Electrical Modeling and Design for 3D System Integration E-Book

Er-Ping Li

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 373

Veröffentlichungsjahr: 2012

Beschreibung

New advanced modeling methods for simulating the electromagnetic properties of complex three-dimensional electronic systems Based on the author's extensive research, this book sets forth tested and proven electromagnetic modeling and simulation methods for analyzing signal and power integrity as well as electromagnetic interference in large complex electronic interconnects, multilayered package structures, integrated circuits, and printed circuit boards. Readers will discover the state of the technology in electronic package integration and printed circuit board simulation and modeling. In addition to popular full-wave electromagnetic computational methods, the book presents new, more sophisticated modeling methods, offering readers the most advanced tools for analyzing and designing large complex electronic structures. Electrical Modeling and Design for 3D System Integration begins with a comprehensive review of current modeling and simulation methods for signal integrity, power integrity, and electromagnetic compatibility. Next, the book guides readers through: * The macromodeling technique used in the electrical and electromagnetic modeling and simulation of complex interconnects in three-dimensional integrated systems * The semi-analytical scattering matrix method based on the N-body scattering theory for modeling of three-dimensional electronic package and multilayered printed circuit boards with multiple vias * Two- and three-dimensional integral equation methods for the analysis of power distribution networks in three-dimensional package integrations * The physics-based algorithm for extracting the equivalent circuit of a complex power distribution network in three-dimensional integrated systems and printed circuit boards * An equivalent circuit model of through-silicon vias * Metal-oxide-semiconductor capacitance effects of through-silicon vias Engineers, researchers, and students can turn to this book for the latest techniques and methods for the electrical modeling and design of electronic packaging, three-dimensional electronic integration, integrated circuits, and printed circuit boards.

Leseprobe

Table of Contents

Cover

IEEE Press

Title page

Copyright page

Foreword

Preface

CHAPTER 1 Introduction

1.1 INTRODUCTION OF ELECTRONIC PACKAGE INTEGRATION

1.2 REVIEW OF MODELING TECHNOLOGIES

1.3 ORGANIZATION OF THE BOOK

CHAPTER 2 Macromodeling of Complex Interconnects in 3D Integration

2.1 INTRODUCTION

2.2 NETWORK PARAMETERS: IMPEDANCE, ADMITTANCE, AND SCATTERING MATRICES

2.3 RATIONAL FUNCTION APPROXIMATION WITH PARTIAL FRACTIONS

2.4 VECTOR FITTING (VF) METHOD

2.5 MACROMODEL SYNTHESIS

2.6 STABILITY, CAUSALITY, AND PASSIVITY OF MACROMODEL

2.7 MACROMODELING APPLIED TO HIGH-SPEED INTERCONNECTS AND CIRCUITS

2.8 CONCLUSION

CHAPTER 3 2.5D Simulation Method for 3D Integrated Systems

3.1 INTRODUCTION

3.2 MULTIPLE SCATTERING METHOD FOR ELECTRONIC PACKAGE MODELING WITH OPEN BOUNDARY PROBLEMS

3.3 NOVEL BOUNDARY MODELING METHOD FOR SIMULATION OF FINITE-DOMAIN POWER-GROUND PLANES

3.4 NUMERICAL SIMULATIONS FOR FINITE STRUCTURES

3.5 MODELING OF 3D ELECTRONIC PACKAGE STRUCTURE

3.6 CONCLUSION

CHAPTER 4 Hybrid Integral Equation Modeling Methods for 3D Integration

4.1 INTRODUCTION

4.2 2D INTEGRAL EQUATION EQUIVALENT CIRCUIT (IEEC) METHOD

4.3 3D HYBRID INTEGRAL EQUATION METHOD

4.4 CONCLUSION

CHAPTER 5 Systematic Microwave Network Analysis for 3D Integrated Systems

5.1 INTRINSIC VIA CIRCUIT MODEL FOR MULTIPLE VIAS IN AN IRREGULAR PLATE PAIR

5.2 PARALLEL PLANE PAIR MODEL

5.3 CASCADED MULTIPORT NETWORK ANALYSIS OF MULTILAYER STRUCTURE WITH MULTIPLE VIAS

APPENDIX: PROPERTIES OF THE AUXILIARY FUNCTION Wmn(x, y)

CHAPTER 6 Modeling of Through-Silicon Vias (TSV) in 3D Integration

6.1 INTRODUCTION

6.2 EQUIVALENT CIRCUIT MODEL FOR TSV

6.3 MOS CAPACITANCE EFFECT OF TSV

6.4 CONCLUSION

Index

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Copyright © 2012 by Institute of Electrical and Electronics Engineers. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.

Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data:

Li, Er-Ping.

 Electrical modeling and design for 3D system integration : 3D integrated circuits and packaging, signal integrity, power integrity and EMC / Er-Ping Li.

p. cm.

 ISBN 978-0-470-62346-6 (hardback)

1. Three-dimensional integrated circuits. I. Title.

 TK7874.893.L53 2011

 621.3815–dc23

2011028946

Foreword

Today, the modeling of electrical interconnects and packages is very important from both a practical and a theoretical point of view. High performance and high speed especially require a great deal of skill. An ever-increasing number of practical designs fall into this class.

The fact that we now have powerful design tools increases our ability to solve a larger number of real-world problems for many different issues. This greatly helps solve most of the important problems for a large class of geometries. However, the ever-increasing performance of the technology requires a continuous evolution of the skills in modeling techniques. A key performance issue is the reduction in effort and computing time for very large problems. Clearly, better design tools and techniques lead to better designs. Over the years, we also could observe that the opposite is true, namely that the more challenging problems lead to improved tools as well as better technical solutions. A consequence of this process is the continuous bootstrapping of the tools and techniques as well as the designers’ skills.

This book represents an educational tool for modelers as well as for tool designers. It offers an unusual combination of the latest techniques for the electromagnetic (EM) modeling of packages and signal interconnections, including the challenging via problems. In fact, it is much more detailed than some of the introductory texts which are available today on the subject. It considers all aspects such as the analysis methods for the construction of macromodels which are stable, causal, and passive. Such models are widely in use today, and the passivity issue impacts the accuracy in both the frequency and time domains, while instability is unacceptable in the time domain. Also, key aspects of the modeling are the noise interactions between the multitude of wires and signal planes which are present in a typical design. All these aspects are considered in detail from an electromagnetic point of view, and sophisticated solution techniques are given. It is evident from this book that addressing the modern 3D packaging technology is an integral part of what makes the book relevant.

We are fortunate to find in this book the contributions of an author who is both experienced and knowledgeable in this field. Dr. Er-Ping Li is an internationally well-known contributor to the field of electromagnetic solutions in the area of interest. He has been a Principal Scientist and Director of Electronics and Photonics at A*STAR (Agency for Science, Technology and Research) Institute of High Performance Computing in Singapore. From 2010 he holds an appointment as Chair Professor in Zhejiang University, China. He is a Fellow of the IEEE and a Fellow of the Electromagnetic Academy, USA. He received numerous international awards and honors in recognition of his professional work.

ALBERT E. RUEHLI, PhD, Life Fellow of IEEE

Emeritus, IBM T. J. Watson Research Center, Yorktown, NY, USA

Adjunct Professor, EMC Lab,

Missouri University of Science and Technology, Rolla, MO, USA

Preface

The requirements of higher bandwidth and lower power consumption of electronic systems render the integration of circuits and electronic packages more and more complex. In particular, the introduction of three-dimensional (3D) structures based on through-silicon via (TSV) technology provides a potential solution to reduce the size and to increase the performance of these systems. As a consequence, the electromagnetic compatibility (EMC) between circuits, signal integrity (SI), and power integrity (PI) in electronic integration are of vital importance. For this reason, the electronic circuits and packaging systems must be designed by taking into account the trade-offs between cost and performance. This requires ever more accurate modeling techniques and powerful simulation tools to achieve these goals. Incredible progress in electromagnetic field modeling has been achieved in the world. My research group has invested considerable efforts to develop novel simulation techniques over the last decade. Nevertheless, the present modeling techniques may be still far from perfect; for example, the modeling of multiphysics relevant to 3D integration is still far behind the requirements of the available technology.

