-01 -2.36E-01 -2.19E-01 -1.94E-01 -1.66E-01 -1.44E-01 -1.21E-01 -8.80E-02 -5.90E-02 -2.40E-02 6.00E-03 4.40E-02 8.30E-02 1.23E-01 1.65E-01 2.09E-01 2.52E-01 2.97E-01 3.45E-01 3.83E-01 4.35E-01 4.83E-01 5.29E-01 5.74E-01 6.24E-01 6.64E-01 385 7.39E-01 7.80E-01 8.25E-01 8.55E-01 8.59E-01 8.44E-01 8.21E-01 7.86E-01 7.36E-01 7.18E-01 6.99E-01 6.46E-01 6.45E-01 6.44E-01 6.22E-01 5.75E-01 5.60E-01 5.29E-01 5.58E-01 5.53E-01 5.40E-01 5.02E-01 4.63E-01 4.26E-01 3.93E-01 3.56E-01 3.24E-01 2.92E-01 2.59E-01 2.28E-01 1.98E-01 1.72E-01 1.46E-01 1.20E-01 9.40E-02 7.00E-02 4.60E-02 2.30E-02 2.00E-03 -1.20E-02 -3.60E-02 -5.40E-02 -6.90E-02 -7.80E-02 -8.70E-02 -8.60E-02 2.33E-01 1.44E-01 1.08E-01 1.40E-02 -7.90E-02 -1.50E-01 -2.08E-01 -2.63E-01 -3.03E-01 -3.14E-01 -3.47E-01 -3.59E-01 -3.51E-01 -3.90E-01 -4.18E-01 -4.35E-01 -4.46E-01 -4.30E-01 -4.34E-01 -4.80E-01 -5.18E-01 -5.52E-01 -5.74E-01 -5.85E-01 -5.95E-01 -6.04E-01 -6.04E-01 -6.09E-01 -6.08E-01 -6.03E-01 -5.95E-01 -5.89E-01 -5.81E-01 -5.73E-01 -5.63E-01 -5.50E-01 -5.36E-01 -5.21E-01 -5.02E-01 -4.87E-01 -4.66E-01 -4.41E-01 -4.12E-01 -3.83E-01 -3.48E-01 -3.07E-01 1.23E+09 1.26E+09 1.28E+09 1.31E+09 1.33E+09 1.36E+09 1.38E+09 1.41E+09 1.43E+09 1.46E+09 1.48E+09 1.51E+09 1.53E+09 1.56E+09 1.58E+09 1.61E+09 1.63E+09 1.66E+09 1.68E+09 1.70E+09 1.73E+09 1.75E+09 1.78E+09 1.80E+09 1.83E+09 1.85E+09 1.88E+09 1.90E+09 1.93E+09 1.95E+09 1.98E+09 2.00E+09 2.03E+09 2.05E+09 2.08E+09 2.10E+09 2.13E+09 2.15E+09 2.18E+09 2.20E+09 2.23E+09 2.25E+09 2.27E+09 2.30E+09 2.32E+09 2.35E+09 -3.12E-01 -1.67E-01 -1.99E-01 -1.21E-01 -2.62E-01 -1.86E-01 -5.40E-02 8.30E-02 2.03E-01 3.41E-01 3.77E-01 4.17E-01 5.18E-01 5.03E-01 4.29E-01 2.96E-01 1.34E-01 -2.50E-02 -1.55E-01 -2.41E-01 -2.53E-01 -2.93E-01 -2.96E-01 -2.43E-01 -1.60E-01 -4.90E-02 8.80E-02 1.01E-01 1.74E-01 2.70E-01 3.37E-01 3.78E-01 4.27E-01 4.59E-01 4.52E-01 4.05E-01 3.31E-01 2.60E-01 1.89E-01 1.07E-01 2.60E-02 -4.90E-02 -8.60E-02 -1.38E-01 -2.14E-01 -2.88E-01 6.91E-01 6.36E-01 6.07E-01 5.09E-01 5.47E-01 6.78E-01 7.21E-01 7.06E-01 6.70E-01 5.70E-01 3.93E-01 4.08E-01 2.29E-01 3.50E-02 -1.30E-01 -2.53E-01 -3.16E-01 -3.06E-01 -2.47E-01 -1.44E-01 -5.40E-02 2.00E-03 1.24E-01 2.26E-01 3.01E-01 3.45E-01 3.26E-01 2.23E-01 2.69E-01 2.21E-01 1.39E-01 6.60E-02 -2.00E-02 -1.31E-01 -2.57E-01 -3.79E-01 -4.64E-01 -5.23E-01 -5.77E-01 -6.18E-01 -6.40E-01 -6.42E-01 -6.27E-01 -6.66E-01 -6.77E-01 -6.67E-01 386 -6.70E-02 1.10E-02 -1.20E-02 7.50E-02 -3.10E-02 -8.90E-02 -7.50E-02 -3.10E-02 1.20E-02 9.40E-02 1.79E-01 1.44E-01 2.50E-01 3.26E-01 3.67E-01 3.75E-01 3.44E-01 2.78E-01 1.96E-01 9.70E-02 1.50E-02 -3.70E-02 -1.37E-01 -2.25E-01 -2.94E-01 -3.48E-01 -3.74E-01 -3.37E-01 -4.02E-01 -3.97E-01 -3.70E-01 -3.46E-01 -3.22E-01 -2.81E-01 -2.33E-01 -1.82E-01 -1.35E-01 -9.60E-02 -6.30E-02 -3.50E-02 -1.20E-02 3.00E-03 -1.10E-02 1.40E-02 3.00E-02 3.60E-02 -2.60E-01 -2.26E-01 -2.68E-01 -2.80E-01 -3.66E-01 -2.78E-01 -2.04E-01 -1.51E-01 -1.16E-01 -9.30E-02 -1.53E-01 -1.34E-01 -1.39E-01 -2.11E-01 -3.01E-01 -4.09E-01 -5.15E-01 -6.02E-01 -6.63E-01 -6.95E-01 -6.90E-01 -7.04E-01 -6.97E-01 -6.59E-01 -6.03E-01 -5.33E-01 -4.40E-01 -4.16E-01 -3.58E-01 -2.75E-01 -2.12E-01 -1.72E-01 -1.22E-01 -8.00E-02 -5.20E-02 -3.80E-02 -3.80E-02 -4.70E-02 -6.10E-02 -7.80E-02 -1.00E-01 -1.29E-01 -1.41E-01 -1.43E-01 -1.65E-01 -1.88E-01 2.37E+09 2.40E+09 2.42E+09 2.45E+09 2.47E+09 2.50E+09 2.52E+09 2.55E+09 2.57E+09 2.60E+09 2.62E+09 2.65E+09 2.67E+09 2.70E+09 2.72E+09 2.75E+09 2.77E+09 2.80E+09 2.82E+09 2.84E+09 2.87E+09 2.89E+09 2.92E+09 2.94E+09 2.97E+09 2.99E+09 3.02E+09 3.04E+09 3.07E+09 3.09E+09 3.12E+09 3.14E+09 3.17E+09 3.19E+09 3.22E+09 3.24E+09 3.27E+09 3.29E+09 3.32E+09 3.34E+09 3.36E+09 3.39E+09 3.41E+09 3.44E+09 3.46E+09 3.49E+09 -3.46E-01 -4.03E-01 -4.59E-01 -5.10E-01 -5.55E-01 -5.91E-01 -6.19E-01 -6.46E-01 -6.68E-01 -6.90E-01 -7.10E-01 -7.27E-01 -7.43E-01 -7.58E-01 -7.70E-01 -7.76E-01 -7.81E-01 -7.78E-01 -7.82E-01 -7.83E-01 -7.85E-01 -7.74E-01 -7.59E-01 -7.60E-01 -7.45E-01 -7.29E-01 -7.11E-01 -6.94E-01 -6.70E-01 -6.42E-01 -6.31E-01 -6.05E-01 -5.68E-01 -5.59E-01 -5.68E-01 -5.62E-01 -5.33E-01 -4.97E-01 -4.62E-01 -4.25E-01 -3.87E-01 -3.66E-01 -3.73E-01 -3.46E-01 -3.07E-01 -2.66E-01 -6.50E-01 -6.33E-01 -6.09E-01 -5.81E-01 -5.45E-01 -5.07E-01 -4.71E-01 -4.37E-01 -4.04E-01 -3.71E-01 -3.37E-01 -3.04E-01 -2.70E-01 -2.36E-01 -2.01E-01 -1.62E-01 -1.25E-01 -9.00E-02 -6.60E-02 -2.90E-02 6.00E-03 4.30E-02 6.70E-02 9.70E-02 1.29E-01 1.58E-01 1.82E-01 2.08E-01 2.29E-01 2.44E-01 2.52E-01 2.81E-01 2.82E-01 2.66E-01 2.75E-01 3.09E-01 3.39E-01 3.53E-01 3.60E-01 3.62E-01 3.53E-01 3.25E-01 3.25E-01 3.43E-01 3.44E-01 3.32E-01 387 3.90E-02 3.70E-02 3.40E-02 2.80E-02 2.00E-02 1.10E-02 2.00E-03 -7.00E-03 -1.80E-02 -3.00E-02 -4.40E-02 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4.70E+09 4.73E+09 4.75E+09 4.78E+09 4.80E+09 4.83E+09 4.85E+09 4.88E+09 4.90E+09 4.93E+09 4.95E+09 4.98E+09 5.00E+09 -6.75E-01 -6.51E-01 -6.40E-01 -6.37E-01 -6.27E-01 -6.14E-01 -5.92E-01 -5.61E-01 -5.38E-01 -5.38E-01 -5.21E-01 -4.96E-01 -4.69E-01 -4.39E-01 -4.09E-01 5.15E-01 5.27E-01 5.28E-01 5.50E-01 5.70E-01 5.95E-01 6.19E-01 6.35E-01 6.35E-01 6.47E-01 6.75E-01 6.97E-01 7.15E-01 7.28E-01 7.41E-01 389 2.07E-01 2.02E-01 2.03E-01 2.14E-01 2.22E-01 2.30E-01 2.35E-01 2.36E-01 2.36E-01 2.50E-01 2.61E-01 2.68E-01 2.73E-01 2.79E-01 2.84E-01 2.63E-01 2.52E-01 2.54E-01 2.49E-01 2.43E-01 2.34E-01 2.22E-01 2.13E-01 2.12E-01 2.11E-01 1.99E-01 1.88E-01 1.78E-01 1.69E-01 1.60E-01 REFERENCES 1 E. Pettenpaul, et. al., "CAD Models of Lumped Elements on GaAs up to 10 GHz," IEEE Trans. on Microwave Theory Tech., vol. 36, no. 2, pp. 294-304, Feb. 1988. 