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
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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.
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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.
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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.
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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.
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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
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3.74E-02
4.02E-02
4.31E-02
4.61E-02
4.92E-02
5.23E-02
5.56E-02
5.89E-02
6.23E-02
6.58E-02
6.94E-02
7.30E-02
7.67E-02
8.05E-02
8.43E-02
8.82E-02
9.21E-02
9.61E-02
1.00E-01
1.04E-01
1.08E-01
-1.93E-02
-1.50E-02
-1.07E-02
-6.35E-03
-2.03E-03
2.29E-03
6.60E-03
1.09E-02
1.52E-02
1.95E-02
2.37E-02
2.80E-02
3.22E-02
3.64E-02
4.05E-02
4.46E-02
4.87E-02
5.27E-02
5.66E-02
6.05E-02
6.44E-02
6.81E-02
7.18E-02
7.55E-02
7.91E-02
8.25E-02
8.59E-02
8.93E-02
9.25E-02
9.57E-02
9.87E-02
1.02E-01
1.05E-01
1.07E-01
1.10E-01
1.13E-01
1.15E-01
1.17E-01
1.20E-01
1.22E-01
1.24E-01
1.26E-01
1.27E-01
1.29E-01
1.31E-01
1.32E-01
1.33E-01
309
-9.79E-01
-9.80E-01
-9.80E-01
-9.81E-01
-9.81E-01
-9.81E-01
-9.81E-01
-9.80E-01
-9.80E-01
-9.79E-01
-9.77E-01
-9.76E-01
-9.74E-01
-9.72E-01
-9.70E-01
-9.68E-01
-9.65E-01
-9.62E-01
-9.59E-01
-9.56E-01
-9.52E-01
-9.48E-01
-9.44E-01
-9.40E-01
-9.35E-01
-9.30E-01
-9.25E-01
-9.20E-01
-9.15E-01
-9.09E-01
-9.03E-01
-8.97E-01
-8.91E-01
-8.84E-01
-8.78E-01
-8.71E-01
-8.64E-01
-8.57E-01
-8.49E-01
-8.41E-01
-8.34E-01
-8.26E-01
-8.17E-01
-8.09E-01
-8.01E-01
-7.92E-01
-7.83E-01
-6.93E-02
-5.44E-02
-3.95E-02
-2.45E-02
-9.59E-03
5.36E-03
2.03E-02
3.53E-02
5.02E-02
6.51E-02
8.00E-02
9.49E-02
1.10E-01
1.25E-01
1.39E-01
1.54E-01
1.69E-01
1.83E-01
1.98E-01
2.12E-01
2.27E-01
2.41E-01
2.55E-01
2.70E-01
2.84E-01
2.98E-01
3.12E-01
3.26E-01
3.39E-01
3.53E-01
3.66E-01
3.80E-01
3.93E-01
4.06E-01
4.19E-01
4.32E-01
4.45E-01
4.58E-01
4.70E-01
4.83E-01
4.95E-01
5.07E-01
5.19E-01
5.31E-01
5.42E-01
5.54E-01
5.65E-01
1.40E+10
1.40E+10
1.41E+10
1.42E+10
1.42E+10
1.43E+10
1.43E+10
1.44E+10
1.44E+10
1.45E+10
1.45E+10
1.46E+10
1.46E+10
1.47E+10
1.47E+10
1.48E+10
1.49E+10
1.49E+10
1.50E+10
1.50E+10
1.51E+10
1.51E+10
1.52E+10
1.52E+10
1.53E+10
1.53E+10
1.54E+10
1.54E+10
1.55E+10
1.55E+10
1.56E+10
1.57E+10
1.57E+10
1.58E+10
1.58E+10
1.59E+10
1.59E+10
1.60E+10
1.60E+10
1.61E+10
1.61E+10
1.62E+10
1.62E+10
1.63E+10
1.63E+10
1.64E+10
1.65E+10
1.13E-01
1.17E-01
1.21E-01
1.25E-01
1.29E-01
1.34E-01
1.38E-01
1.42E-01
1.46E-01
1.51E-01
1.55E-01
1.59E-01
1.63E-01
1.68E-01
1.72E-01
1.76E-01
1.80E-01