This book presents the material that results from many years of our collective research work in the fields of modeling and simulation of SI, PI, and EMC in electronic package integration and multilayered printed circuit boards. It represents the state-of-the-art in electronic package integration and printed circuit board simulation and modeling technologies. I hope this book can serve as a good basis for further progress in this field in both academic research and industrial applications. The book consists of six chapters: Chapter 1 is written by Er-Ping Li, Chapter 2 by Enxiao Liu and Er-Ping Li, Chapter 3 by Zaw-Zaw Oo and Er-Ping Li, Chapter 4 by Xingchang Wei and Er-Ping Li, Chapter 5 by Yaojiang Zhang, and Chapter 6 by En-Xiao Liu.

Chapter 1 provides a review of progress in modeling and simulation of SI, PI, and EMC scenarios; Chapter 2 focuses on the macromodeling technique used in the electrical and electromagnetic modeling and simulation of complex interconnects in 3D integrated systems; Chapter 3 presents the semianalytical scattering matrix method (SMM) based on the N-body scattering theory for modeling of 3D electronic package and multilayered printed circuit boards with multiple vias. In Chapter 4, 2D and 3D integral equation methods are employed for the analysis of power distribution networks in 3D package integration. Chapter 5 describes the physics-based algorithm for extracting the equivalent circuit of a complex power distribution network in 3D integrated systems and printed circuit boards; Chapter 6 presents an equivalent-circuit model of through-silicon vias (TSV) and addresses the metal-oxide-semiconductor (MOS) capacitance effects of TSVs.

I gratefully acknowledge the technical reviewers of this book, Dr. Albert Ruehli, Emeritus of the IBM Watson Research Center, Yorktown, New York, USA; Prof. Wolfgang Hoefer, A*STAR, Singapore, and Prof. Zhongxiang Shen, Nanyang Technological University, Singapore, who donated their time and effort to review the manuscript. Also acknowledged are the contributors of the book, Dr. Xingchang Wei, Dr. Enxiao Liu, Dr. Zaw Zaw OO, and Dr. Yaojiang Zhang, who did the really hard work. I also wish to express my gratitude to Mary Hatcher at Wiley/IEEE Press for her great help in keeping us on schedule. Finally, I am grateful to my wife and the contributors’ wives, for without their continuing support and understanding, this book would have never been published.

I hope that this book will serve as a valuable reference for engineers, researchers, and postgraduate students in electrical modeling and design of electronic packaging, 3D electronic integration, integrated circuits, and printed circuit boards. Even though much work has been accomplished in this field, I anticipate that many more exciting challenges will arise in this area, particularly in 3D integrated circuits and systems.

ER-PING LI

West Lake, Hangzhou, China

CHAPTER 1

Introduction

1.1 INTRODUCTION OF ELECTRONIC PACKAGE INTEGRATION

The rapid growth and convergence of digital computers and wireless communication have been driving semiconductor technology to continue its evolution following Moore’s law in today’s nanometer regime. Future electronic systems require higher bandwidth with lower power consumption to handle the massive amount of data, especially for large memory systems, high-definition displays, and high-performance microprocessors. Electronic packaging is one of the key technologies to realize a wider bus architecture with high bandwidth operating at higher frequencies. Various packages have been developed toward a higher density structure. In particular, a three-dimensional (3D) integration based on through-silicon via (TSV) [1] arrays technology provides a potential solution to reduce the size and to increase the performance of the systems. Furthermore, nano-interconnects to replace the Cu-based interconnects provides a promising solution for long-term application.

There is a great challenge for further increasing of the signal speed in electronic systems due to the serious electromagnetic compatibility (EMC) problem. Figure 1.1 plots the technology trends versus actuals and survey, and Figure 1.2 shows the trends of microprocessors predicted by the International Technology Roadmap for Semiconductors (ITRS) [2, 3]. From these figures one can see that

Interconnect pitch will continue to decrease to 11.3 nanometer, while the on-chip clock frequency will be increased to 14.3 GHz by 2022. Due to the reduction of the feature size and pitch, more and more circuits are integrated into one electronic package, such as the system in package (SIP) and the 3D integration. This results in a complex and high-density environment inside the electronic systems. At the same time, with the ever-increasing clock frequency (also its high-frequency harmonics), the physical size of the small electronic package becomes electrically large, and so the electromagnetic wave propagation inside such a small structure must be considered.Until 2011, the voltage supply of the microprocessor is continually reduced with an increased power density. The electromagnetic noise will be pronounced due to the increased power density, which then makes the decreased voltage supply unstable. To design a high-speed and stable electronic system, we need better understand the electromagnetic interactions and the EMC issues inside the electronic package.

