Mathematical requirements of physical channe
Table of Contents
Preface xv
1. Introduction: The Importance of Signal Integrity 1
1.1 Computing Power: Past and Future 1
1.2 The Problem 4
1.3 The Basics 5
1.4 A New Realm of Bus Design 7
1.5 Scope of the Book 7
1.6 Summary 8
References 8
2. Electromagnetic Fundamentals for Signal Integrity 9
2.1 Maxwell’s Equations 10
2.2 Common Vector Operators 13
2.2.1 Vector 13
2.2.2 Dot Product 13
2.2.3 Cross Product 14
2.2.4 Vector and Scalar Fields 15
2.2.5 Flux 15
2.2.6 Gradient 18
2.2.7 Divergence 18
2.2.8 Curl 20
2.3 Wave Propagation 23
2.3.1 Wave Equation 23
2.3.2 Relation Between E and H and the Transverse Electromagnetic Mode 25
2.3.3 Time-Harmonic Fields 27
2.3.4 Propagation of Time-Harmonic Plane Waves 28
2.4 Electrostatics 32
2.4.1 Electrostatic Scalar Potential in Terms of an Electric Field 36
2.4.2 Energy in an Electric Field 37
2.4.3 Capacitance 40
2.4.4 Energy Stored in a Capacitor 41
2.5 Magnetostatics 42
2.5.1 Magnetic Vector Potential 46
2.5.2 Inductance 48
2.5.3 Energy in a Magnetic Field 51
2.6 Power Flow and the Poynting Vector 53
2.6.1 Time-Averaged Values 56
2.7 Reflections of Electromagnetic Waves 57
2.7.1 Plane Wave Incident on a Perfect Conductor 57
2.7.2 Plane Wave Incident on a Lossless Dielectric 60
References 62
Problems 62
3. Ideal Transmission-Line Fundamentals 65
3.1 Transmission-Line Structures 66
3.2 Wave Propagation on Loss-Free Transmission Lines 67
3.2.1 Electric and Magnetic Fields on a Transmission Line 68
3.2.2 Telegrapher’s Equations 73
3.2.3 Equivalent Circuit for the Loss-Free Case 76
3.2.4 Wave Equation in Terms of LC 80
3.3 Transmission-Line Properties 82
3.3.1 Transmission-Line Phase Velocity 82
3.3.2 Transmission-Line Characteristic Impedance 82
3.3.3 Effective Dielectric Permittivity 83
3.3.4 Simple Formulas for Calculating the Characteristic Impedance 85
3.3.5 Validity of the TEM Approximation 86
3.4 Transmission-Line Parameters for the Loss-Free Case 90
3.4.1 Laplace and Poisson Equations 91
3.4.2 Transmission-Line Parameters for a Coaxial Line 91
3.4.3 Transmission-Line Parameters for a Microstrip 94
3.4.4 Charge Distribution Near a Conductor Edge 100
3.4.5 Charge Distribution and Transmission-Line Parameters 104
3.4.6 Field Mapping 107
3.5 Transmission-Line Reflections 113
3.5.1 Transmission-Line Reflection and Transmission Coefficient 113
3.5.2 Launching an Initial Wave 116
3.5.3 Multiple Reflections 116
3.5.4 Lattice Diagrams and Over- or Underdriven Transmission Lines 118
3.5.5 Lattice Diagrams for Nonideal Topologies 121
3.5.6 Effect of Rise and Fall Times on Reflections 129
3.5.7 Reflections from Reactive Loads 129
3.6 Time-Domain Reflectometry 134
3.6.1 Measuring the Characteristic Impedance and Delay of a Transmission Line 134
3.6.2 Measuring Inductance and Capacitance of Reactive Structures 137
3.6.3 Understanding the TDR Profile 140
References 140
Problems 141
4. Crosstalk 145
4.1 Mutual Inductance and Capacitance 146
4.1.1 Mutual Inductance 147
4.1.2 Mutual Capacitance 149
4.1.3 Field Solvers 152
4.2 Coupled Wave Equations 153
4.2.1 Wave Equation Revisited 153
4.2.2 Coupled Wave Equations 155
4.3 Coupled Line Analysis 157
4.3.1 Impedance and Velocity 157
4.3.2 Coupled Noise 165
4.4 Modal Analysis 177
4.4.1 Modal Decomposition 178
4.4.2 Modal Impedance and Velocity 180
4.4.3 Reconstructing the Signal 180
4.4.4 Modal Analysis 181
4.4.5 Modal Analysis of Lossy Lines 192
4.5 Crosstalk Minimization 193
4.6 Summary 194
References 195
Problems 195
5. Nonideal Conductor Models 201
5.1 Signals Propagating in Unbounded Conductive Media 202
5.1.1 Propagation Constant for Conductive Media 202
