LOSSY TRANSMISSION LINE MODELING AND SIMULATION
USING SPECIAL FUNCTIONS
by
Bing Zhong
______________________
A Dissertation Submitted to the Faculty of the
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
In Partial Fulfillment of the Requirements
For the Degree of
DOCTOR OF PHILOSOPHY
In the Graduate College
THE UNIVERSITY OF ARIZONA
2006
2
THE UNIVERSITY OF ARIZONA
GRADUATE COLLEGE
As members of the Dissertation Committee, we certify that we have read the dissertation
prepared by Bing Zhong
entitled Lossy Transmission Line Modeling and Simulation Using Special Functions
and recommend that it be accepted as fulfilling the dissertation requirement for the
Degree Doctor of Philosophy
_______________________________________________________________________Date:
05/02/2006
Steven L. Dvorak, Ph.D.
_______________________________________________________________________Date:
05/02/2006
Kathleen Melde, Ph.D.
_______________________________________________________________________Date:
05/02/2006
Olgierd A. Palusinski, Ph.D.
Final approval and acceptance of this dissertation is contingent upon the candidate’s
submission of the final copies of the dissertation to the Graduate College.
I hereby certify that I have read this dissertation prepared under my direction and
recommend that it be accepted as fulfilling the dissertation requirement.
________________________________________________ Date: 05/02/2006
Dissertation Director: Steven L. Dvorak, Ph.D.
3
STATEMENT BY AUTHOR
This dissertation has been submitted in partial fulfillment of requirements for an
advanced degree at The University of Arizona and is deposited in the University Library
to be made available to borrowers under rules of the Library.
Brief quotations from this dissertation are allowable without special permission,
provided that accurate acknowledgment of source is made. Requests for permission for
extended quotation from or reproduction of this manuscript in whole or in part may be
granted by the head of the major department or the Dean of the Graduate College when in
his or her judgment the proposed use of the material is in the interests of scholarship. In
all other instances, however, permission must be obtained from the author.
Bing Zhong
4
ACKNOWLEDGEMENTS
I would like to thank my advisors, Dr. Steven Dvorak and Dr. John Prince, for
their guidance and support in my work.
I would like also to thank my committee members, Dr. Kathleen Melde, and Dr.
Olgierd A. Palusinski for reviewing my dissertation.
I am very grateful to Dr. Tingdong Zhou, Dr. Zhaohui Zhu, and Dr. Mehdi M.
Mechaik for their previous work on this research.
I am indebted to the Semiconductor Research Corporation (SRC) who provided
the funding for this project. I would like to thank all members of Center for Electronic
Packaging Research. They are Betsey Lyons, Xing Wang, Dr. Yi Cao, Dawei Fu, Zhen
Zhou, Guang Chen, and Lin Zhu. Best wishes to Dawei Fu on recovery!
Finally, I appreciate support from my wife and my daughter.
5
To my family
&
In memory of Dr. John L. Prince
6
TABLE OF CONTENTS
LIST OF FIGURES .......................................................................................................... 9
LIST OF TABLES .......................................................................................................... 11
ABSTRACT ................................................................................................................... 12
CHAPTER 1 INTRODUCTION................................................................................... 14
CHAPTER 2 BACKGROUND...................................................................................... 24
2.1 Introduction ........................................................................................................... 24
2.2 TEM Assumption .................................................................................................. 24
2.3 Interconnect Models.............................................................................................. 25
2.3.1 Lumped Element Model ................................................................................ 25
2.3.2 Distributed Element Model ........................................................................... 26
2.4 Telegrapher’s Equation ........................................................................................ 27
2.5 Interconnect Properties Characterized by Distributed Elements .................... 34
2.5.1 Frequency Dependent Resistance and Inductance Parameters................. 35
2.5.2 Frequency Dependent Capacitance and Conductance Parameters .......... 39
2.5.3 Examples of The Frequency Dependent Line Parameters ......................... 40
2.6 Propagation functions for typical lines ............................................................... 44
2.7 The Internal Relationships Between the Distributed Elements ........................ 45
CHAPTER 3 GENERAL ALGORITHMS FOR LOSSY INTERCONNECT
MODELING AND SIMULATION ............................................................................... 47
3.1 General Transmission Line Macromodeling Techniques ................................. 48
3.1.1 Direct Discretization of Transmission Lines................................................ 48
3.1.2 Convolution of the Impulse Response .......................................................... 50
3.1.3 Recursive Convolution Method .................................................................... 55
3.1.3.1 Pade Approximations of the Characteristic Admittance and Propagation
Function................................................................................................................. 56
3.1.3.2 The Recursive Convolution Scheme ......................................................... 58
3.1.3.3 Properties of Recursive Convolution Methods ......................................... 60
3.1.4 Method of Characteristics ............................................................................. 61
3.1.4.1 Brainin’s Method and Chang’s Method .................................................... 62
3.1.4.2 W-Element ................................................................................................ 64
3.1.5 Spectral Methods............................................................................................ 66
3.1.6 Least-Square Approximation........................................................................ 67
3.1.6.1 Standard Least-Square Algorithms ........................................................... 68
7
TABLE OF CONTENTS - Continued
3.1.6.2 The Vector Fitting Algorithm ................................................................... 71
3.2 Model Order Reduction Algorithms Based on Moment Matching .................. 73
3.2.1 Single Moment Matching .............................................................................. 76
3.2.1.1 Direct Single Moment Matching............................................................... 76
3.2.1.2 Matrix Rational Approximation................................................................ 77
3.2.2 Multiple Moment Matching .......................................................................... 78
CHAPTER 4 SIMULATION OF LOSSY TRANSMISSION LINES WITH
FREQUENCY INDEPENDENT LINE PARAMETERS USING SPECIAL
FUNCTIONS……. .......................................................................................................... 81
4.1 Frequency-Domain Expressions for the Responses of Lossy Transmission
Lines with FILP........................................................................................................... 83
4.2 Time-Domain Expressions for the Far-End Voltage Responses of Lossy
Transmission Lines with FILP................................................................................... 84
4.2.1 The Unit-Step Voltage Response................................................................... 84
4.2.2 The Exponentially Decaying Signal Voltage Response............................... 86
4.2.3 Ramp Signal Voltage Response..................................................................... 87
4.3 Solutions for the Three Key Integrals for the Voltage Responses .................... 88
4.4 Time-Domain Expressions for the Near-End Current Responses of Lossy
Transmission Lines with FILP................................................................................... 94
4.4.1 Unit-Step Signal Current Response.............................................................. 94
4.4.2 The Exponentially Decaying Signal Current Response .............................. 95
4.4.3 The Ramp Signal Current Response ............................................................ 96
4.4.4 Solutions for the Key Integrals for the Current Responses ....................... 96
4.5 Numerical Validations of the Voltage and Current Responses....................... 100
4.5.1 Definitions for the Source Signals............................................................... 101
4.5.2 Validation of the Voltage Responses .......................................................... 103
4.5.3 Validation of the Current Responses.......................................................... 109
CHAPTER 5 MODELING AND SIMULATION OF LOSSY TRANSMISSION
LINES WITH FREQUENCY DEPENDENT LINE PARAMETERS USING
DISPERSIVE HYBRID PHASE-POLE MACROMODELS ................................... 114
5.1 Analysis of the Properties of Lossy Transmission Lines with FDLP ............. 116
5.2 Development of a Frequency-Domain DHPPM for Frequency-Dependent
Lossy Interconnects................................................................................................... 122
5.3 Validation of the Frequency-Domain DHPPM ................................................ 126
5.4 Development of a Time-Domain DHPPM for Frequency-Dependent Lossy
Interconnects.............................................................................................................. 141
5.4.1 Expression for the Triangular Impulse Response..................................... 142
5.4.2 Expression for the Trapezoidal Response.................................................. 144
5.5 The Time-Domain DHPPM for Coupled Lines................................................ 145
8
TABLE OF CONTENTS - Continued
5.6 Application of the Closed-Form Results to the Simulation of Transmission
Lines with FDLP........................................................................................................ 149
CHAPTER 6 CONCLUSION AND FUTURE WORK ............................................ 161
APPENDIX A GLOSSARY OF TERMS.................................................................... 164
REFERENCES .............................................................................................................. 165
9
LIST OF FIGURES
Figure 1.1 Pin count and off-chip frequencies over the next decade ................................ 15
Figure 1.2 Diagram of a node-to-node bus ....................................................................... 17
Figure 1.3 Equivalent circuit of the node-to-node bus...................................................... 17
Figure 2.1 Some typical interconnection structures: a) microstrip, b) stripline, c) coplanar
line............................................................................................................................. 26
Figure 2.2 The lumped element model for the interconnection. ....................................... 26
Figure 2.3 The simplified distributed element model for the interconnection.................. 27
Figure 2.4 The field lines on a coaxial TEM transmission line ........................................ 28
Figure 2.5 The equivalent circuit for a short section of a short section of an
interconnection. ......................................................................................................... 31
Figure 2.6 The relationship between the frequency and the skin depth for a copper
conductor................................................................................................................... 38
Figure 2.7 A comparison of the per unit length resistance R, inductance L, conductance G,
and capacitance C for some common interconnect structures .................................. 43
Figure 2.8 A Comparison of absolute values of the propagation functions for three