Figure 1.1 2008 ITRS update—technology trends versus actuals and survey [2].

Figure 1.2 The trends of microprocessor predicted by the International Technology Roadmap for Semiconductors (ITRS).

The EMC researches related to the high-speed circuit systems have a long history, which can be classified into different levels according to the size of the interested objects, which includes the system level, printed circuit board (PCB) level, electronic package level, and component level. The increasing clock frequency makes the size of tiny structures on the chip be comparable with the wavelength of interest. The fluctuation of electromagnetic wave cannot be ignored any more. Therefore, we must accurately model the electromagnetic wave behavior for all scales of the high-speed circuit systems. In the near future, the nanoscale integrated circuits (ICs) will be characterized by using the electric and magnetic fields instead of the conventional voltage and current. EM in micro-E is becoming a hot topic in both academic community and industrial applications.

The EMC analysis for high-speed electronics includes lots of issues, such as the ground bounce, cross talk, conducted emission, radiated emission, conducted immunity, and radiated immunity. The interaction between on-board capacitance and on-chip capacitance causes an antiresonance which induces a peak in the total power distribution network (PDN) impedance as shown in Figure 1.3. Figure 1.4 shows a typical multilayered advanced electronic package which consists of two main electrically functional systems: the PDN and the signal distribution network (SDN). The passive structures are composed of three main categories: (1) traces or transmission lines, typically microstrip lines or striplines, (2) vias used as vertical interconnections, and (3) conductor plates serving as power or ground planes. Because of the complexity of an advanced package, it is difficult to model the entire SDN or PDN simultaneously. Yet, we need to consider the impact of the PDN on the SDN in order to characterize the SDN more accurately. Many researchers have proposed various approaches to study the electrical properties of the above passive structures [4–44].

Figure 1.3 Example of antiresonances in total PDN impedances for various on-chip capacitance values [3].

Figure 1.4 A schematic diagram of a multilayered electronic package [37].

A typical EMC problem residing in this PDN of the electronic package is illustrated in Figure 1.5. In Figure 1.5, the power and ground planes are used to supply DC power for the circuits integrated in the electronic package. The signal traces are often laid out in different layers of power-ground planes. Their return currents flow on the power-ground planes just below them. When the traces pass through different layers, their return currents also exchange from one plane to another plane, as shown in Figure 1.5. Accordingly, a vertical displacement current is induced between different planes for the continuity of the return currents. This displacement current will excite electromagnetic field noise, which then propagates inside the power-ground planes and couples to other signal traces passing through the same layer. At the same time, this noise also leaks to the surrounding area of the electronic package through the periphery and gaps of the power-ground planes. These interferences will be further amplified if the noise’s spectrum covers any inherent resonant frequency of the cavity-like power-ground planes.

Figure 1.5 Noise coupling inside and emission from the power-ground planes.

To achieve first-pass design success, we must employ an advanced modeling and simulation technique to analyze the electrical performance of the 3D electronic packages, PCB, and chips at the system level. However, both industry and academia communities face the great challenges in developing the electrical design and simulation tools due to the multiscale nature of the problem, the strong local and global electromagnetic coupling, and the complexity of 3D integration systems. ITRS has summarized the state of the art of current semiconductor industry development, where the major challenges for simulation and modeling are listed as [2] mixed-signal co-design and simulation environment, rapid turnaround modeling and simulation, electrical (power disturbs, electromagnetic interference (EMI), signal and power integrity associated with higher frequency/current and lower voltage switching), system-level co-design, electronic design automation (EDA) for “native” area array to meet the roadmap projections, and models for reliability prediction. Therefore, advanced modeling techniques, which stand up to the challenges imposed by the complexity of nanoscale silicon chips and their interconnections including 3D ICs, 3D packaging, and PCB [45–47], are in great demand.

1.2 REVIEW OF MODELING TECHNOLOGIES

Modeling of transmission lines has a long history and is well documented in many textbooks [4]. So in the following, we will mainly review the modeling of vias and power-ground planes for electronic packaging and PCBs. Such modeling methods can be roughly classified into three categories: (1) lumped circuit approaches, (2) full-wave approaches, and (3) hybrid circuit coupled full-wave approaches.