1.84E-01
1.89E-01
1.93E-01
1.97E-01
2.01E-01
2.05E-01
2.09E-01
2.12E-01
2.16E-01
2.20E-01
2.24E-01
2.27E-01
2.31E-01
2.34E-01
2.38E-01
2.41E-01
2.44E-01
2.47E-01
2.51E-01
2.54E-01
2.57E-01
2.59E-01
2.62E-01
2.65E-01
2.67E-01
2.70E-01
2.72E-01
2.75E-01
2.77E-01
2.79E-01
1.34E-01
1.35E-01
1.36E-01
1.37E-01
1.38E-01
1.38E-01
1.39E-01
1.39E-01
1.39E-01
1.39E-01
1.39E-01
1.39E-01
1.39E-01
1.38E-01
1.38E-01
1.37E-01
1.36E-01
1.35E-01
1.34E-01
1.33E-01
1.32E-01
1.30E-01
1.29E-01
1.27E-01
1.25E-01
1.24E-01
1.22E-01
1.20E-01
1.17E-01
1.15E-01
1.13E-01
1.10E-01
1.08E-01
1.05E-01
1.02E-01
9.94E-02
9.65E-02
9.35E-02
9.05E-02
8.73E-02
8.41E-02
8.08E-02
7.74E-02
7.40E-02
7.05E-02
6.69E-02
6.33E-02
310
-7.74E-01
-7.65E-01
-7.56E-01
-7.46E-01
-7.37E-01
-7.27E-01
-7.17E-01
-7.07E-01
-6.97E-01
-6.87E-01
-6.76E-01
-6.66E-01
-6.55E-01
-6.45E-01
-6.34E-01
-6.23E-01
-6.12E-01
-6.01E-01
-5.89E-01
-5.78E-01
-5.67E-01
-5.55E-01
-5.44E-01
-5.32E-01
-5.20E-01
-5.08E-01
-4.96E-01
-4.84E-01
-4.72E-01
-4.60E-01
-4.48E-01
-4.36E-01
-4.23E-01
-4.11E-01
-3.99E-01
-3.86E-01
-3.74E-01
-3.61E-01
-3.48E-01
-3.36E-01
-3.23E-01
-3.10E-01
-2.97E-01
-2.85E-01
-2.72E-01
-2.59E-01
-2.46E-01
5.77E-01
5.88E-01
5.98E-01
6.09E-01
6.20E-01
6.30E-01
6.40E-01
6.50E-01
6.60E-01
6.70E-01
6.80E-01
6.89E-01
6.98E-01
7.07E-01
7.16E-01
7.25E-01
7.33E-01
7.42E-01
7.50E-01
7.58E-01
7.66E-01
7.73E-01
7.81E-01
7.88E-01
7.95E-01
8.02E-01
8.09E-01
8.15E-01
8.22E-01
8.28E-01
8.34E-01
8.40E-01
8.46E-01
8.51E-01
8.57E-01
8.62E-01
8.67E-01
8.72E-01
8.76E-01
8.81E-01
8.85E-01
8.89E-01
8.93E-01
8.97E-01
9.01E-01
9.04E-01
9.07E-01
1.65E+10
1.66E+10
1.66E+10
1.67E+10
1.67E+10
1.68E+10
1.68E+10
1.69E+10
1.69E+10
1.70E+10
1.70E+10
1.71E+10
1.72E+10
1.72E+10
1.73E+10
1.73E+10
1.74E+10
1.74E+10
1.75E+10
1.75E+10
1.76E+10
1.76E+10
1.77E+10
1.77E+10
1.78E+10
1.78E+10
1.79E+10
1.80E+10
1.80E+10
1.81E+10
1.81E+10
1.82E+10
1.82E+10
1.83E+10
1.83E+10
1.84E+10
1.84E+10
1.85E+10
1.85E+10
1.86E+10
1.87E+10
1.87E+10
1.88E+10
1.88E+10
1.89E+10
1.89E+10
1.90E+10
2.81E-01
2.83E-01
2.84E-01
2.86E-01
2.88E-01
2.89E-01
2.90E-01
2.92E-01
2.93E-01
2.94E-01
2.95E-01
2.95E-01
2.96E-01
2.97E-01
2.97E-01
2.97E-01
2.97E-01
2.97E-01
2.97E-01
2.97E-01
2.97E-01
2.97E-01
2.96E-01
2.95E-01
2.95E-01
2.94E-01
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
5.97E-02
5.59E-02
5.22E-02
4.83E-02
4.45E-02
4.06E-02
3.66E-02
3.27E-02
2.86E-02
2.46E-02
2.05E-02
1.65E-02
1.23E-02
8.22E-03
4.09E-03
-6.09E-05
-4.22E-03
-8.37E-03
-1.25E-02
-1.67E-02
-2.08E-02
-2.50E-02
-2.91E-02
-3.32E-02
-3.73E-02
-4.14E-02
-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
311
-2.33E-01
-2.20E-01
-2.07E-01
-1.94E-01
-1.81E-01
-1.68E-01
-1.55E-01
-1.42E-01
-1.28E-01
-1.15E-01
-1.02E-01
-8.88E-02
-7.56E-02
-6.24E-02