different types of interconnect lines. ......................................................................... 44
Figure 3.1 The direct discretization of a single lossy transmission line ........................... 49
Figure 3.2 Method of Characteristics (MC) equivalent circuit. ........................................ 63
Figure 3.3. A simple interconnect model. ......................................................................... 75
Figure 3.4 The single Moment Matching Techniques (MMT) case ................................. 79
Figure 3.5 The multiple Moment Matching Techniques (MMT) case ............................. 80
Figure 4.1 The cross-section of a microstrip line with W=5µm, t=2µm, T=10µm, l=5cm,
and εr=4.4. ............................................................................................................... 101
Figure 4.2 Time-domain triangle input ........................................................................... 101
Figure 4.3 Time-domain trapezoidal input...................................................................... 102
Figure 4.4The SPICE simulation scheme for the transmission line with FILP .............. 106
Figure 4.5 Comparison of the unit-step responses of the FILP between HSPICE and the
closed-form ILHI results. ........................................................................................ 106
Figure 4.6 Comparison of the TIR of the FILP lossy line between HSPICE and the
closed-form ILHI results ......................................................................................... 107
Figure 4.7 Comparison of the exponential decaying response of the frequencyindependent lossy line between HSPICE and the closed-form ILHI results........... 107
Figure 4.8 Comparison of the exponential decaying sine response of the frequencyindependent lossy line between HSPICE and the closed-form ILHI results........... 108
Figure 4.9 An enlarged part of Figure 4.7 showing the numerical issues in the HSPICE
results ...................................................................................................................... 108
Figure 4.10 Comparison of the unit-step signal current response of the frequencyindependent lossy line between HSPICE and the closed-form ILHI results........... 110
Figure 4.11 Comparison of ramp signal current response of the frequency-independent
lossy line between HSPICE and the closed-form ILHI results ............................... 110
10
LIST OF FIGURES - Continued
Figure 4.12 Comparison of the exponential decaying signal current responses of the
frequency-independent lossy line between HSPICE and the closed-form ILHI results
for three cases.......................................................................................................... 112
Figure 5.1 The time-domain output waveforms obtained for a MCM line for a 10ps
triangle impulse input.............................................................................................. 119
Figure 5.2 The frequency-dependent behaviors of the (a) R and (b) L parameters for a
typical on-chip, MCM, and PCB lossy line. ........................................................... 128
Figure 5.3 The magnitudes and phases of in the propagation function, the residue series
data of HPPM and DHPPM for the on-chip lossy interconnect.............................. 131
Figure 5.4 A comparison between the propagation function for an on-chip interconnect
modeled by different macromodels in terms of (a) magnitude, (b) phase, and (c)
absolute error........................................................................................................... 137
Figure 5.5 A comparison between the propagation function for an on-chip interconnect
modeled by different macromodels in terms of (a) magnitude, and (b)absolute error
................................................................................................................................. 138
Figure 5.6 A comparison between the propagation function for an MCM interconnect
modeled by different macromodels in terms of (a) magnitude, (b) phase, and (c)
absolute error........................................................................................................... 141
Figure 5.7 The general DHPPM algorithm flow chart for the simulation of lossy
transmission line with FDLP................................................................................... 149
Figure 5.8 The time-domain output waveforms obtained from different models for a 10ps
trapezoidal inputs .................................................................................................... 154
Figure 5.9 The time-domain output waveforms obtained from different models for a
100ps trapezoidal input ........................................................................................... 155
Figure 5.10 The comparison of the DHPPM results, HSPICE results, and IFFT results for
a 10ps triangle impulse input .................................................................................. 156
Figure 5.11 The comparison of the DHPPM results, IFFT results, and HSPICE results for
a 10ps triangle impulse input .................................................................................. 157
Figure 5.12 The comparison of two HSPICE results, IFFT results, and DHPPM with 10
poles results ............................................................................................................. 158
Figure 5.13 The Far End Active (FEA) line voltage comparisons of the DHPPM results
and the IFFT results for a 20ps triangle impulse on a coupled line ........................ 159
Figure 5.14 The Far End Passive (FEP) line voltage comparison of the DHPPM results
and the IFFT results for a 20ps triangle impulse on a coupled line ........................ 160
11
LIST OF TABLES
Table 2.1 Experimental data of relative dielectric permittivity εr and loss tangent for FR-4.
................................................................................................................................... 40
Table 5.1 The RMS errors of different macromodels for on-chip and MCM cases ....... 134
Table 5.2 Frequency-dependent R and L parameters for a microstrip line calculated using
UAPDSE [30] where W=5µm, t=2µm, T=10µm, l=5cm, and εr=4.4..................... 153
Table 5.3 Frequency-dependent line parameters for the MCM coupled transmission lines
as listed in [29]. ....................................................................................................... 153
12
ABSTRACT
A new algorithm for modeling and simulation of lossy interconnect structures
modeled by transmission lines with Frequency Independent Line Parameters (FILP) or
Frequency Dependent Line Parameters (FDLP) is developed in this research. Since
frequency-dependent RLGC parameters must be employed to correctly model skin effects
and dielectric losses for high-performance interconnects, we first study the behaviors of
various lossy interconnects that are characterized by FILP and FDLP. Current general
macromodeling methods and Model Order Reduction (MOR) algorithms are discussed.
Next, some canonical integrals that are associated with transient responses of lossy
transmission lines with FILP are presented. By using contour integration techniques,
these integrals can be represented as closed-form expressions involving special functions,
i.e., Incomplete Lipshitz-Hankel Integrals (ILHIs) and Complementary Incomplete
Lipshitz-Hankel Integrals (CILHIs). Various input signals, such as ramp signals and the
exponentially decaying sine signals, are used to test the expressions involving ILHIs and
CILHIs. Excellent agreements are observed between the closed-form expressions
involving ILHIs and CILHIs and simulation results from commercial simulation tools.
We then developed a frequency-domain Dispersive Hybrid Phase-Pole Macromodel
(DHPPM) for lossy transmission lines with FDLP, which consists of a constant RLGC
propagation function multiplied by a residue series. The basic idea is to first extract the
dominant physical phenomenology by using a propagation function in the frequency
domain that is modeled by FILP. A rational function approximation is then used to
13
account for the remaining effects of FDLP lines. By using a partial fraction expansion
and analytically evaluating the required inverse Fourier transform integrals, the timedomain DHPPM can be decomposed as a sum of canonical transient responses for lines
with FILP for various excitations (e.g., trapezoidal and unit-step). These canonical
transient responses are then expressed analytically as closed-form expressions involving
ILHIs, CILHIs, and Bessel functions. The DHPPM simulator can simulate transient
results for various input waveforms on both single and coupled interconnect structures.
Comparisons between the DHPPM results and the results produced by commercial
simulation tools like HSPICE and a numerical Inverse Fast Fourier Transform (IFFT)
show that the DHPPM results are very accurate.
14
CHAPTER 1
INTRODUCTION
The rapid developments that are taking place in the modern electronic industry
has brought about an era where system level integration, such as System-On-Chip (SOC)
and System-On-Package (SOP) is possible. In such SOC and SOP systems, different
modules like RF modules, Analog to Digital Converters (ADC), Digital Signal
Processing (DSP) units, and Central Processing Units (CPU) are integrated into a single
chip or package. On the other hand, the advancements in design and manufacturing
capabilities for each module results in more challenges for researchers in this field.
According to the International Technology Roadmap for Semiconductor (ITRS) [1], the
off-chip frequencies for high performance semiconductor products over the next decade
should reach as high as 2 GHz, while the total number of pins will be tens of thousands
(Figure 1.1). What’s more, CPU frequencies can be much higher than this. For example,
the development of CPUs is guided by the empirical Moore’s Law, which states that the
performance of CPUs will double, while the feature size will shrink to another generation
every 18 months. Current popular CPU products have operating frequencies over 3GHz
and feature sizes that are less than 90nm.
The increases in the operating frequency have made the lengths of many
interconnects comparable to the wavelength of propagating signals. Likewise, the
decreases in the feature sizes have increased the coupling and crosstalk. Therefore,
complicated Electromagnetic Compatibility (EMC) and Electromagnetic Inference (EMI)
problems arise because of these trends.
15
Highperformance
pin count
(Pins)
Highperformance
offchip
frequency (Hz)
10000
9000
8000
7000
6000
5000
4000
3000
2000
1000
0
1998
year
2003
2008
2013
2018
Figure 1.1 Pin count and off-chip frequencies over the next decade
The rapid developments that have taken place are backed by the rapid evolutions
in Electronic Design Automation (EDA) tools. Electronic packaging design and signal
integrity analysis have become indispensable steps in the electronic system design cycle.
EDA tools that incorporate the capabilities of electronic packaging design and signal
integrity analysis have made chip package co-design a feasible solution to the EMC and
EMI problems. In such design environments, it is desirable for design engineers to be
able to evaluate the signal propagation quality of the system without physically
manufacturing the chip and performing measurement.
Such a design philosophy heavily relies on robust, accurate models of the
interconnect system, seamless incorporations of such models into current modeling and
simulation tools, and swift implementation of such modeling and simulation algorithms
in current EDA tools. Numerous algorithms have been implemented in interconnect
16
system modeling and simulation tools. According to [2,3], these methods can be placed in
two main categories. The first methods are based on full-wave analysis of interconnect
systems given the geometry, material properties, and operation frequency. Another
category is based on transmission line modeling of the interconnect system. The
transmission line models can be generated from either the full-wave analysis of the
interconnect system or the measurements [4] from a Vector Network Analyzer (VNA).
The full-wave analysis tools, e.g., Finite Element Method (FEM), FiniteDifference-Time-Domain (FDTD) method, Method of Moments (MoM), Boundary