For its simplification and ease of understanding, at the beginning of the research, lumped circuit approaches have been used for the electrical modeling of electronic packages. Such examples are shown in Figures 1.6 and 1.7. Empirical and analytical formulae for via capacitance and inductance can be easily found in many handbooks. Quasi-static numerical methods have also been introduced to calculate the lumped circuit values in T or PI types of via models [5, 6]. These methods allow rapid computation, but often suffer from accuracy problems. The authors in Reference 7 proposed a model of a magnetic-frill array and utilized microwave network theory for analysis of vias in multilayered packages. But it is a single via model which is difficult to be generalized to multiple vias. The equivalent circuits of much complex via array can be extracted by using de-embedding method [8]. Distributed circuit approaches have also been widely used for package modeling, such as the partial element equivalent circuit (PEEC) method [9, 10].

Figure 1.6 A typical transmission line model on a printed circuit board [3]. PRBS: pseudo-random binary sequence. ODT: on die termination.

Figure 1.7 Power distribution network [3].

Second, full-wave methods both in the time and frequency domains have been employed to study the packaging problems. The commonly used full-wave commercial simulators include Ansoft HFSS and CST Microwave Studio, which are based on finite element method (FEM) [11, 12] and finite integral technique, respectively. Other full-wave algorithm includes the finite-difference time-domain (FDTD) method [13] and the transmission line matrix method (TLM) [14]. Recently, the integral equation-based full-wave method begins to attract more attention and had been employed. The advantage of the integral equation method is that it can use the suitable Green’s function to present the effect of the complex environment, so that the unknowns are only placed on discontinuities inside the PDN. This can give an efficient simulation. According to the different Green’s functions used, the integral equation methods can be classified into (a) two-dimensional (2D) integral equations, including 2D mode method [15] and image method for rectangular power and ground planes, and 2D transverse magnetic (TM) integral equation for arbitrarily shaped power and ground planes [16–18]; (b) 3D integral equations, including 3D cavity mode method for rectangular power and ground planes [19] and parallel plate mode method for arbitrarily shaped power and ground planes [20, 21]. For most real applications the parallel plates have regular shapes, such as rectangles, circles, or triangles, a closed form of the Green’s functions can be formulated which results in an impedance formula in terms of the summation of infinite number of resonant modes [22–24]. This 2D integral equation method is sometimes called the cavity resonator method. Segmentation techniques may be applied to extend the cavity resonator method for parallel plates with irregular shapes.

These full-wave methods are versatile and able to solve a wide range of problems, but at the expense of large memory usage and long CPU time, especially for those 3D full-wave methods. Although the overall size of the electronic packages is small enough to apply these full-wave methods, the high aspect ratio of the power and ground planes and the tiny structures, such as the signal traces and narrow slots, result in a huge number of meshing. This makes these full-wave methods very expensive in terms of computing time and memory requirement.

Third, to avoid the computational cost of these full-wave methods and the geometrical limitations of the analytical methods, a more efficient approach is to combine both methods together, so that we can benefit from both analytic and numerical techniques. The coupled circuit-field approaches are also widely used to model the electronic packages in order to leverage the advantages of both circuit and full-wave approaches. An important approach under this category is rooted in the theory of modal decomposition and the salient features of electronic packages. The transmission lines and power-ground planes in an electronic package convey different modes, that is, transmission line modes and parallel plate modes. Modal decomposition can be used to decouple these two modes, which are then solved independently. These two modes are finally recombined to reflect the original problem. The coupling between the transmission line mode and the parallel-plate mode often occurs due to the vias. The current flowing in the via excites the parallel-plate mode field, while the transmission line experiences the loading effect of the power-ground plane in the presence of the via.