2 M. Engles and R. H. Jansen, “Modeling and Design of Novel Passive MMIC Components with Three and More Conductor Layers”, IEEE MTT-S Digest, pp. 1293-6, 1994. 3 I. Wolff and H. Kapusta, “Modeling of Circular Spiral Inductors for MMICs”, IEEE MTT-S Digest, pp. 123-126, 1987. 4 H. Bryan, “Printed Inductors and Capacitors”, Tele-Tech and Electronic Industries, p. 68, 1955. 5 P. Li, “A New Closed Form Formula for Inductance Calculation in Microstrip Line Spiral Inductor Design”, IEEE Electrical Performance of Electrical Packaging, pp. 58-60, 1996. 6 K. C. Gupta, R. Garg, I. Bahl, and P. Bhartia, Microstrip Lines and Slotlines, Boston: Artech House, 1996. 7 S. S. Gevorgian, et. al., "CAD Models for Multilayered Substrate Interdigital Capacitors," IEEE Trans. on Microwave Theory Tech., vol. 44, no. 6, pp. 896904, June 1996. 8 V. K. Sadhir, I. J. Bahl and D. A. Willems, “CAD Compatible Accurate Models of Microwave Passive Lumped Elements for MMIC Applications,” Int. J. of Microwave and Millimeter-Wave Computer-Aided Engineering, vol. 4, no. 2, pp. 148-62, April 1994. 9 D. Lovelace, N. Camilleri and G. Kannell, "Silicon MMIC Inductor Modeling for High Volume, Low Cost Applications," Microwave Journal, pp. 60-71, August 1994. 10 J. Zhao, Frye, R.C., Dai, W.W.-M., Tai, K.L, “S parameter-based experimental modeling of high Q MCM inductor with exponential gradient learning algorithm”, IEEE Trans. CPMT-B, vol.20, no.3, pp.202-10, August 1997. 390 11 J. Strickland, Time Domain Reflectometry Measurements, Tektronix, Inc., Beaverton, Oregon, Aug. 1979. 12 J. C. Toscano, A. Elshabini-Raid, “Wide-Band Characterization of Multilayer Thick Film Structures Using a Time-Domain Technique”, IEEE Trans. on Instrumentation and Measurement, vol. 38, no. 2, pp. 515-520, April 1989. 13 A. Elshabini-Raid and J. C. Toscano, “Wideband Characterization and Modeling of Thick Film Inductors”, Proceedings ISHM, pp. 73-78, 1987. 14 T. Becks and I. Wolff, “Analysis of 3-D Metallization Structures by a Full-Wave Spectral Domain Approach”, IEEE Trans. on Microwave Theory and Tech., vol. 40, no. 12, pp. 2219-27, December 1992. 15 R. Bunger, and F. Arndt, “Efficient MPIE Approach for the Analysis of Three Dimensional Microstrip Structures in Layered Media”, IEEE Trans. on Microwave Theory and Tech., vol. 45, no. 8, pp. 1141-53, August 1997. 16 A. Nakatani, S. A. Maas and J. Castaneda, “Modeling of High Frequency MMIC Passive Components,” IEEE MTT-S Int. Microwave Symp. Digest, pp. 1139-1141, June 1989. 17 K. Naishadham and T. W. Nuteson, “Efficient Analysis of Passive Microstrip Elements in MMICs,” Int. J. of Microwave and Millimeter-Wave Computer-Aided Engineering, vol. 4. vo.3, pp. 148-62, April 1994. 18 R. Sorrentino, “Numerical Methods for Passive Components,” IEEE MTT-S Int. Microwave Symp. Digest, pp. 619-622, 1988. 19 D. G. Swanson Jr., “Simulating EM Fields,” IEEE Spectrum, vol. 28, pp. 34-37, Nov. 1991. 