Element Method (BEM), feature the meshing of the interconnect system and analysis of
the electromagnetic field using Maxwell Equations. The resulting models usually have
either a large number of unknowns or very complicated expressions that prohibited them
from being easy adopted into current state-of-the-art simulation tools. Although
numerous attempts have been reported for incorporating the full-wave analysis
algorithms into current simulation tools such as Simulation Program with Integrated
Circuit Emphasis (SPICE), e.g., Partial Element Equivalent Circuit (PEEC), the scale of
the problems that such tools can solve is still limited.
Another simplified model that is useful for the analysis of the interconnect
systems is the transmission line model. The transmission line model is based on the
Transverse Electromagnetic Mode (TEM) assumption. Under this assumption, the
interconnect system is modeled by transmission lines with per-unit-length RLGC
parameters, i.e., resistance, inductance, conductance, and capacitance. These RLGC
parameters serve as bridges between complicated electromagnetic fields and circuit
17
simulation tools, thereby enabling the simulation of interconnect systems in SPICE
environments. An example of such a model for a node-to-node bus is shown in [5]. The
equivalent circuit model is shown in Figure 1.3.
200mm PCB line
Socket
Strait
Socket
200mm PCB line
MCM single line
MCM single line
1mm short-line
1mm short-line
Driver
Load
Figure 1.2 Diagram of a node-to-node bus
R1
R2
L1
L2
R3
30
3.78
0.138
0.138
3.78
C2 LUMPED
C1
0.215
V1
PCB_line
GC_single _trace
40
LUMPED
GC_bottom_via
2.6
LUMPED
0.1
80
R7
LUMPED
C10
1.3
GC_top_via
2.6
C4
1.3
L4
0.0125
L3
0.9
0.9
R6
R5
1G
1G
PCB_line
R4
0.0125
200
LUMPED
C3
0.5
R9
L5
0.0125
L6
0.9
R11
1G
1G
R15
R16
140
0.0125
200
PCB_line
LUMPED
2.6
GC_top_via
LUMPED
C5
0.1
C9
Vdd
140
0.9
R10
R12
0.5
C7
1.5
C8
1.5
R14
L8
L7
3.78
0.138
0.138
R13 GC_bottom_via
3.78
2.6
GC_single _trace
LUMPED
C6
0.215
Figure 1.3 Equivalent circuit of the node-to-node bus
40
LUMPED
18
The net consists of a driver, a receiver, 2 chip to multi-chip module (MCM) lines,
2 MCM lines, 2 printed circuit board (PCB) lines, 2 sockets, and a strait between these
two sockets. The chip to MCM lines are modeled with lumped elements, and each MCM
line is modeled with lossy transmission lines with FILP and are named as GC_top_via,
GC_single_trace, and GC_bottom_via. Each printed circuit board line is modeled with a
lossy transmission line with FDLP named as PCB_line. The socket is modeled with
lumped elements, and the strait is modeled with lumped elements and a 8cm PCB_line.
It can be seen from Figure 1.2 and Figure 1.3 that different types of models are
used to model modern interconnect systems. First, the on-chip interconnect systems are
often modeled by lumped elements as shown in Figure 1.3. For the off-chip interconnects
that are modeled by transmission lines, the type of transmission line model that is used
depends on the dimensions of the interconnect. The first stage is a lossless transmission
line model consisting of per-unit-length inductance and capacitance parameters. As the
cross sectional dimensions decrease, the conductive losses on the interconnect can no
longer be ignored and a resistance R has to be incorporated in the transmission line
model. Since the resistance R corresponds to an electrical field in the direction of
propagation, the quasi-TEM assumption, instead of the TEM mode, has to be taken.
With ever-increasing operating frequencies and ever-shrinking interconnect sizes,
high-frequency effects, such as skin, edge, and proximity effects, and dielectric loss
become more and more important in transient simulations of interconnect systems. Many
signal integrity problems or phenomena such as crosstalk, Simultaneous Switching Noise
(SSN), line impedance mismatch, timing analysis, and Inter Symbol Interference (ISI),
19
are related to the accurate RLGC modeling of lossy transmission lines. FrequencyDependent Line Parameters (FDLP) have to be employed to properly model these effects.
However, these FDLP model lossy interconnect in the frequency domain and there are no
closed-form time-domain counter parts for these FDLP models. Since active devices such
as Field Effect Transistors (FET) in electronic systems are typically modeled in the timedomain due to their non-linear behavior, it is hard to incorporate lossy transmission lines
into a time-domain simulator.
Two different approaches are taken to better model large interconnect network
represented by lossy transmission lines. The Model Order Reduction (MOR) algorithms
are employed to reduce the large model orders of large networks of interconnect
structures into small order models by extracting the major propagation effects and
excluding the minor propagation effects. Such algorithms include, but are not limited to
the Krylov-subspace formulation, Pade via Lanczos [6] [7], Passive Reduced-Order
Interconnect
Macromodeling
Algorithm
(PRIMA)
[8],
Integrated
Congruent
Transformation [9], Split Congruent Transform[10], and Arnoldi [11]. The second types
of approach, the macromodeling techniques, are widely applied to solve the problem of
directly modeling lossy transmission lines with FDLP by using simplified models such as
sums of pole and residue pairs. Examples of such techniques include the direct
discretization of transmission lines, numerical convolution of transfer functions [12,13],
Method of Characteristics (MoC) [14-18], basis function macromodels [19,20], compact
finite-difference-based approximations[21], and integrated congruence transform[9].
20
However, classical macromodels, e.g., least-square approximation or Pade
approximation methods, still suffer from large numbers of poles and residues, i.e., high
model orders. The Method of Characteristics (MoC) suggests a pre-extraction of the
propagation delay term from the propagation function before the numerical
approximations such as least-square approximation are employed. It has been shown that
the pre-extraction of the propagation delay will significantly reduce the macromodel
order for electrically long lines [17,22]. Furthermore, the pre-extraction of the
propagation delay also guarantees the causality of the macromodel, which means the
signal response won’t appear on the far-end of the transmission line before the time-offlight in MoC-based methods. The MoC method can also lead to equivalent circuits that
are easy to incorporate in SPICE environments. Due to these features, MoC methods are
successfully employed in commercial simulation tools such as HSPICE.
When the system operation frequency increases up to several GHz, the frequencydependent effects of the line parameters cause severe changes in the propagation function
and leads to large numbers of poles and residues in MoC-based macromodels.
Furthermore, recent studies indicate close inter-relationships between the FDLP. It turns
out that the frequency-dependent R and L, as well as G and C, are related through a
Hilbert Transform. A macromodel that doesn’t take these inter-dependent effects into
account may lead to non-causal simulation results.
In order to overcome the difficulties associated with modern transmission line
modeling and simulation, a new modeling and simulation algorithm is developed in this
dissertation. We first develop a new method for the simulation of lossy transmission lines
21
with FILP. We start with the frequency-domain Telegrapher’s Equation and solve
Telegrapher’s Equation in the frequency domain to obtain a closed-form expression for
the response of lossy transmission lines with FILP. Previous researchers have
successfully derived the impulse response and a special case of the unit-step response for
the lossy transmission lines with FILP. The time-domain response involves Bessel
functions of zeroth and first orders. In this research, we introduce integrals of Bessel
function called Incomplete Lipshitz-Hankel Integrals (ILHIs) and Complementary
Incomplete Lipshitz-Hankel Integrals (CILHIs). Therefore, various input response of the
lossy transmission line with FILP can be represented in closed-form expressions
involving ILHIs and CILHIs. Examples are given to show the unit-step response, the
ramp response, and the exponentially decaying signal responses can all be expressed as
close-form expressions involving ILHIs and CILHIs. These closed-form results are then
compared with commercial simulation tools such as the HSPICE W-element model for
lossy transmission lines with FILP and excellent agreement is observed between the two
methods. Furthermore, both the voltage and current responses are derived for the
simulation of lossy transmissions lines with FILP under excitation of various complex
source waveforms such as triangle impulse and exponentially decaying sine signals.
Next, a Dispersive Hybrid Phase Pole Macromodel (DHPPM) is developed for the
modeling and simulation of lossy transmission lines with FDLP. Given tabular data for
the frequency-dependent RLGC parameters, we first take the resistance and conductance
at DC and the inductance and capacitance at the highest frequency point to form a
consistent RLGC line model. Then, a propagation function with frequency-independent
22
RLGC parameters is extracted from the propagation function for the lossy transmission
line with FDLP and the remaining effects are modeled by a residue series that is
approximated by a robust Vector Fitting algorithm. It has been shown that the resistance
and conductance at DC and the inductance and capacitance at highest frequency point
form a self-consistent causal model, because the extraction of the inductance and
capacitance from the propagation function ensures that proper propagation delay is
realized during the transient simulations. What’s more, the inclusion of the resistance at
DC accounts for part of the dispersion on high-lossy line. Practical results show that the
DHPPM yields well-behaved residue series that can be easily approximated by rational
functions using far-fewer terms than for the cases of classical macromodels and MoCbased macromodels.
The DHPPM algorithms can be successfully applied to on-chip interconnects
modeled by lossy transmission lines with FDLP. However, for the off-chip interconnect
structures, the large cross sections of the interconnects and low loss of the lines lead to
DHPPM model orders that approach those of MoC-based macromodels. However, much
advantage is still observed when comparing the DHPPM with classical macromodel for
off-chip lines. In our DHPPM-based simulator, transient simulations for both single and
coupled lossy transmission lines with FDLP are realized.
The organization of this dissertation is as follows. In Chapter 2, we discuss the
background knowledge that is required for this research. In Chapter 3, a general
discussion of the macromodeling and MOR algorithms is given. In Chapter 4, we develop
the transient response formulas for the simulation of lossy transmission lines with FILP.
23
ILHIs and CILHIs are introduced to form the closed-form expressions for lossy
transmission lines with FILP for various input excitations, such as the unit-step signal,
the ramp signal, and the exponentially decaying signal. In Chapter 5, we develop the
DHPPM for the modeling and simulation of lossy transmission lines with FDLP.
Examples are then given to show the capabilities of the DHPPM. Finally, in Chapter 6,
conclusions are made about our algorithm and the future work is discussed.
24
CHAPTER 2
BACKGROUND
2.1 Introduction
In the last chapter, the significance of interconnect modeling in the design cycle
was outlined. In this chapter, we now discuss several types of interconnect models of
various complexities, i.e., lumped-element and distributed-element models. In the
distributed element category, we will address lossless-line models, lossy-line models with
Frequency-Independent Line Parameters (FILP), and lossy-line models with FrequencyDependent Line Parameters (FDLP). Then the relationship between the lumped element
models and the distributed element models is discussed. The properties of lossy line
models for real interconnect structures like on-chip lossy lines, Multiple Chip Module
(MCM) lines, and Printed Circuit Board (PCB) lines are then discussed. Finally the