5.1.2 Skin Depth 204
5.2 Classic Conductor Model for Transmission Lines 205
5.2.1 Dc Losses in Conductors 206
5.2.2 Frequency-Dependent Resistance in Conductors 207
5.2.3 Frequency-Dependent Inductance 213
5.2.4 Power Loss in a Smooth Conductor 218
5.3 Surface Roughness 222
5.3.1 Hammerstad Model 223
5.3.2 Hemispherical Model 228
5.3.3 Huray Model 237
5.3.4 Conclusions 243
5.4 Transmission-Line Parameters for Nonideal Conductors 244
5.4.1 Equivalent Circuit Impedance and Propagation Constant 244
5.4.2 Telegrapher’s Equations for a Real Conductor and a Perfect Dielectric 246
References 247
Problems 247
6. Electrical Properties of Dielectrics 249
6.1 Polarization of Dielectrics 250
6.1.1 Electronic Polarization 250
6.1.2 Orientational (Dipole) Polarization 253
6.1.3 Ionic (Molecular) Polarization 253
6.1.4 Relative Permittivity 254
6.2 Classification of Dielectric Materials 256
6.3 Frequency-Dependent Dielectric Behavior 256
6.3.1 Dc Dielectric Losses 257
6.3.2 Frequency-Dependent Dielectric Model: Single Pole 257
6.3.3 Anomalous Dispersion 261
6.3.4 Frequency-Dependent Dielectric Model: Multipole 262
6.3.5 Infinite-Pole Model 266
6.4 Properties of a Physical Dielectric Model 269
6.4.1 Relationship Between ε_ and ε__ 269
6.4.2 Mathematical Limits 271
6.5 Fiber-Weave Effect 274
6.5.1 Physical Structure of an FR4 Dielectric and Dielectric Constant Variation 275
6.5.2 Mitigation 276
6.5.3 Modeling the Fiber-Weave Effect 277
6.6 Environmental Variation in Dielectric Behavior 279
6.6.1 Environmental Effects on Transmission-Line Performance 281
6.6.2 Mitigation 283
6.6.3 Modeling the Effect of Relative Humidity on an FR4 Dielectric 284
6.7 Transmission-Line Parameters for Lossy Dielectrics and Realistic Conductors 285
6.7.1 Equivalent Circuit Impedance and Propagation Constant 285
6.7.2 Telegrapher’s Equations for Realistic Conductors and Lossy Dielectrics 291
References 292
Problems 292
7. Differential Signaling 297
7.1 Removal of Common-Mode Noise 299
7.2 Differential Crosstalk 300
7.3 Virtual Reference Plane 302
7.4 Propagation of Modal Voltages 303
7.5 Common Terminology 304
7.6 Drawbacks of Differential Signaling 305
7.6.1 Mode Conversion 305
7.6.2 Fiber-Weave Effect 310
Reference 313
Problems 313
8. Mathematical Requirements for Physical Channels 315
8.1 Frequency-Domain Effects in Time-Domain Simulations 316
8.1.1 Linear and Time Invariance 316
8.1.2 Time- and Frequency-Domain Equivalencies 317
8.1.3 Frequency Spectrum of a Digital Pulse 321
8.1.4 System Response 324
8.1.5 Single-Bit (Pulse) Response 327
8.2 Requirements for a Physical Channel 331
8.2.1 Causality 331
8.2.2 Passivity 340
8.2.3 Stability 343
References 345
Problems 345
9. Network Analysis for Digital Engineers 347
9.1 High-Frequency Voltage and Current Waves 349
9.1.1 Input Reflection into a Terminated Network 349
9.1.2 Input Impedance 353
9.2 Network Theory 354
9.2.1 Impedance Matrix 355
9.2.2 Scattering Matrix 358
9.2.3 ABCD Parameters 382
9.2.4 Cascading S-Parameters 390
9.2.5 Calibration and Deembedding 395
9.2.6 Changing the Reference Impedance 399
9.2.7 Multimode S-Parameters 400
9.3 Properties of Physical S-Parameters 406
9.3.1 Passivity 406
9.3.2 Reality 408
9.3.3 Causality 408
9.3.4 Subjective Examination of S-Parameters 410
References 413
Problems 413
10. Topics in High-Speed Channel Modeling 417
10.1 Creating a Physical Transmission-Line Model 418
10.1.1 Tabular Approach 418
10.1.2 Generating a Tabular Dielectric Model 419
10.1.3 Generating a Tabular Conductor Model 420
10.2 NonIdeal Return Paths 422
10.2.1 Path of Least Impedance 422
10.2.2 Transmission Line Routed Over a Gap in the Reference Plane 423
10.2.3 Summary 434