20 M. Stubbs, L. Chow, and G. Howard, “Simulation tool accurately models MMIC passive elements”, Microwaves and RF, vol.27, no.1, p.75-6, 78-9, Jan. 1988 21 D.I. Wu, D.C. Chang, and B.L. Brim, “Accurate numerical modeling of microstrip junctions and discontinuities”, International Journal of Microwave and Millimeter-Wave Computer-Aided Engineering, vol.1, no.1, pp.48-58, Jan. 1991 22 C. Amrani, M. Drissi, V. F. Hanna, and J. Citerne, “Theoretical and Experimental Investigation of Some General Suspended Stripline Discontinuities”, IEEE MTT-S Digest, pp. 409-412, 1992. 23 A. Hill and V. K. Tripathi, "Analysis and Modeling of Coupled Right Angle Microstrip Bend Discontinuities," in Proc. IEEE MTT-S Digest, pp. 1143-1146, 1989. 391 24 M. Rittweger and I. Wolff, “Analysis of Complex Passive (M)MIC-Components Using the Finite Difference Time-Domain Approach”, IEEE MTT-S Digest, pp. 1147-1150, 1990. 25 M. Rittweger, et al., “Miniaturization of MMIC Inductors Using a 3D FDTD Approach with a SI Method”, IEEE MTT-S Digest, pp. 1297-1300, 1994. 26 D. Lovelace, N. Camilleri and G. Kannell, "Silicon MMIC Inductor Modeling for High Volume, Low Cost Applications," Microwave Journal, pp. 60-71, August 1994. 27 W. Heinrich, et al., “MMIC Spiral Inductor Modeling”, Microwave Journal, pp. 286-290, May 1996. 28 M. Naghed and I. Wolff, "Equivalent Capacitances of Coplanar Waveguide Discontinuities and Interdigitated Capacitors Using a Three-Dimensional Finite Difference Method," IEEE Trans. on Microwave Theory Tech., vol. 38, no. 12, pp. 1808-1815, December 1990. 29 P. Pogatzki, et al., “A Comprehensive Evaluation of Quasi-Static 3D-FD Calculations for more that 14 CPW Structures – Lines, Discontinuities and Lumped Elements”, IEEE MTT-S Digest, pp. 1289-1292, 1994. 30 R. Kulke, et. al., "Enhancement of Coplanar Capacitor Models and Verification up to 67 GHz for (M)MIC Circuit Design," Proc. 24th European Microwave Conference, pp. 258-62, 1996. 31 A. Bailey, et al., “Miniature LTCC Filters for Digital Receivers”, IEEE MTT-S Digest, pp. 999-1002, 1997. 32 J. Gipprich, L. Dickens, B. Hayes, and F. Sacks, “A Compacy 8-14 GHz LTCC Stripline Coupler Network for High Efficiency Power Combining with Better Than 82% Combining Efficiency”, IEEE MTT-S Digest, pp. 1583-1586, 1995. 33 G. L. Creech, et al., “Artificial Neural Networks for Accurate Microwave CAD Applications”, IEEE MTT-S Digest, pp. 733-736, 1996. 34 J. W. Bandler, et al., “Microstrip Filter Design Using Direct EM Field Simulation”, IEEE Trans. on Microwave Theory and Tech., vol. 42, no.7, pp. 1353-1359, July 1994. 35 R. Poddar and M. Brooke, “Accurate, High Speed Modeling of Integrated Passive Devices in Multichip Modules,” IEEE Topical Meeting on Electrical Performance of Electronic Packaging, pp. 184-6, 1996. 