internal relationships between the RLGC parameters for lossy interconnects are
addressed.
2.2 TEM Assumption
In this dissertation, TEM wave or quasi-TEM wave propagation is assumed for all
the interconnection models. The term TEM stands for Transverse Electro-Magnetic wave,
which means that the electric and magnetic fields are both perpendicular to the direction
of the wave propagation. Two requirements must be met to make TEM waves possible:
I) There must be two or more conductors forming the cross section of the line.
25
II) The material surrounding the conductors must be homogenous.
Signals on typical interconnects are not purely TEM waves because of nonhomogenous dielectrics and conductor losses. However, for frequencies below a few
GHz, the waves that propagate are nearly TEM in nature. So they are called quasi-TEM
waves and treated like TEM waves. Quasi-TEM wave propagation assumes that the
transverse field values are much larger than the longitudinal field values. In the following
chapter, all the propagating waves are assumed to be TEM.
2.3 Interconnect Models
2.3.1 Lumped Element Model
The term interconnection in the modern electronics industry refers to “the
conductive path required to achieve connection from a circuit element to the rest of the
circuit” [23]. Interconnections can be divided into several categories according to the
application, i.e., on-chip interconnections, off-chip interconnections, and board-level
interconnections. As stated in the definition, the interconnections primarily serve as
connections between the circuits and are often considered as perfect conductors that
reside in perfect dielectrics, i.e., no loss or distortion of signals should exist on the
interconnections. Some typical interconnection structures [24] in electronic systems are
shown in Figure 2.1: the microstrip, the stripline, and the coplanar line.
26
a)
b)
c)
Figure 2.1 Some typical interconnection structures: a) microstrip, b) stripline, c) coplanar
line.
Figure 2.2 The lumped element model for the interconnection.
Most interconnections are made from conductors like Al or Cu. They are buried in
or attached to materials like SiO2, Si, Duroid, Telflon, or FR-4. The finite conductivity of
the conductors, the self and mutual inductances of the conductors, the capacitance formed
between the conductors, and the dielectric loss will all cause signal distortion. At low
frequencies, or for short lines, this distortion can be modeled by a R-L-C network. The
lumped element model is then used to model the interconnections as shown in Figure 2.2.
2.3.2 Distributed Element Model
The selection of the lumped element model or the distributed element model is
decided by the signal propagation delay and the signal rise time [24]. The general rule of
thumb for the microstrip case is that the distributed element model is needed if the rise
27
time of the signal ( t r ) finishes before reflections from the far end of the line return back
to the source. Let’s define the time of flight as
Td = l / v0 ,
(2.1)
where l is the length of the interconnection, and v0 is the propagation speed of the signal
on the interconnection. The distributed element model should be applied if
tr < 2Td .
(2.2)
The simplified distributed model of the interconnection is made up of a series of unit
sections. Each unit section has an inductance and capacitance. The simplified distributed
element model is shown in Figure 2.3.
2.4 Telegrapher’s Equation
The detailed distributed model of the interconnection not only has L and C
parameters, but also includes R and G parameters. The definitions for these parameters
are as follows [25]:
R – series resistance per unit length.
L – series inductance per unit length.
Figure 2.3 The simplified distributed element model for the interconnection.
28
C – shunt capacitance per unit length.
G – shunt conductance per unit length.
In this model, the R parameter represents the conductive loss of the
interconnection, the L parameter stands for the inductance of the interconnection, the C
parameter represents the capacitance between the interconnection and the ground, and the
G parameter stands for the dielectric loss in the material surrounding the
interconnections. The physical definitions of the transmission line parameters are shown
by an example. Suppose a uniform transmission line of unit length with fields E and H
is shown in Figure 2.4, where the cross-sectional surface area of the line is S, the voltage
between the conductors is V0 e ± jβ z and the current on one of the conductors is I 0 e ± jβ z .
C2
C1
E
H
S
Figure 2.4 The field lines on a coaxial TEM transmission line
29
The time-average stored magnetic energy for this section of line is given as [25]
µ
Wm =
*
H ⋅ H ds.
∫
4
S
(2.3)
Since circuit theory states that Wm = L I 0 / 4 , we can identify the self-inductance per
2
unit length as
L=
µ
2
I0
∫
S
*
H ⋅ H ds,
(2.4)
where µ is the permeability of the material between the two conductors. Likewise, the
time-average stored electric energy per unit length can be found as
*
E ⋅ E ds.
ε
4 ∫S
We =
(2.5)
Since circuit theory states that We = C V0 / 4 , the capacitance per unit length can be
2
represented as
C=
ε
V0
2
∫
S
*
E ⋅ E ds,
(2.6)
where ε is the permittivity of the material between the two conductors. The power loss
per unit length due to the finite conductivity of the metallic conductors can be
approximated by
Pc =
Rs
2
∫
C1 + C2
*
H ⋅ H dl ,
(2.7)
30
where Rs = 1/ σδ s is the surface resistance of the conductors, and C1 + C2 is the
integration paths over the conductor boundaries. Then circuit theory gives Pc = R I 0 / 2 .
2
The series resistance R per unit length of the line is
Rs
R=
I0
2
∫
C1 + C2
*
H ⋅ H dl.
(2.8)
The time-averaged power dissipated per unit length in a lossy dielectric is
Pd =
ωε " *
∫
2
S
E ⋅ E ds,
(2.9)
where ε " is the imaginary part the complex permittivity as
ε = ε '− jε " = ε '(1 − j tan δ ).
(2.10)
Since circuit theory gives Pd = G V0 / 2 , the shunt conductance per unit length can be
2
written as
G=
ωε " *
V0
2
∫
S
E ⋅ E ds.
(2.11)
Once the static electric and magnetic fields are known, then (2.4), (2.6), (2.8), and
(2.11) can be used to find the desired per unit length RLGC parameters.
The equivalent circuit for an electrically short section of an interconnection is
shown in Figure 2.5.
31
i(z,t)
+
v(z,t)
_
z
∆z
i(z + ∆ z, t)
i(z,t)
R∆ z
+
+
L∆ z
v(z,t)
C∆ z
G∆ z
v(z + ∆ z, t)
_
_
∆z
Figure 2.5 The equivalent circuit for a short section of a short section of an
interconnection.
Kirchhoff’s voltage law can be applied to the equivalent circuit for the
interconnection in the time domain, i.e.,
v( z , t ) − v( z + ∆z , t ) = R∆z ii ( z , t ) + L∆z i
∂i ( z , t )
∂t
(2.12)
Likewise, Kirchhoff’s current law can also be applied to the equivalent circuit:
i ( z , t ) − i ( z + ∆z , t ) = G∆z iv( z , t ) + C ∆z i
∂v( z , t )
∂t
(2.13)
Dividing (2.12) and (2.13) by ∆z and taking the limit as ∆z → 0 will lead to the Partial
Differential Equations (PDE):
32
∂v( z , t )
∂i ( z , t )
= − Ri ( z , t ) − L
∂z
∂t
(2.14)
∂i ( z , t )
∂v( z , t )
= −Gv( z , t ) − C
∂z
∂t
(2.15)
These are the time-domain Telegrapher’s equations or the transmission line equations.
Applying a Fourier Transform on both sides of (2.14) and (2.15) will generate the
frequency-domain Telegrapher’s equations:
dV ( z )
= − RI ( z ) − jω LI ( z )
dz
(2.16)
dI ( z )
= −GV ( z ) − jωCV ( z )
dz
(2.17)
These two equations can be solved together to give Helmholtz Equations for V(z) and
I(z):
d 2V ( z )
− γ 2V ( z ) = 0 ,
dz 2
(2.18)
d 2 I ( z)
− γ 2 I ( z ) = 0,
2
dz
(2.19)
γ = α + j β = ( R + jω L)(G + jωC )
(2.20)
where
is the complex propagation constant, which is a function of frequency. For lossless
transmission lines, R=G=0, thus,
α =0,
(2.21)
33
γ = j β = jω LC .
(2.22)
If the transmission line is lossy, the attenuation factor is no longer zero. Instead, it can be
calculated by matching the real and imaginary parts of (2.20)
α=
( RG − ω 2 LC ) + ( RG − ω 2 LC )2 + ω 2 ( RC + GL)2
.
2
(2.23)
Also the phase constant is
β=
−( RG − ω 2 LC ) + ( RG − ω 2 LC ) 2 + ω 2 ( RC + GL) 2
.
2
(2.24)
A quick check of letting R=0 and G=0 will reduce (2.23) and (2.24) to (2.21) and (2.22).
These are the expressions for a single lossy line. For multiple coupled transmission lines,
the R, L, G, C parameters must be expressed as matrices. Therefore, the expression for
the attenuation constant and the phase constant will be more complex.
From (2.18) and (2.19), we can solve the travelling wave solutions as
V ( z ) = V0+ e −γ z + V0− eγ z ,
(2.25)
I ( z ) = I 0+ e−γ z + I 0− eγ z ,
(2.26)
where the e−γ z term represents the wave propagation in the +z direction, and the eγ z term
represents the wave propagation in the -z direction. By applying (2.25) into (2.16), we
can find the I-V relationship on a transmission line
I ( z) =
γ
V0+ e −γ z − V0− eγ z  .
R + jω L 
The characteristic impedance of a transmission line, Z0 , is defined as
(2.27)
34
Z0 =
R + jω L
γ
=
R + jω L
.
G + jωC
(2.28)
Since a lossless transmission line has R=0, G=0, the characteristic impedance is
simplified as
Z0 =
L
.
C
(2.29)
2.5 Interconnect Properties Characterized by Distributed
Elements
Telegrapher’s Equations in (2.16) and (2.17) are derived for the lossy
transmission line with Frequency-Independent Line Parameter (FILP). However, as
electronic system clock frequencies approach multiple GHz, with rise and fall times
shrinking to less than 0.1ns, harmonic components with frequencies up to 100GHz need
to be taken into account in a signal integrity analysis. At high frequencies, the RLGC
parameters defined from (2.3) to (2.9) become frequency-dependent variables.
Telegrapher’s Equations in (2.16) and (2.17) still hold for lossy transmission lines with
Frequency-Dependent Lossy Parameters (FDLP) as long as the quasi-TEM assumption is
still valid in the frequency range, i.e., the transverse electric and magnetic field
components are still much larger than the electric and magnetic field components in the
propagation direction.
35
In the case of the multiple transmission lines, the RLGC parameters have to be
expressed in matrix forms, representing the self and mutual components respectively.
Thus, Telegrapher’s Equations have to be rewritten as
dV ( z )
= − R(ω ) I ( z ) − jω L(ω ) I ( z ),
dz
(2.30)
dI ( z )
= −G (ω )V ( z ) − jωC (ω )V ( z ),
dz
(2.31)
where the RLGC parameters represented by the bold fonts represent the matrix form for
multiple transmission line components.
The frequency-dependent behaviors of the RLGC parameters have great impacts
on the modeling and simulation of interconnect systems. A detailed discussion about the
behaviors of the frequency-dependent RLGC parameter is given here to address the
situation.
2.5.1 Frequency Dependent Resistance and Inductance Parameters
In ideal interconnect systems with Perfect Electrical Conductors (PEC), the direct
current (DC) can exist at any location in the cross-section of a PEC, while the highfrequency time-varying electrical currents exist only within an infinitively thin surface of
the conductor. In practical interconnect systems with good conductors like copper,
aluminum, or poly-silicon, the DC generates voltage drop between two ends of a good
conductor due to the resistance of a conductor. The DC resistance of a conductor is
defined as
36
RDC = ρ
l
l
=
,
S σS
(2.32)
where ρ is the resistivity of a conductor, σ is the conductivity of a conductor, l is the
length of a conductor, and S is the cross-section area of a conductor.
In a good conductor, the high-frequency time-varying electrical currents exist
within a finite thin surface of the conductor, which is like a skin. The time-varying
electrical currents decay exponentially from the surface of a good conductor to the inner
center of the conductor. The distribution of the electrical current in a good conductor is
also related to the frequency. As the frequency increases from low to high, the electrical
currents crowd toward the surface of a good conductor. This effect is called the skin
effect. The distribution that the time-varying electrical currents exist on the surface of a
good conductor is described by the skin depth. The skin depth of a good conductor can be
derived from the propagation function. The electromagnetic wave that propagates inside a
good conductor must satisfy the Helmholtz Equation for electrical field as [25]
σ 