-4.92E-02
-3.59E-02
-2.27E-02
-9.44E-03
3.81E-03
1.71E-02
3.03E-02
4.36E-02
5.68E-02
7.01E-02
8.33E-02
9.65E-02
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
9.10E-01
9.13E-01
9.16E-01
9.19E-01
9.21E-01
9.23E-01
9.25E-01
9.27E-01
9.29E-01
9.31E-01
9.32E-01
9.33E-01
9.34E-01
9.35E-01
9.36E-01
9.37E-01
9.37E-01
9.37E-01
9.37E-01
9.37E-01
9.37E-01
9.36E-01
9.36E-01
9.35E-01
9.34E-01
9.33E-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
1.90E+10
1.91E+10
1.91E+10
1.92E+10
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
2.47E-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
-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
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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
-9.06E-01
-9.11E-01
-9.15E-01
-9.19E-01
-9.23E-01
-9.26E-01
-9.29E-01
-9.31E-01
-9.34E-01
-9.36E-01
-9.37E-01
-9.38E-01
3.48E+09
3.52E+09
3.56E+09
3.60E+09
3.64E+09
3.68E+09
3.70E+09
3.71E+09
3.73E+09
3.75E+09
3.77E+09
3.79E+09
3.81E+09
3.83E+09
3.85E+09
3.87E+09
3.89E+09
3.91E+09
3.93E+09
3.95E+09
3.97E+09
3.99E+09
4.01E+09
4.03E+09
4.05E+09
4.06E+09
4.08E+09
4.10E+09
4.12E+09
4.14E+09
4.16E+09
4.18E+09
4.20E+09
4.22E+09
4.24E+09
4.26E+09
4.28E+09
4.30E+09
4.32E+09
4.34E+09
4.36E+09
4.38E+09
4.39E+09
4.41E+09
4.43E+09
4.45E+09
4.47E+09
1.21E-01
1.18E-01
1.16E-01
1.14E-01
1.11E-01
1.09E-01
1.08E-01
1.06E-01
1.05E-01
1.03E-01
1.02E-01
1.00E-01
9.89E-02
9.74E-02
9.58E-02
9.42E-02
9.26E-02
9.10E-02
8.94E-02
8.77E-02
8.60E-02
8.43E-02
8.26E-02
8.08E-02
7.90E-02
7.72E-02
7.54E-02
7.36E-02
7.18E-02
6.99E-02
6.81E-02
6.62E-02
6.43E-02
6.24E-02
6.05E-02
5.86E-02
5.67E-02
5.47E-02
5.28E-02
5.09E-02
4.90E-02
4.70E-02
4.51E-02
4.32E-02
4.13E-02
3.94E-02
3.75E-02
-1.54E-02
-1.70E-02
-1.85E-02
-2.00E-02
-2.14E-02
-2.26E-02
-2.32E-02
-2.38E-02
-2.43E-02
-2.48E-02
-2.53E-02
-2.57E-02
-2.61E-02
-2.65E-02
-2.69E-02
-2.72E-02
-2.74E-02
-2.77E-02
-2.79E-02
-2.80E-02
-2.81E-02
-2.82E-02
-2.82E-02
-2.82E-02
-2.81E-02
-2.80E-02
-2.79E-02
-2.77E-02
-2.75E-02
-2.72E-02
-2.68E-02
-2.64E-02
-2.60E-02
-2.55E-02
-2.50E-02
-2.44E-02
-2.37E-02
-2.30E-02
-2.23E-02
-2.15E-02
-2.06E-02
-1.97E-02
-1.87E-02
-1.77E-02
-1.66E-02
-1.54E-02
-1.42E-02
329
2.43E-02
4.98E-03
-1.45E-02
-3.41E-02
-5.38E-02
-7.36E-02
-8.35E-02
-9.34E-02
-1.03E-01
-1.13E-01
-1.23E-01
-1.34E-01
-1.44E-01
-1.54E-01
-1.64E-01
-1.74E-01
-1.84E-01
-1.94E-01
-2.04E-01
-2.14E-01
-2.24E-01
-2.35E-01
-2.45E-01
-2.55E-01
-2.65E-01
-2.75E-01
-2.85E-01
-2.95E-01
-3.05E-01
-3.16E-01
-3.26E-01
-3.36E-01
-3.46E-01
-3.56E-01
-3.66E-01
-3.76E-01
-3.86E-01
-3.96E-01
-4.06E-01
-4.16E-01
-4.26E-01
-4.36E-01
-4.46E-01
-4.55E-01
-4.65E-01
-4.75E-01
-4.85E-01
-9.39E-01
-9.40E-01
-9.40E-01
-9.39E-01
-9.38E-01
-9.37E-01
-9.36E-01