The complete modal decomposition and recombination approach has been demonstrated by several researchers. Current or voltage controlled sources are used to link these two modes. A general modal recombination approach was presented in Reference 26 for coupled striplines and nonideal power-ground planes, while the parallel-plate mode associated with the power-ground planes has been studied by many researchers. 2D full-wave methods have been extensively employed in the literature to model the power-ground planes. The 2D integral equation method is also called the contour integral method and has been used in Reference 16 to study general parallel-plate structures with arbitrary shapes. Another 2D approach, called the 2D FDTD, has also been used to model parallel plates [27]. Discretization of the metal plates by the finite-difference method was interpreted as a 2D distributed LC circuit, and a rigorous derivation is given in Reference [27]. The 2D distributed RLCG (resistance, inductance, capacitance, and conductance) circuit network, which is widely used in the literature to represent the power and ground planes, can be considered as an extension of the LC network derived from the finite-difference method. Instead of using Simulation Program with Integrated Circuit Emphasis (SPICE)-like solvers to simulate the large equivalent circuit network of power-ground planes, the latency insertion method is proposed in Reference 28 to perform fast transient simulation of large RLC networks. Moreover, a transmission matrix method reported in Reference 29 divides the 2D distributed RLCG circuit network into many interdependent blocks, and each block is formulated as a transmission (ABCD) matrix. Cascading those transmission matrices produces a fast way to obtain the desired impedance of the power-ground plane. A multilayered finite-difference method (MFDM) was recently proposed in Reference 26. The 2D finite element method (2D FEM) is also used to simulate power-ground planes [30] and had been integrated into the commercial software Ansoft SIWave. In addition, the radial transmission line theory has been applied to derive an admittance matrix to account for the effect of the parallel plates [31]. However, image theory [32] is needed to model the reflection from the edges of finite-sized substrates. Image theory is elegant for modeling the boundary with a regular shape but is cumbersome for modeling arbitrary shapes of the edges of PCBs or packages. In the model decomposition and recombination approach, a single via can be represented by a PI type of equivalent circuit. The capacitance and inductance in the PI circuit are usually computed by analytical formulae or quasi-static solvers. Recently, an elegant analytical formula was derived for the via barrel-plate capacitance [6].

1.3 ORGANIZATION OF THE BOOK

This book is organized in six chapters. Chapter 1 provides an overview of the state-of-art of electrical modeling and simulation techniques for electrical packaging systems. Chapter 2 focuses on the macromodeling technique widely used in the electrical and electromagnetic modeling and simulation of complex interconnects in 3D integrated systems. Macromodels are generated by employing the vector fitting (VF) method to perform rational-function approximation of scattering or admittance network parameters of high-speed complex interconnects and passive circuits. Subsequently, the macromodel can be synthesized as an equivalent circuit, which is compatible with the SPICE circuit simulator and can be combined with other external linear or nonlinear circuits to perform signal and power integrity analysis or other electrical performance analysis of electronic systems. The stability, causality, and passivity assessment and enforcement of the macromodel are also discussed in detail. Finally, numerical examples of macromodeling are presented and discussed.

In Chapter 3, the semianalytical scattering matrix method (SMM) based on the N-body scattering theory is presented for modeling of 3D electronic package and multilayered PCBs with multiple vias. Using the modal expansion of fields in a parallel-plate waveguide, the formula derivation of the SMM is presented in detail. In the conventional SMM, the power-ground planes are assumed to be infinitely large so they cannot capture the resonant behavior of the real-world packages. In particular, the SMM method has been extended to solve the finite domain of power-ground planes in coupling with a novel boundary modeling method proposed by the author’s group. This method has demonstrated its unique features which is capable to efficiently handle the complex real-world 3D package integration and PCB structures.

In Chapter 4, 2D and 3D integral equation methods are employed for the analysis of PDN in 3D package integration. The 2D integral equation method provides a comprehensive way for one to quickly extract the equivalent circuits of the PDN, and then substitute them into a SPICE-like simulator to perform the signal and power integrity analysis. The 3D integral equation method provides a more accurate solution for both the emission and susceptibility issues of the PDN. Both of the 2D and 3D integral equation methods are optimized by making a full use of the structural features of the PDN.

Chapter 5 is based on the physical-based algorithm to extract the equivalent circuit of the complex PDN in 3D integrated systems and PCBs. An intrinsic via circuit model is first derived through rigorous electromagnetic analysis for an irregular plate pair with multiple vias in a PCB. The derivation of the intrinsic via circuit model naturally leads to a new impedance definition of plate pair or power-bus, which is expressed in terms of cylindrical waves. The new plate pair impedance has clear physical meaning and makes possible signal/power integrity co-simulations. Numerical and measurement examples have indicated that while the new impedance gives almost the same results to the conventional one in a plate pair with few vias, it can correctly predict the resonant frequency shift in the case of a plate pair with a large amount of vias.