E = 0,
∇ 2 E + σω 2 µε  1 − j
ωε 

(2.33)
where E is the electrical field density. The propagation function is then defined as
γ = α + j β = jω µε 1 − j
σ
.
ωε
(2.34)
The conductivity of a good conductor σ is much larger than ωε in the square root term.
Thus the propagation function can be written as
37
σ
ωµσ
= (1 + j )
.
jωε
2
γ = α + j β ≈ jω µε
(2.35)
The skin depth is then defined as
δs =
1
α
=
2
ωµσ
.
(2.36)
The skin depth is defined as the depth where the field decays by an amount of 1/e or
36.8%. The relationship between the skin depth and the frequency is plotted in Figure 2.6
for a copper conductor with conductivity of 3.087e7 S/m, which is widely used in onchip, Multi-Chip Module (MCM), and Printed Circuit Board (PCB) lines. It can be seen
in Figure 2.6 that the skin depth is near 1 µ m at 10GHz, which is approximately equal to
the cross-sectional dimensions for a typical on-chip interconnect structure.
As the
frequency increases beyond 10GHz, the skin depth will be even smaller than the width or
depth of the on-chip interconnects. Physically, this means that the electrical currents are
not distributed evenly inside the interconnect beyond 10GHz and the V-I relationship
cannot be modeled by a constant R parameter. This situation becomes even more serious
when the frequency increases up to 46GHz. Thus a constant RLGC parameter model will
result in inaccurate simulation results for the high-frequency, on-chip interconnect
designs. For the MCM and PCB lines with much larger cross-sections [26], the skin depth
approaches the cross-section dimensions of conductors at much lower frequencies.
Correspondingly, lossy transmission lines with FDLP models have to be adopted in a
circuit simulator environment at much lower frequencies in the MCM and PCB cases
than in the on-chip case.
38
The relationship between the frequency and the resistance and inductance per unit
length parameters R and L are described by the internal impedance concept [27]. The
internal impedance is defined as
Z i (ω ) = R(ω ) + jω Li (ω ) = A + B jω = RDC
f
(1 + j ) , f > f0 ,
f0
(2.37)
where R(ω ) and Li (ω ) are the conductor resistance and internal inductance, ω = 2π f is
the angular frequency, and f 0 is the turning frequency where the R parameter starts to
Figure 2.6 The relationship between the frequency and the skin depth for a copper
conductor
39
increase as the result of the skin effect. By matching the real and imaginary parts of
(2.37), one can find the expressions for A, B, R(ω ) and Li (ω ) as
A = RDC ,
RDC
,
π f0
B=
Li (ω ) =
RDC
2π
f0 f
(2.38)
,
R(ω ) = RDC + RDC
(2.39)
f
.
f0
The expressions in (2.39) shows that the resistance parameter R(ω ) increases at the rate
of
f as the frequency increases, while the internal inductance Li (ω ) decreases at the
rate of 1/
f as the frequency increases. Note that the total inductance of a good
conductor is made up of the internal inductance Li (ω ) and the external inductance Le (ω )
as
L(ω ) = Li (ω ) + Le ,
(2.40)
where the external inductance Le is constant with respect to frequencies. Thus the
inductance L(ω ) will drop to the external inductance Le as the frequency increases.
2.5.2 Frequency Dependent Capacitance and Conductance Parameters
The frequency dependent capacitance and conductance parameters are closely
related to the complex permittivity. Studies [28] show that both the real and imaginary
parts of the complex permittivity are frequency-dependent. The loss tangent of a material
is not a constant with respect to the frequency, either. For example, the dielectric constant
40
of the FR-4 material has been listed as Table 2.1 with respect to frequencies [28]. It is
observed that from 100Hz to 10 GHz, the relative dielectric constant ε r reduces from 4.8
to 4.4, while the loss tangent increases from 0.009 to 0.025.
As can be seen from (2.6) and (2.11), the conductance and capacitance values of a
transmission line is directly proportional to the real and imaginary part of the dielectric
constant. Thus, frequency-dependent conductance and capacitance values are need in the
correct modeling and simulation of lossy interconnects especially at high frequencies.
f (Hz)
100
1000
10000
100000
1MHz
10MHz
100MHz
1000MHz
10000MHz
εr
4.80
4.75
4.70
4.65
4.60
4.55
4.50
4.45
4.40
tanδ
0.009
0.012
0.015
0.018
0.020
0.022
0.024
0.025
0.025
Table 2.1 Experimental data of relative dielectric permittivity εr and loss tangent for FR-4.
2.5.3 Examples of The Frequency Dependent Line Parameters
Examples are given here to show the impacts of the frequency-dependent losses
on different types of interconnects. Figure 2.7 shows the frequency-dependent R, L, G, C
parameters of a typical on-chip microstrip line with a 1.83 µ m by 1.23 µ m cross section
[26], a Multi-Chip-Module (MCM) line [29], and a Printed Circuit Board (PCB) line
[29]. The frequency-dependent RLGC parameters of the on-chip interconnect are
calculated using an integral equation method described in [30], while those of the MCM
and PCB lines are listed in [29].
41
Due to the small cross section dimensions of the on-chip interconnect, the skin
depth is larger than the height or width of the interconnect. For this case, it is safe to
assume that the currents are distributed evenly inside the interconnect, thereby allowing a
frequency-independent, lossy transmission line model with constant RLGC parameters to
be used up to a few GHz. However, it is observed in Figure 2.7 that the R and L
parameters of the on-chip line change for frequencies above 10 GHz. These changes
impact the modeling of the interconnect.
In the cases of the other two lines, the effects of the frequency-dependent losses
on the interconnect can be summarized as follows. The increasing R parameter, which is
proportional to the square root of frequency at high frequencies, is related to the skin
effects that result from the eddy currents flowing in the conductor (see Figure 2.7(a)).
Note that the R and L parameters are related by the Hilbert transform in a causal system.
Therefore, the changes in the R parameter dictate changes in the L parameters. To reflect
the changes in the R parameters, it is observed that the L parameters of the MCM and
PCB lines decrease and settle at finite values as the frequency increases in Figure 2.7.
The changes in the G parameters are associated with the dielectric losses resulting from
imperfect dielectrics and rotating dipoles. Note that the G and C parameters are also
linked by the Hilbert transform in a causal system. But the small changes in the G
parameters are usually neglected and set to zero in the on-chip and MCM interconnects.
However, the dielectric losses in a PCB line are too large to be ignored. Thus, the G
parameter of the PCB line cannot be neglected and this parameter increases as the
42
frequency increases. Correspondingly, an increase in the G parameter reflects a decrease
in the C parameter of the PCB line, as shown in Figure 2.7(c).
(a)
(b)
43
(c)
(d) G_MCM=0, G_On_Chip=0
Figure 2.7 A comparison of the per unit length resistance R, inductance L, conductance G,
and capacitance C for some common interconnect structures
44
2.6 Propagation functions for typical lines
To evaluate the impact of the frequency-dependent interconnect losses on signal
integrity, we first define the propagation function in the frequency domain as
{
exp [ −lγ ( s) ] = exp −l
[ R(s ) + sL( s)][G( s) + sC ( s)]} ,
(2.41)
where s = jω is the complex Laplace frequency, and l is the line length. We plot the
magnitudes of the propagation functions for the aforementioned on-chip, MCM, and PCB
lines in Figure 2.8. It is observed in Figure 2.8 that the relatively long frequency-
Figure 2.8 A Comparison of absolute values of the propagation functions for three
different types of interconnect lines.
45
dependent PCB line exhibits the maximum attenuation effect for signals propagating on
the interconnect. This can also be interpreted as the frequency-dependent, lossy PCB line
has the smallest Band Width (BW) for signal propagation, while the on-chip line provides
the largest BW for signal propagation. In addition to the BW limitation, the PCB line
with frequency-dependent losses also yields the most dispersion for propagating signals.
The dispersion effects are best illustrated in the time domain.
2.7 The Internal Relationships Between the Distributed
Elements
Although the frequency-dependent RLGC parameters in Fig 2.7 vary significantly
with respect to frequency, they are inter-related with each other. As stated in [31], for a
causal real system, if we define the parallel admittance and the series impedance matrices
of the frequency-dependent lossy interconnect as
Y ( s ) = G ( s ) + sC ( s ),
(2.42)
Z ( s ) = R( s ) + sL( s ),
then the real and imaginary parts of Y ( s ) and Z ( s ) should satisfy the Kramer-Kronig
conditions through the Hilbert Transform, i.e.,
X (ω ) = jω L(ω ) = H ( R(ω ) ) =
1
π∫
∞
0
dR(ω ') ω '+ ω
ln
d ω ',
ω '− ω
dω '
(2.43)
where the scalar R(ω ) is an element of the matrix R( s ) , s = jω , H represents the Hilbert
Transform[31].
However,
in
practical
implementations
of
(2.43),
numerical
differentiation has to be applied for the derivative and numerical integration has to be
46
applied to the integral in (2.43). In [32], a combined numerical differentiation and
numerical integration algorithm is developed, i.e.,
X (ωm ) =
1
dR(ω ')
(k ) F (ω 'k −1 , ω 'k , ωm ),
dω '
k =1
N
∑
π
F (ω 'k −1 , ω 'k , ωm ) = (ω 'k + ωm ) ln ω 'k + ωm − (ω 'k − ωm ) ln ω 'k − ωm
− (ω 'k + ωm ) ln ω 'k −1 + ωm + (ω 'k −1 − ωm ) ln ω 'k −1 − ωm ,
R(ωk ') − R(ωk −1 ')
dR(ω ')
,
≈
d ω ' ωk
ωk − ωk −1
(2.44)
(2.45)
(2.46)
where k is the frequency sampling point. A similar expression can also be used to relate
G (ω ) and C (ω ) . The above method provides an algorithm to check the immittance
consistency in the sampled frequency-dependent RLGC parameters.
47
CHAPTER 3
GENERAL ALGORITHMS FOR LOSSY
INTERCONNECT MODELING AND SIMULATION
In Chapter 2, the challenges of the interconnect modeling and simulation research
are outlined. In order to overcome these challenges, numerous techniques are adopted in
current state-of-the-art simulators. In [2], these techniques are generally divided into two
categories. The first category contains the methods that are based on macromodeling each
individual transmission line set. The second category represents the methods that are
based on model-order reduction of the entire linear sub-network containing lumped as
well as distributed sub-networks. In this chapter, the macromodeling techniques for
general lossy interconnect modeling and simulations are first discussed [2]. Then, some
model-order-reduction based techniques are reviewed. Specifically, explicit and implicit
moment matching based methods are introduced. Comparisons are made between several
typical macromodeling methods in terms of model order number and efficiency. In
particular, standard macromodeling methods based on moment matching and the Method
of Characteristics (MoC) are compared and contrasted with the explicit moment matching
method in terms of stability, passivity, and causality.
In both model-order reduction and macromodeling techniques that are applied to
lossy transmission lines with Frequency Dependent Line Parameters (FDLP), an
important issue is the maximum frequency. In high-speed digital designs, the extent of
48
the frequency spectrum is determined by the rise and fall times of the signal propagating
on the interconnects. A conservative criteria for the desired maximum frequency is
f max ≈ 0.35 / tr ,
(3.1)
where f max is the maximum frequency of interest in the simulation, and tr is the rise time
of the signal. By this criteria, if a trapezoidal signal with a rise time of 0.01ns propagates
on a transmission line, then the maximum frequency of interest is 35GHz. However, this
criteria will not produce accurate results for high-speed, high-frequency interconnect
designs. Thus, a stricter criteria is suggested as
f max ≈ 1/ tr .
(3.2)
In this dissertation, the maximum frequency is chosen according (3.2) if not specified.
3.1 General Transmission Line Macromodeling Techniques
Macromodeling techniques can be sub-divided into several methods or algorithms
as shown in [2]. These techniques include but are not limited to the direct discretization
of transmission lines, numerical convolution of transfer functions [12,13], Method of
Characteristics (MoC) [14-18], basis function macromodels [19,20], compact finitedifference-based approximations[21], and the integrated congruence transform[9].
3.1.1 Direct Discretization of Transmission Lines
Telegrapher’s Equations in Chapter 2 have been established as a good model for
lossy transmission lines. Note that Telegrapher’s Equation can be applied to both
transmission lines with Frequency Independent Lossy Parameters (FILP), as well as
transmission lines with Frequency Dependent Lossy Parameters (FDLP).
49
R∆z
L∆z
C ∆z
R∆z
G ∆z...
L∆z
R∆z
C ∆z
G ∆z
L∆ z
C ∆z
G ∆z
Figure 3.1 The direct discretization of a single lossy transmission line
In the direct discretization method, the transmission lines are represented by a
series of lumped element sections as shown in Figure 3.1, where ∆z is a small fraction of
the wavelength λ , i.e.,
∆z
λ.
(3.3)
This method can be easily implemented in current circuit simulators like SPICE
since all the elements in the model are already defined in SPICE. In fact, this model is
still widely used in commercial simulation tools like HSPICE as the U-element.
However, the direct discretization method suffers from several drawbacks. First,
as indicated in Figure 3.1, the number of sections of a transmission line is directly
proportional to the length of the line. Therefore, the direct discretization method is very
inefficient for modeling long lines with large numbers of sections. Secondly, the
increasing operation frequency in digital systems makes the minimum wavelength of the
signal smaller and smaller. Therefore, in order to satisfy (3.3), ∆z also has to be scaled
down. Thus, the number of sections used to model a lossy transmission line keeps
increasing and the simulation requirements become severe. Another problem is that the
lumped RLGC element in the direct discretization model introduces poles to the model.
50
Sometimes oscillations are observed in the transient simulation results from the direct
discretization method when the number of sections becomes large, as reported in [33].
3.1.2 Convolution of the Impulse Response
The signal response for a linear system can be expressed using the convolution
operation,
t
f (t ) = v(t )* h(t ) = ∫ v(τ )h(t − τ )dτ ,
(3.4)
0
where v(t ) is the input signal, such as trapezoidal, unit-step, or sine signal, f (t ) is the
signal response, and h(t ) is the impulse response of the system (e.g. a lossy transmission
line with FILP).
A method that employs the convolution of the impulse response for the simulation
of lossy transmission lines with FILP was suggested in [12]. This method starts with a
single uniform lossy transmission line with FILP. Here we consider a single uniform
lossy transmission line with FILP with the following initial condition as
v(0, t ) = v1 (t ), v(l , t ) = v2 (t ),
i (0, t ) = i1 (t ),
i (l , t ) = −i2 (t ),
(3.5)
v( x, 0) = v0 ( x), i ( x, 0) = i0 ( x),
where l is the length of the line and x is any location on the line, i.e., 0 ≤ x ≤ l . This
transmission line structure should satisfy the time-domain Telegrapher’s Equations in
(2.16) and (2.17). The classical solutions to Telegrapher’s Equations are adopted to solve
the second order Ordinary Differential Equations (ODE) as given in (2.18) and (2.19).
51
The time-domain convolution equations (constitutive relationships) obtained for lossy
transmission lines with FILP are
v1 (t )* hY (t ) − i1 (t ) = v2 (t )* hγ Y (t ) + i2 (t )* hγ (t ),
(3.6)
v2 (t ) * hY (t ) − i2 (t ) = v1 (t ) * hγ Y (t ) + i1 (t )* hγ (t ),
(3.7)
where * denotes the convolution operation defined in (3.4), and hY (t ) , hγ Y (t ) , and hγ (t )
are the three impulse-responses associated with a lossy transmission line with FILP.
First, we define some parameters as
L
,
C
(3.8)
T = l LC ,
(3.9)
Y0 =
1 R
G
β =  + .
2 L C 
(3.10)
Then, hY (t ) , hγ Y (t ) , and hγ (t ) , which are the inverse Laplace Transforms of H Y ( s ) ,
H γ Y ( s ) , and H γ ( s ) , can be expressed as
-1
hY (t ) = L
-1 
∞
1
Y
0 0
[ H Y ( s)] = ∫
G + sC st
e ds,
R + sL
(3.11)
∞
hγ (t ) = L  H γ ( s )  = ∫ exp  − x ( R + sL)(G + sC )  e st ds,
(3.12)
0
∞
1
hγ Y (t ) = L  H γ ( s ) H Y ( s )  = ∫
Y
-1 
0
0
G + sC
exp  − x sC ( R + sL)  e st ds,
R + sL
(3.13)
52
where L-1 is the inverse Laplace Transform operator and s = jω is the complex
frequency. The expressions in (3.11), (3.12), and (3.13) also have physical significance.
For example, the expression in (3.11) is proportional to the time-domain characteristic
admittance, the expression in (3.12) models the time-domain propagation function, and
the expression in (3.13) models the current response of a lossy transmission line with
FILP.
As described in [13], the time-domain expressions for (3.11), (3.12) and (3.13)
can be written as [34,35]
hY (t ) = Y0 e − β t  β ( I1 ( β t ) − I 0 ( β t ) ) + δ (t )  ,
(3.14)