-9.36E-01
-9.35E-01
-9.33E-01
-9.32E-01
-9.31E-01
-9.30E-01
-9.28E-01
-9.26E-01
-9.25E-01
-9.23E-01
-9.21E-01
-9.19E-01
-9.16E-01
-9.14E-01
-9.12E-01
-9.09E-01
-9.06E-01
-9.03E-01
-9.00E-01
-8.97E-01
-8.94E-01
-8.91E-01
-8.87E-01
-8.84E-01
-8.80E-01
-8.76E-01
-8.72E-01
-8.68E-01
-8.64E-01
-8.60E-01
-8.55E-01
-8.50E-01
-8.46E-01
-8.41E-01
-8.36E-01
-8.31E-01
-8.25E-01
-8.20E-01
-8.14E-01
-8.09E-01
4.49E+09
4.51E+09
4.53E+09
4.55E+09
4.57E+09
4.59E+09
4.61E+09
4.63E+09
4.65E+09
4.67E+09
4.69E+09
4.71E+09
4.73E+09
4.74E+09
4.76E+09
4.78E+09
4.80E+09
4.82E+09
4.84E+09
4.86E+09
4.88E+09
4.90E+09
4.92E+09
4.94E+09
4.96E+09
4.98E+09
5.00E+09
5.02E+09
5.04E+09
5.06E+09
5.08E+09
5.10E+09
5.13E+09
5.16E+09
5.19E+09
5.22E+09
5.25E+09
5.28E+09
5.31E+09
5.34E+09
5.37E+09
5.40E+09
5.42E+09
5.45E+09
5.48E+09
5.51E+09
5.54E+09
3.56E-02
3.37E-02
3.18E-02
3.00E-02
2.81E-02
2.63E-02
2.45E-02
2.27E-02
2.10E-02
1.92E-02
1.75E-02
1.59E-02
1.42E-02
1.26E-02
1.10E-02
9.46E-03
7.95E-03
6.48E-03
5.05E-03
3.66E-03
2.32E-03
1.02E-03
-2.25E-04
-1.42E-03
-2.57E-03
-3.66E-03
-4.70E-03
-5.68E-03
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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
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364
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365
1.59E-01
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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 -1.88E-01
1.53E+09
1.25E-01
-4.60E-02 6.69E-01
-6.24E-01 3.48E-01
-7.85E-01 -2.72E-01 -1.32E-01
1.56E+09
1.76E-01
-1.30E-02 5.75E-01
-6.99E-01 2.57E-01
-8.32E-01 -2.51E-01 -8.20E-02
1.58E+09
2.36E-01
6.00E-03
4.71E-01
-7.55E-01 1.59E-01
-8.68E-01 -2.25E-01 -4.20E-02
1.61E+09
2.99E-01
9.00E-03
3.63E-01
-7.92E-01 5.30E-02
-8.87E-01 -1.97E-01 -9.00E-03
1.63E+09
3.63E-01
-6.00E-03 2.53E-01
-8.07E-01 -5.80E-02 -8.92E-01 -1.69E-01 1.80E-02
1.66E+09
4.22E-01
-3.40E-02 1.45E-01
-8.06E-01 -1.72E-01 -8.77E-01 -1.42E-01 3.90E-02
1.68E+09
4.73E-01
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4.60E+09
-2.26E-01 -8.76E-01 -3.60E-02 -5.30E-02 7.16E-01
-4.80E-01 5.80E-02
1.18E-01
4.63E+09
-2.92E-01 -8.59E-01 9.00E-03
-6.50E-02 6.42E-01
-5.63E-01 7.30E-02
1.12E-01
4.65E+09
-3.53E-01 -8.33E-01 4.50E-02
-8.60E-02 5.48E-01
-6.35E-01 8.80E-02
1.03E-01
4.68E+09
-4.10E-01 -8.03E-01 7.20E-02
-1.14E-01 4.31E-01
-6.94E-01 1.05E-01
9.20E-02
4.70E+09
-4.63E-01 -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
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-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
-5.90E-02
-7.40E-02
-9.00E-02
-1.06E-01
-1.24E-01
-1.43E-01
-1.67E-01
-1.79E-01
-1.96E-01
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-3.87E-01
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-4.25E-01