10.3 Vias 434
10.3.1 Via Resonance 434
10.3.2 Plane Radiation Losses 437
10.3.3 Parallel-Plate Waveguide 439
References 441
Problems 442
11. I/O Circuits and Models 443
11.1 I/O Design Considerations 444
11.2 Push–Pull Transmitters 446
11.2.1 Operation 446
11.2.2 Linear Models 448
11.2.3 Nonlinear Models 453
11.2.4 Advanced Design Considerations 455
11.3 CMOS receivers 459
11.3.1 Operation 459
11.3.2 Modeling 460
11.3.3 Advanced Design Considerations 460
11.4 ESD Protection Circuits 460
11.4.1 Operation 461
11.4.2 Modeling 461
11.4.3 Advanced Design Considerations 463
11.5 On-Chip Termination 463
11.5.1 Operation 463
11.5.2 Modeling 463
11.5.3 Advanced Design Considerations 464
11.6 Bergeron Diagrams 465
11.6.1 Theory and Method 470
11.6.2 Limitations 474
11.7 Open-Drain Transmitters 474
11.7.1 Operation 474
11.7.2 Modeling 476
11.7.3 Advanced Design Considerations 476
11.8 Differential Current-Mode Transmitters 479
11.8.1 Operation 479
11.8.2 Modeling 480
11.8.3 Advanced Design Considerations 480
11.9 Low-Swing and Differential Receivers 481
11.9.1 Operation 481
11.9.2 Modeling 482
11.9.3 Advanced Design Considerations 483
11.10 IBIS Models 483
11.10.1 Model Structure and Development Process 483
11.10.2 Generating Model Data 485
11.10.3 Differential I/O Models 488
11.10.4 Example of an IBIS File 490
11.11 Summary 492
References 492
Problems 494
12. Equalization 499
12.1 Analysis and Design Background 500
12.1.1 Maximum Data Transfer Capacity 500
12.1.2 Linear Time-Invariant Systems 502
12.1.3 Ideal Versus Practical Interconnects 506
12.1.4 Equalization Overview 511
12.2 Continuous-Time Linear Equalizers 513
12.2.1 Passive CTLEs 514
12.2.2 Active CTLEs 521
12.3 Discrete Linear Equalizers 522
12.3.1 Transmitter Equalization 525
12.3.2 Coefficient Selection 530
12.3.3 Receiver Equalization 535
12.3.4 Nonidealities in DLEs 536
12.3.5 Adaptive Equalization 536
12.4 Decision Feedback Equalization 540
12.5 Summary 542
References 545
Problems 546
13. Modeling and Budgeting of Timing Jitter and Noise 549
13.1 Eye Diagram 550
13.2 Bit Error Rate 552
13.2.1 Worst-Case Analysis 552
13.2.2 Bit Error Rate Analysis 555
13.3 Jitter Sources and Budgets 560
13.3.1 Jitter Types and Sources 561
13.3.2 System Jitter Budgets 568
13.4 Noise Sources and Budgets 572
13.4.1 Noise Sources 572
13.4.2 Noise Budgets 579
13.5 Peak Distortion Analysis Methods 583
13.5.1 Superposition and the Pulse Response 583
13.5.2 Worst-Case Bit Patterns and Data Eyes 585
13.5.3 Peak Distortion Analysis Including Crosstalk 594
13.5.4 Limitations 598
13.6 Summary 599
References 599
Problems 600
14. System Analysis Using Response Surface Modeling 605
14.1 Model Design Considerations 606
14.2 Case Study: 10-Gb/s Differential PCB Interface 607
14.3 RSM Construction by Least Squares Fitting 607
14.4 Measures of Fit 615
14.4.1 Residuals 615
14.4.2 Fit Coefficients 616
14.5 Significance Testing 618
14.5.1 Model Significance: The F-Test 618
14.5.2 Parameter Significance: Individual t-Tests 619
14.6 Confidence Intervals 621
14.7 Sensitivity Analysis and Design Optimization 623
14.8 Defect Rate Prediction Using Monte Carlo Simulation 628
14.9 Additional RSM Considerations 633
14.10 Summary 633
References 634
Problems 635
Appendix A: Useful Formulas Identities Units and Constants 637
Appendix B: Four-Port Conversions Between T- and S-Parameters 641
Appendix C: Critical Values of the F-Statistic 645
Appendix D: Critical Values of the T-Statistic 647
Appendix E: Causal Relationship Between Skin Effect Resistance and Internal Inductance for Rough Conductors 649
Appendix F: Spice Level 3 Model for 0.25 μm MOSIS Process 653
Index 655