βt
hγ (t ) = e − β t u (t − T )
I1 ( β t 2 − T 2 ) + δ (t − T )  ,
t2 − T 2


(3.15)




t
hγ Y (t ) = Y0 e− β t u (t − T ) β 
I1 ( β t 2 − T 2 ) − I 0 ( β t 2 − T 2 )  + δ (t − T )  ,
2
2
 t −T



(3.16)
where I 0 and I1 are modified Bessel functions of zeroth and first orders, and δ (t ) and
u (t ) are delta and unit-step functions, respectively.
In computer simulation programs like SPICE, the expressions in (3.6), (3.7),
(3.14), (3.15), (3.16), and the source waveforms v1 (t ) are first discretized in the time
domain. Then numerical convolutions are carried out to obtain the unknown variables
v2 (t ) , i1 (t ) , and i2 (t ) at each simulation step. It is observed that at each simulation step,
the numerical convolution operation requires all the information for the source signal and
53
the propagation function at all previous simulation steps. Thus, if the simulation step
number is n, then the total simulation has a computation complexity of order of n 2 ,
which is written as O(n2 ) .
The research in [12,13] successfully derived the impulse response or impulse
characteristics of lossy transmission lines with FILP. Then, the unit-step response or the
transient characteristics were derived in [15]. In [15], the transient characteristics of the
propagation function and the characteristic admittance are written as
t

I (b t 2 − T 2 ) 
pγ (t ) = e−α + a ∫ e − β t 1
dt  u (t − T ),
2
2
t
T
−


T
(3.17)
t
 −βt

G − βτ
pY (t ) = Y0  e I 0 (bt ) + ∫ e I 0 (bτ )dτ  ,
C0


(3.18)
where β is defined in (3.10) and

1G
+ RY0  l ,
2  Y0

(3.19)
1 G
− RY0 ,
2 Y0
(3.20)
1 R G
− .
2 L C
(3.21)
α= 
a=
b=
A close examination of (3.17) reveals that the transient characteristics of the propagation
function are made up of two parts. The first part is an exponentially decaying function
e−α , while the second part is an integral involving the modified Bessel function of first
order. This result is also validated in [26] (see Equation (5)).
54
Many practical interconnects often have a zero conductance parameter as G = 0 .
Furthermore, an approximation based on using the asymptotic expansion of the modified
Bessel function can be employed in (3.17) and (3.18) [15]. Thus, simplified expressions
can be derived for (3.17) and (3.18) when G = 0 and the asymptotic expansion to the
Bessel function is adopted, i.e.,


2a
pγ (t ) ≈ 1 −
 u (t − T ),
2π b(t + d ) 

(3.22)
pY (t ) = Y0 e− β t I 0 ( β t ),
(3.23)
where
d=
4a 2
.
2π b(1 − e− a ) 2
(3.24)
The approximation in (3.22) is reported to have an accuracy of within 1% over the full
time range [15].
The expressions in (3.17) and (3.18) show that the unit-step response of a lossy
transmission line with FILP involves with an integral of the modified Bessel function of
the first order when the conductance per-unit-length parameter is not set to zero. This
integral has only been previously solved by numerical method like numerical integration.
As will be shown in Chapter 4, the integrals in (3.17) and (3.18) can be represented in
terms of Incomplete Lipshitz-Hankel Integrals (ILHI).
In practical simulations, the source signal is typically represented as a
complicated function of time, e.g. sinusoidal or trapezoidal waveforms. Since it is
55
difficult to obtain the transient responses of many different source signals separately, the
source signal is often discretized as a piece-wise linear function of time, i.e., between
each time step, the source signal is expressed as a linear function of time. Therefore, in
order to speed up the simulation procedure and avoid the use of numerical convolution, it
is desirable to have the ramp response of a lossy transmission line with FILP.
The method of convolution with the impulse response so far can only be applied
to lossy transmission lines with FILP. For lossy transmission lines with FDLP, the RLGC
parameters become frequency-dependent and the closed-form representations of the
characteristic admittance (3.14), the propagation function (3.15), and the current response
(3.16) are no longer valid. Therefore, advanced macromodeling methods have to be
adopted to handle lossy transmission lines with FDLP.
3.1.3 Recursive Convolution Method
The convolution method suffer from a common drawback that the convolution
operation needs to extend over the entire past history. In order to improve the efficiency
of the regular convolution procedure suggested in (3.6) and (3.7), a recursive convolution
technique is suggested in [36]. It was shown from the previous section that the key
variables in the convolution method are the characteristics admittance Y in (3.11), the
propagation function e− lγ in (3.12), and their products or the current response. The
recursive convolution method first approximates these key variables with rational
functions through a Pade approximation method. Then a recursive relationship is
developed between the present-time-step impulse response and the previous-time-step
56
impulse response by utilizing the time-domain Pade approximation results. A detailed
explanation is given here based on a single lossy transmission line with FILP.
3.1.3.1 Pade Approximations of the Characteristic Admittance and Propagation
Function
As indicated in [37], the high frequency components of the propagation function
and the characteristic admittance decide the early-time response of a lossy transmission
line with FILP, while the low-frequency components control the late-time response. In
order to obtain accurate early-time responses that are crucial to the convolution operation
over a small time interval, an expansion of 1/s instead of s is used to rewrite the
characteristic admittance as
G
1+ y
G + sC
C
C ,
Y (s) =
=
R + sL
L 1+ R y
L
(3.25)
where y = 1/ s . Y ( s ) is the expanded into a Maclaurin series in y about y=0 or s = ∞ as
Y (s) ≈
C
C
Yp ( s ) =
1 + m1 y + m2 y 2 + ... + mn y n ) ,
(
L
L
(3.26)
where the nth moment is defined as
 G   R 
d N 1 + y  / 1 + y 
1
 C   L 
.
mN =
N!
dy N
(3.27)
A symbolic differentiation is adopted here to calculate the derivative in (3.27). Then a
Pade approximation is introduced to approximate the characteristic admittance Y ( s ) as
57
Y (s) =
G + sC
C aN s N + aN −1 y N −1 + ... + a1 y + 1
,
≈
R + sL
L bN s N + bN −1 y N −1 + ... + b1 y + 1
(3.28)
where a1 to aN and b1 to bN are unknown coefficients of the Pade Approximation.
The expression in (3.26) should be equal to equation (3.28) for the first 2 N − 1 terms.
Therefore, a matrix relationship can be found by comparing the two equations:
 aN
1 − b
N

 m1

 ...
 mN −1
m1
m2
...
mN

mN −1 

... mn 

...
... 
... m2 N − 2 
...
 bN 
b 
 N -1  =
 ... 


 b1 
 mN 
m 
-  N +1  ,
 ... 


 m2 N −1 
(3.29)
and
 a1   1
 a   m
 2 = 1
 ...   ...

 
 aN −1   mN − 2
0
1
...
mN −3
... 0 
... 0 
... ...

... 1 
 b1   m1 
b   m 
 2  +  2 .
 ...   ... 