-4.55E-01
-4.86E-01
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-4.75E-01
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-6.21E-01
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-6.86E-01
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-2.31E-01
-2.50E-01
-2.68E-01
-2.85E-01
-3.00E-01
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-3.26E-01
-3.38E-01
-3.51E-01
-3.63E-01
-3.72E-01
-3.81E-01
-3.89E-01
-3.96E-01
-4.02E-01
-4.09E-01
-4.08E-01
-4.05E-01
-4.11E-01
-4.13E-01
-4.14E-01
-4.06E-01
-4.08E-01
-4.08E-01
-4.04E-01
-3.99E-01
-3.95E-01
-3.85E-01
-3.69E-01
-3.52E-01
-3.48E-01
-3.16E-01
-2.77E-01
-2.68E-01
-2.77E-01
-2.73E-01
-2.60E-01
-2.40E-01
-2.14E-01
-1.78E-01
-1.32E-01
-1.20E-01
-1.11E-01
-8.80E-02
-5.10E-02
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
4.63E+09
-2.30E-01
-2.06E-01
-1.80E-01
-1.60E-01
-1.54E-01
-1.42E-01
-1.36E-01
-1.43E-01
-1.59E-01
-1.80E-01
-2.08E-01
-2.43E-01
-2.78E-01
-3.13E-01
-3.52E-01
-3.88E-01
-4.23E-01
-4.51E-01
-4.75E-01
-5.08E-01
-5.43E-01
-5.75E-01
-6.05E-01
-6.32E-01
-6.57E-01
-6.78E-01
-6.97E-01
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-7.68E-01
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-7.35E-01
-7.46E-01
-7.41E-01
-7.35E-01
-7.24E-01
-7.10E-01
-6.96E-01
3.08E-01
2.78E-01
2.49E-01
2.10E-01
1.76E-01
1.39E-01
9.60E-02
4.90E-02
8.00E-03
-3.10E-02
-6.60E-02
-9.30E-02
-1.12E-01
-1.26E-01
-1.37E-01
-1.39E-01
-1.38E-01
-1.29E-01
-1.28E-01
-1.29E-01
-1.20E-01
-1.07E-01
-9.00E-02
-7.00E-02
-4.70E-02
-2.50E-02
0.00E+00
2.70E-02
5.50E-02
8.00E-02
1.08E-01
1.36E-01
1.62E-01
1.91E-01
2.18E-01
2.48E-01
2.78E-01
3.06E-01
3.28E-01
3.36E-01
3.58E-01
3.87E-01
4.15E-01
4.42E-01
4.66E-01
4.92E-01
388
-7.04E-01
-7.14E-01
-7.24E-01
-7.24E-01
-7.15E-01
-7.09E-01
-6.96E-01
-6.75E-01
-6.44E-01
-6.09E-01
-5.70E-01
-5.27E-01
-4.82E-01
-4.39E-01
-3.97E-01
-3.56E-01
-3.16E-01
-2.79E-01
-2.51E-01
-2.17E-01
-1.81E-01
-1.48E-01
-1.16E-01
-8.70E-02
-5.90E-02
-3.40E-02
-1.20E-02
9.00E-03
2.80E-02
4.50E-02
6.10E-02
7.50E-02
8.80E-02
1.02E-01
1.13E-01
1.24E-01
1.33E-01
1.39E-01
1.42E-01
1.46E-01
1.63E-01
1.75E-01
1.84E-01
1.91E-01
1.98E-01
2.04E-01
-9.00E-03
3.70E-02
8.30E-02
1.37E-01
1.80E-01
2.29E-01
2.81E-01
3.34E-01
3.80E-01
4.23E-01
4.60E-01
4.90E-01
5.13E-01
5.30E-01
5.43E-01
5.51E-01
5.56E-01
5.55E-01
5.57E-01
5.60E-01
5.59E-01
5.55E-01
5.47E-01
5.38E-01
5.26E-01
5.14E-01
5.01E-01
4.88E-01
4.75E-01
4.61E-01
4.47E-01
4.34E-01
4.21E-01
4.09E-01
3.96E-01
3.82E-01
3.69E-01
3.56E-01
3.44E-01
3.45E-01
3.38E-01
3.26E-01
3.14E-01
3.01E-01
2.89E-01
2.77E-01
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
-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
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