392 36 R. Poddar and M. Brooke, “Accurate, High Speed Modeling of Integrated Passive Devices,” IMAPS ATW Passive Component Technology, Braselton, GA, March 1997. 37 R. Poddar and M. Brooke, “Integrated Passive Device Design Based Upon Design and Modeling of Test Structures,” IMAPS Next Generation Package Design Workshop, South Carolina, June 1997. 38 G. Strang, Introduction to Applied Mathematics, Wellesley, MA: WellesleyCambridge Press, 1986. 39 A. E. Ruehli, “Equivalent Circuit Models for Three Dimensional Multiconductor Systems,” IEEE Trans. Microwave Theory Tech., vol. MTT-22, pp. 216-221, Mar. 1974. 40 H. Heeb and A. E. Ruehli, “Three-Dimensional Interconnect Analysis Using Partial Element Equivalent Circuits,” IEEE Trans. On Circuits and Systems-I: Fund. Theory and Applications, vol. 39, no. 11, pp. 974-82, Nov. 1992. 41 D. M. Pozar, "Microwave Engineering", Reading, MA, Addison-Wesley, 1990. 42 L. T. Pillage, and R. A. Rohrer, “Asymptotic Waveform Evaluation for Timing Analysis”, IEEE Trans. On CAD, vol. 9, no. 4, pp. 352-66, April 1990. 43 L. T. Pillage, R. A. Rohrer, and C. Visweswariah, “Electronic Circuit and System Simulation Methods”, New York, NY, McGraw-Hill, 1995. 44 W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical Recipes in C, The Art of Scientific Computing, New York: Cambridge University Press, 1988. 45 Hspice Users Manual, Meta Software, May 1996. 46 Y. Eo and W. R. Eisenstadt, “High-Speed VLSI Interconnect Modeling Based on S-Parameter Measurements,” IEEE Trans. On Components, Hybrids, and Manufacturing Tech., vol. 16, no. 5, pp. 555-562, 1993. 47 J. C. Rautio, “Synthesis of Lumped Models from N-Port Scattering Parameter Data,” IEEE Trans. on Microwave Theory and Tech., vol. 42, no. 3, pp. 535-537, March 1994. 48 R. Poddar, E. Moon, N. Jokerst, and M. Brooke, “Accurate, Rapid High Frequency Empirically Based Predictive Modeling of Arbirary Geometry Planar Resistive Passive Devices,” submitted for publication in IEEE Trans. on Components, Packaging, and Manufacturing Technology, Part B. 393 49 W. H. Haydl, “Properties of Meander Coplanar Transmission Lines,” IEEE Microwave and Guided Wave Letters, vol. 2 no. 11, pp. 439-441, Nov. 1992. 50 A. E. Ruehli and H. Heeb, “Circuit Models for Three-Dimensional Geometries Including Dielectrics,” IEEE Trans. Microwave Theory Tech., vol. 40, no. 7, pp. 1507-16, July 1992. 51 D. Gibson, R. Poddar, G. S. May, and M. A. Brooke, “Statistically Based Parametric Yield Prediction for Integrated Circuits”, IEEE Transactions on Semiconductor Manufacturing, Nov. 1997. 52 G. H. Golub and C. F. Van Loan, Matrix Computations, Baltimore, MD: The Johns Hopkins University Press, 1989. 53 M. Kamath, “Answering Your HOTLINE Questions…,” META-SOFWARE Journal, vol. 1, no. 2, June 1994. 54 L. Spruiell, “Optimization Error Surfaces,” META-SOFWARE Journal, vol. 1, no. 4, December 1994. 394