 

bN −1   mN −1 
(3.30)
Equation (3.29) and equation (3.30) give rises to an approximated model for the
characteristic admittance. If we rewrite (3.28) as
C a N s N + aN −1 y N −1 + ... + a1 y + 1
C s N + a1s N −1 + ... + aN
,
Y (s) ≈
=
L bN s N + bN −1 y N −1 + ... + b1 y + 1
L s N + b1s N −1 + ... + bN
(3.31)
and define
P( s ) = s N + b1s N −1 + ... + bN ,
Q( s ) = s N + a1s N −1 + ... + aN ,
then the Heaviside theorem can be applied to Y ( s ) as
(3.32)
58
Y (s) ≈
n
qi
C Q( s)
C
1
=
+
 ∑
L P( s)
L
1 s − pi

,

(3.33)
where pi are the roots to the polynomial equation P ( s ) = 0 and
qi =
Q( pi ) − P( pi )
,
P ( pi )
(3.34)
where P ( pi ) is the derivative of P ( s ) at s = pi .
Similarly, the propagation constant can also be expanded as
γ (s) =
( sL + R )( sC + G ) = s
  G
LC + s LC  1 +
  C
 R
y  1 +
 L
∞
m


= s LC + s LC 1 + ∑ ii − 1 ,
 i =1 s

 G  R 
d N 1 + y  1 + y 
1
 C  L 
.
mN =
N!
dy N
 
y  − 1
 
(3.35)
(3.36)
Then Pade Approximation is applied to the propagation function as
H γ ( s ) = e −γ ( s )l ≈ e − s
LCl − m1 LCl
e
n

qi 
1 + ∑
.
1 s − pi 

(3.37)
Note that the macromodel order n is different from the model order for the characteristic
admittance.
3.1.3.2 The Recursive Convolution Scheme
The frequency-domain expressions in (3.33) and (3.37) can be easily converted
into time-domain expressions as a Dirac function δ (t ) plus the sum of n exponential
decaying functions of t as
59
n
hγ (t ) ≈ ∑ qi e pi (t −T ) + δ (t − T ),
(3.38)
i =1
where T is the propagation delay caused by the first term in (3.37).
Recursive convolution is used to reduce the computational complexity from
O(n2 ) in the conventional convolution case to O(n) by utilizing the exponential
properties in the macromodel of (3.33) and (3.37). For example, if a transmission line
with FILP is matched at both the source and load ends, then the transfer function h(t )
can be simplified as a time-domain propagation function hγ (t ) . As defined in (3.4), the
convolution operations between the current at the far-end of the line i2 (t ) and the
propagation function hγ (t ) in (3.6) at time t = tn +1 can be written as
f (tn +1 ) = i2 (t ) * hγ (t ) |t =t n +1
=
tn +1
∫
0
 n

i2 (τ )  ∑ qi e pi (tn+1 −T −τ ) + δ (tn +1 − T − τ ) dτ
 i =1

tn
n
0
i =1
= ∫ i2 (τ )∑ qi e pi (tn +1 −T −τ ) dτ +
tn +1
∫
tn
n
i2 (τ )∑ qi e pi ( tn +1 −T −τ ) dτ + i2 (tn +1 − T )
(3.39)
i =1

 p ( t −t )
 n  tn +1
pi ( tn −T −τ )
pi ( tn +1 −T −τ )
i n +1 n
= ∑ e
(
)
d
τ
i
(
τ
)
q
e
d
τ
+
i
q
e
τ


 ∑ ∫ 2
i
∫0 2 i
i =1 
i
=
1



t


 n
+i2 (tn +1 − T ),
tn
n
Now a trapezoidal approximation is applied to the second term in (3.39) as
tn +1
∫ i (τ )q e
2
tn
i
pi ( tn +1 −T −τ )
dτ =
hn
qi i2 (tn − T )e pi hn + i2 (tn +1 − T )  ,
2
(3.40)
where hn is the simulation step, which is defined as hn = tn +1 − tn . Thus, the final solution
for the convolution operation can be written as
60
tn
n 

i2 (t ) * hγ (t ) |t =t n +1 = ∑  e pi (tn +1 −tn ) ∫ i2 (τ )qi e pi (tn −T −τ ) dτ 
i =1 

0

n
h
+ ∑ n qi i2 (tn )e pi hn + i2 (tn +1 )  + i2 (tn +1 − T ).
i =1 2
tn
Note that only the convolution operation
∫ i (τ )q e
2
i
pi ( tn −T −τ )
(3.41)
dτ is unknown in (3.41). At
0
time t = tn +1 , expression (3.41) states that the current convolution operation can be
obtained from the convolution operation at the previous simulation step result at time
t = tn . This forms the recursive convolution relationship between the current time step
and the previous time step so that the convolution integration does not need to be
expanded over all the past time history.
3.1.3.3 Properties of Recursive Convolution Methods
It is observed in the previous section that at each simulation time step, only the
simulation result of the last simulation time step is needed to obtain the current step
result. Thus, for the nth simulation time step, it is necessary to perform the calculation n
times to complete the simulation, i.e., the computational complexity is O(n) .
Another important feature in the recursive convolution method is the extraction of
the propagation delay factor e− sl
LC
from the propagation function before the Pade
Approximation. It will be seen later that this is one of the key steps in the Method of
Characteristics (MoC). It is fair to categorize the recursive convolution method as one of
the generalized MoC method.
61
The recursive convolution methods successfully reduced the computation
complexity from O(n2 ) to O(n) . However, several aspects of this method still need to be
improved. First, the Pade Approximation method outlined in this chapter doesn’t
guarantee stable poles and residues. It was observed in [36] that sometime poles on the
right hand sided of the Laplace transform plane are generated in (3.33) and (3.37), i.e.,
the poles in (3.33) and (3.37) may have positive real parts, which may produce
exponentially-increasing results in the time domain. Also, the Pade Approximation
cannot guarantee the passivity of the poles and residues. Non-passive but stable poles still
have negative real parts, but may lead to oscillating time-domain results when the
macromodel is connected with other circuits. Finally, although it is theoretically possible
to apply the recursive convolution method to lossy transmission lines with FDLP,
recursive convolution for lossy transmission lines with FDLP has not been realized so far.
3.1.4 Method of Characteristics
The Method of Characteristics (MoC) was first developed in [14] to model
lossless transmission lines where only frequency-independent L and C parameters exists.
Later, the work in [38] extended the MoC to lossy transmission with FILP. Then, the Welement model [15] was introduced to successfully model lossy transmission lines with
FDLP. Similar work is also reported in [16]. In [32], a network synthesis approach is
taken to generate circuit models for lossy transmission lines with FDLP. In [5,39], the
impulse response is approximated by a sharp Triangle Impulse Response (TIR) and an
convolution operation based on TIR and MoC relationships is established to obtain
transient results of lossy transmission lines with FDLP.
62
The input to the MoC method is usually tabulated frequency-domain RLGC
parameters. However, MoC models can also be obtained from the time-domain results for
different types of interconnects like in the case of the Hybrid Phase-Pole Macromodel
(HPPM) [17]. By expressing the TIR of complicated interconnect structures in terms of
pole-residue pairs, the generated HPPM can be used to efficiently simulate the system for
various input waveforms.
In this section, the basic of the MoC method is first discussed. Then, one typical
MoC model, the W-element, is explored in more detail.
3.1.4.1 Brainin’s Method and Chang’s Method
From the frequency-domain Telegrapher’s Equation, the I-V characteristics for a
single lossy transmission line with FILP can be represented as
1 + e −2γ ( s )l −2e− γ ( s )l  V1 
 I1 
1
=
,

 
−2 γ ( s ) l 
)  −2e −γ ( s )l 1 + e−2γ ( s )l  V2 
 I 2  Z 0 (1 − e
(3.42)
where I1 and V1 are the frequency-domain near-end current and voltage, I 2 and V2 are
the far-end current and voltage as defined in (3.5), Z 0 is the characteristic impedance
which is the inverse of the characteristic admittance defined in (3.28), and γ ( s ) is
defined in (3.35). The expression in (3.42) can be rewritten as
V1 = Z 0 I1 + e−γ ( s )l  2V2 − e− γ ( s )l ( Z 0 I1 + V1 )  ,
V2 = Z 0 I 2 + e −γ ( s )l  2V1 − e −γ ( s )l ( Z 0 I 2 + V2 )  .
(3.43)
63
i1 (t)
I1
i 2 (t)
Z0
Z0
+
v 1 (t)
V1
v 2 (t)
+
W1
W2
_
x=0
I2
V2
_
x=d
Figure 3.2 Method of Characteristics (MC) equivalent circuit.
Based on these results, two equivalent Voltage Controlled Voltage Sources (VCVS) W1
and W2 can be created where
W1 = e −γ l  2V2 − e −γ l ( Z 0 I1 + V1 ) 
(3.44)
W2 = e −γ l  2V1 − e −γ l ( Z 0 I 2 + V2 ) 
(3.45)
If we define the voltage at the two ports as
V1 = W1 + Z 0 I1 ,
V2 = W2 + Z 0 I 2 ,
(3.46)
then the transmission line can be replaced by a model as shown in Figure 3.2. Combining
the equations (3.46) and (3.43) yields a recursive relations for W1 and W2 as
W1 = e −γ d [ 2V2 − W2 ] ,
W2 = e −γ d [ 2V1 − W1 ].
(3.47)
When the transmission line is lossless and its propagation constant γ ( s ) is pure
imaginary, the inverse Laplace Transform can be applied to the equations in (3.47),
thereby yielding
64
w1 (t + τ ) = 2v2 (t ) − w2 (t )
(3.48)
w2 (t + τ ) = 2v1 (t ) − w1 (t )
(3.49)
Equations (3.43), (3.48), and (3.49) forms a transient simulation scheme for lossless
transmission lines as defined in [14].
For lossy transmission line simulations, the propagation constant γ ( s ) is not
purely imaginary anymore. In [40], a Pade synthesis of the characteristic impedance and
the complex propagation function was adopted and this approach leads to a recursive
convolution method, with the exception of the recursive convolution operation.
3.1.4.2 W-Element
Another application of the Method of Characteristics is the W-element in HSPICE
[15]. The basic algorithms in the W-element are the same as those in the MoC technique
that was described in the previous section except that the Voltage Controlled Voltage
Sources W1 and W2 are replaced by Current Controlled Current Sources WIf and WIb .
However, the major differences between the MoC and the W-element method are the
application of the interpolation-based complex rational approximation method, the
difference approximation to the characteristic admittance and the propagation function,
and the indirect numerical integration that were employed in the W-element method.
Given samples of complex open-loop transfer function H ( jω ) for a lossy
transmission line (i.e., the propagation function) at the set of frequency points {0, ω1 ,
ω2 ,… ωK }, an interpolation-based complex rational approximation method can be
employed to approximate H ( jω ) as
65
M
am
a ω
= H ∞ + ∑ m cm ,
m =1 1 + jω / ωcm
m = 0 ωcm + jω
M
H ( jω ) ≈ H ( jω ) = H ∞ + ∑
(3.50)
where am and ωcm are the pole and residue pairs. The interpolation-based complex
rational approximation method guarantees that the poles have negative real parts and
H ( jω ) matches H ( jω ) at the set of frequency points {0, ω1 , ω2 ,… ωK }. Note that the
propagation delays are extracted from the propagation function before the approximation.
The next step is the introduction of the state-space model to the approximated
transfer function for the step invariance case as
M