Chapter 6 presents a compact wideband equivalent-circuit model for electrical modeling of TSVs and addresses the metal-oxide-semiconductor (MOS) capacitance effects of TSVs.

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CHAPTER 2

Macromodeling of Complex Interconnects in 3D Integration

2.1 INTRODUCTION

Complex passive interconnects widely used in printed circuit boards (PCBs), electronic packages, and other electronic systems, are best characterized in the frequency domain. The frequency-dependent characteristics of interconnects require descriptions by either scattering (S) parameters, admittance (Y) parameters, or impedance (Z) parameters. Compared to the admittance parameters, the scattering parameters are well defined to represent networks consisting of high-speed interconnects because they are bounded quantities. Thus, the scattering parameters are primarily used in this chapter. Although the dispersive nature of interconnects is amicable to a frequency-domain representation, nonlinear circuits coexisting with the interconnect structures in electronic systems require time-domain description. System-level simulation for signal and power integrity analysis and electrical performance verification must address the problem of mixed time-frequency domain simulation required by an electronic system with both nonlinear circuits and linear passive interconnects.

Many approaches have been proposed in the literatures to address the mixed frequency-time domain problem. A straightforward approach to solve this problem was to employ the inverse fast Fourier transform (IFFT) and convolution method [1]. However, this approach suffers from excessive computational cost in the convolution process. Another approach to solve this mixed domain problem was the complex frequency hopping (CFH) method by moment matching [2, 3]. The difficulty faced by this approach is that for every moment, a corresponding derivative of each parameter must be computed using numerical integration across the entire time domain. This process has to be repeated on multiple frequency expansion points, which can be cumbersome for high-order approximation, or a network with a large number of ports.

An efficient macromodeling approach based on sampled frequency-domain data has been discussed in References 4–10. The macromodeling approach uses direct rational function approximation instead of moment matching to tackle the mixed domain problem. The macromodel obtained by rational function approximation can be used in conjunction with recursive convolution [11] to simulate interconnects together with nonlinear devices efficiently. Alternatively, the resultant macromodel can be converted to equivalent lumped circuits, which can be incorporated into the industrial standard SPICE (Simulation Program with Integrated Circuit Emphasis) circuit simulator to perform electrical analysis [12].

Many researchers have applied different methods to performing the rational function approximation. A section-by-section approximation approach was proposed in Reference 4, which partitioned the frequency band of the data into small sections to avoid numerically ill-conditioning problems. The drawback of this approach is that the final model has an artificially large number of poles accumulated from the approximation of each section. The matrix equations in Reference 6 introduced unnecessary ill conditioning to the approximation by using ω2 terms in the polynomials at the numerator and denominator. An improved approach was proposed in Reference 5 to compute pole-zero pairs recursively. But this method is only valid for real poles, which restricts its application to resistance-inductance (RL) and resistance-capacitance (RC) circuits.

A robust method for rational function approximation is the vector fitting (VF) method developed by Gustavsen and Semlyen in Reference 13. The VF method has advantages over other fitting methodologies [14]. Most conventional fitting methods rely on nonlinear optimization algorithms that are complex and may converge to a local minimum. Conversely, the VF method relies on the solution of two linear least-squares problems and thus obtains the optimal solution rather directly. At the same time, the VF method does not suffer much from the numerical stability problem, even when the bandwidth of interest is wide. Furthermore, one single run of the VF method can achieve the rational function approximation of all the elements in a transfer function matrix with a common set of poles. Therefore, the VF method has been widely adopted to generate stable macromodels.

This chapter focuses on the macromodeling techniques which are widely used in the electrical and electromagnetic modeling and simulation of complex interconnects. Macromodels are generated by employing the VF method to perform rational-function approximation of scattering or admittance network parameters of high-speed complex interconnects and passive circuits. Subsequently, the macromodel can be synthesized as an equivalent circuit, which is compatible with the SPICE circuit simulator and can be combined with other external linear or nonlinear circuits to perform signal integrity and power integrity analysis or other electrical performance analysis of electronic systems. The stability, causality, and passivity assessment and enforcement of the macromodel are also discussed in detail. Finally, numerical examples of macromodeling are presented and discussed.

2.1.1 Scope of Macromodeling