 y (tn ) = H ∞ x(tn ) + ∑ zm (tn ),
m =0

 z (t ) = a (1 − e−ωcmTn ) x(t ) + e −ωcmTn z (t ),
m
n −1
m n −1
 m n
(3.51)
where Tn = tn − tn −1 is the simulation step size, x, y and zm are the excitation, response,
and state variables. At each simulation step tn , x(tn ) , zm (tn −1 ) , am , ωcm , and H ∞ are
known variables, so the transient simulator needs to update zm (tn ) and y (tn ) step by step.
A companion model is built from (3.51) to make transmission line models compatible
with SPICE.
The W-element method employs the interpolation-based complex rational
approximation method to find stable and well-behaved pole and residues pair for the
system transfer function. The state-variable model is employed to make the transient
simulation well-controlled and compatible with SPICE. The pre-extraction of the
propagation delay from the transfer function reduces the macromodel order. What’s
66
more, the generalized state-variable model is applicable to lossy transmission lines with
both FILP and FDLP. However, it will be observed later that the interpolation-based
complex rational approximation method may result in late-time oscillations for transient
simulations of lossy interconnects with FDLP up to a hundred GHz. Like other MoC
methods, the interpolation-based complex rational approximation method cannot be
proven to be a passive operation.
3.1.5 Spectral Methods
Another important method for modeling transmission line is the spectral method
or the basis function expansion method. The basis functions can be Chebyshev
polynomials [19] [20] or wavelets [41,42]. The basic idea of the basis function expansion
is to expand the port voltage and current variables in the Telegraphers’ Equations using
basis function, e.g., Chebyshev polynomials [43]
M
v( x, t ) = ∑ pm (t )Tm ( x),
m=0
M
(3.52)
i ( x, t ) = ∑ qm (t )Tm ( x),
m=0
where Tm ( x) is a Chebyshev polynomial of mth degree, and pm (t ) and qm (t ) are the
unknown variables. The derivatives of the port voltages and currents with respect to
location in the Telegraphers’ Equations can also be expanded in the form of Chebyshev
polynomials as
67
M
∂
v( x, t ) = ∑ p m (t )Tm ( x),
∂x
m=0
M
∂
i ( x, t ) = ∑ qm (t )Tm ( x),
∂x
m =0
(3.53)
where p m (t ) , and qm (t ) are unknown expansion coefficients. One important property of
the Chebyshev polynomials is that the unknown variables pm (t ) , qm (t ) , p m (t ) , and
qm (t ) have the following interdependencies
1
( p m−1 (t ) − p m+1 (t ) ) ,
2m
1
qm (t ) =
( qm−1 (t ) − qm+1 (t ) ) .
2m
pm (t ) =
(3.54)
By making use of the orthogonality properties of the Chebyshev polynomials, the
Telegraphers’ Equations can be expressed as Ordinary Differential Equations (ODE) that
can be solved in the time-domain.
The spectral method separates the port voltage and currents into products of the
unknown time-domain coefficients and spatial Chebyshev polynomial expressions. It is
therefore possible to model transmission lines with spatial variations, e.g., an IC
packaging pin that shrinks from a large cross section to a small cross section. However,
as pointed out in [2], the Chebyshev polynomial expansion cannot guarantee the passivity
of the generated model.
3.1.6 Least-Square Approximation
An efficient numerical approximation of frequency-domain data by using rational
function expansions in terms of pole-residue pairs has been a long-term goal in the field
68
of modeling and simulation [44-46][63,64] . Usually the frequency-domain data is given
at a set of sampling points {0, ω1 , ω2 ,… ωK }. The macromodel model order may be
smaller than the number of sampling points K so that there are more equations than
unknowns. Therefore, it is impossible to find a solution to exactly satisfy all the
constraints. However, it is possible to obtain a solution that has a minimum deviation
from the given data, which is a typical least-square problem.
One of the reasons for such research is that the pole-residue pairs can be easily
converted into the time domain as a sum of exponential decaying functions, or they can
be easily synthesized as lumped elements in parallel networks.
The rational
approximation is usually applied to the transfer function as in (3.50). Note that rational
functions here can be in the forms of either real poles or complex conjugate poles. The
least-squares based method is a numerical approximation that cannot be proven to be
passive. However, the poles can be forced to have negative real parts or reside in the Left
Hand Side (LHS) of the complex plane to make the macromodel stable.
Two least-square based algorithms are discussed here. The first algorithm is the
standard least-square algorithms [44-46]. Another least-square algorithms is the Vector
Fitting (VectFit) algorithm developed in [47,63,64].
3.1.6.1 Standard Least-Square Algorithms
The first step in the standard Least-Square approximation method is to
approximate the real part of the transfer function as [43]
69
M
P(s )
Re [ H ( s )] ≈ Re  H ( s )  =
=
Q( s)
∑ p (ω
m =0
M
i
∑ q (ω
m =1
i
2 m
)
,
(3.55)
2 m
)
where s = jω , pi and qi are the unknown coefficients. Since (3.55) should be satisfied
at all frequency sampling points, a matrix relationship can be established
1 0

2
1 ω1

 1 ω 2
K

...
0
... ω12 M
...
... ωK2 M
 p0 
    Re  H (0)  

  
−ω12 Re  H (ω1 )  ... −ω12 M Re  H (ω1 )    pM   Re  H (ω1 )  

 .
 q  = 
...

 1  


2
2
M
−ωK Re  H (ωK )  ... −ωK Re  H (ωK )      Re  H (ωK )  
   

 qM 
(3.56)
0
...
0
Note here that the order number M is smaller than the number of sampling points K. Once
the coefficients pi and qi are obtained, then solving the equation Q( s ) = 0 will result in
the poles of the macromodel. During this step, all Right Hand Side (RHS) or pure
imaginary poles are removed to ensure the stability of the macromodel. In order to obtain
the residues, the real and imaginary parts of the transfer function are matched with the
macromodel as
70
1

1
ωc1

1/ ωc1

2
1
1 + (ω1 / ωc1 )



1/ ωc1

1
 1+ ω / ω 2
( K c1 )


ω1 / ωc1
0
2
1 + (ω1 / ωc1 )




ωK / ωc1
0
2
 1 + (ωK / ωc1 )
...
...
...
...
...


ωcM

1/ ωcM

 Re  H (0)  

 
2 

1 + ( ω1 / ωcM ) 
 Re  H (ω1 )  



 

H
∞



1/ ωcM
  a1ωc1  

=  Re  H (ωK )   .
2
1 + (ωK / ωcM )    

  aM ωcM   Im  H (ω1 )  
ω1 / ωcM


2 
1 + ( ω1 / ωcM ) 





 Im  H (ω K )  

ωK / ωcM 
2
1 + (ωK / ωcM ) 
1
(3.57)
The classical Least-Square method provides a good approximation of the transfer
function when the number of frequency sampling points is small. However, when the
frequency increases to the GHz range and the number of sampling points becomes large,
the accuracy of this method is severely degraded. What’s more, the classical LeastSquare method suffers from poor convergence in terms of the macromodel order number.
A higher macromodel order number may not lead to better approximation accuracy. As
we will show later, an improved Vector Fitting algorithm [47] provides a better solution
to the rational function approximation problem.
Another important issue with the direct Least-Square approximation method is the
large number of pole-residue pairs when the interconnect structure is long. In fact, the
number of pole-residue pairs is directly proportional to the length of the interconnect
structure.
71
3.1.6.2 The Vector Fitting Algorithm
In [47,63,64], an improved least-square approximation algorithm is proposed with
a passivity checking routine. The Vector Fitting algorithm starts with the pole
identification stage. A set of starting poles an and an unknown rational function σ ( s ) are
introduced to form an matrix relationship as
 N cn

+ d + sh 
∑

σ ( s ) f ( s )   n =1 s − an
,

≈
N

cn
 σ ( s)  
+1 
 ∑
 n =1 s − an

(3.58)
where cn , cn , d , and h are unknown variables. Note that h can be set to zero. By
substituting the expressions for σ ( s ) in the expressions for σ ( s ) f ( s ) in (3.58), we obtain
the relationship
 N cn

cn
d
sh
+
+
−
+ 1 f ( s ) ≈ f ( s ).
∑
∑
n =1 s − an
 n =1 s − an 
N
(3.59)
Matrix operation in the form of Ax = B can then be formed from (3.59) where the kth
row vectors are
 1
Ak = 
 sk − a1
...
1
sk − aN
x = [ c1 ... cN
1 sk
d
− f (s k )
− f (s k ) 
...
,
sk − a1
sk − a N 
h c1 ... cN ] ,
Bk = f ( sk ) ,
T
(3.60)
(3.61)
(3.62)
72
and k corresponds to the sampling point. By solving the linear equations Ax = B using
the least-square method, we can obtain all the coefficients.
Next, the sum form of the approximation functions are expressed as product
forms, i.e.,
N
c
σ (s) = ∑ n + 1 =
n =1 s − an
N
∏ ( s − z )
n
n =1
N
∏(s − a )
n =1
,
(3.63)
n
N +1
∏
c
σ ( s ) f ( s) = ∑ n + d + sh = h nN=1
n =1 s − an
N
( s − zn )
∏(s − a )
n =1
,
(3.64)
n
where zn and zn are the zeros of σ ( s ) and σ ( s ) f ( s ) that can be calculated from cn , cn ,
d , and h .
In order to obtain zn and zn , the standard linear solution forms a matrix as
H = A − IcT ,
(3.65)
where A is the diagonal matrix containing the starting poles, I is a column vector of ones,
and cT is a row vector containing the residues of σ ( s ) . In standard linear solutions, the
zeros of σ ( s ) f ( s ) and σ ( s ) are the eigen values of the matrix in (3.65). By finding the
eigen values of (3.65), we can obtain the residues zn and zn for σ ( s ) f ( s ) and σ ( s )
respectively.
Since the expressions in (3.63) and (3.64) have the same poles, we can divide
(3.64) by (3.63) and obtain
73
N +1
f ( s) = h
∏(s − z )
n
n =1
N
∏ ( s − z )
n =1
.
(3.66)
n
Note that the poles of f ( s ) are the zeros of σ ( s ) , i.e., zn . Once the poles of f ( s ) are
obtained, then the zeros of f ( s ) can be calculated from the poles using linear algebra.
Therefore, the Vector Fitting algorithm delicately converts the non-linear polesearching problem into a linear zero calculation problem. Efficient approximations are
obtained using a reasonable number of pole and residue pairs. Furthermore, the
convergence of the Vector Fitting algorithm has proved to very good, i.e., high-order
macromodels have better accuracy than low-order macromodels.
3.2 Model Order Reduction Algorithms Based on Moment
Matching
In addition to the macromodeling techniques that were discussed in the previous
section, there is another category of modeling and simulation methods called the model
order reduction algorithms. The model order reduction algorithms are based on Moment
Matching Techniques (MMT). MMT use a reduced-order function to approximate the
transfer function H(s). Many algorithms can be used in MMT and they can be classified
into two types:
I) Approaches based on explicitly matching the moments to a reduced-order
model.
74
II) Approaches based on implicitly matching the moments to a reduced order
model.
In the first type of approach, there are single moment matching algorithm like the
Asympotic Waveform Evaluation (AWE) method [48] and the multiple moment
matching methods like Complex Frequency Hopping (CFH) [2,49], matrix rational
approximation [2], and Truncated Balance Realization (TBR) [50].
In the second type of approach, indirect moment matching methods are adopted.
These algorithms are based on Krylov-subspace formulations and Congruent
Transformations. These methods includes but are not limited to the Krylov-subspace
formulation, Pade via Lanczos [6] [7], Passive Reduced-Order Interconnect
Macromodeling Algorithm (PRIMA) [8], Integrated Congruent Transformation [9], Split
Congruent Transform [10], and Arnoldi [11].
The basic ideas of the MMT are explained here. The response of a system for a
specified signal can be expressed in the time domain as a sum of exponentially decaying
waves
p
f (t ) = f p (t ) + f np (t ) ≈ ∑ Rα exp( sα t )u (t ) + f np (t ),
(3.67)
α =1
while in the frequency domain, we have
p
Rα
+ Fnp ( s ),
α =1 s − sα
F ( s ) = Fp ( s ) + Fnp ( s ) = ∑
(3.68)
where fnp(t) and Fnp(s) are the non-pole components in the time and frequency domains,
respectively. The u(t) term denotes a unit-step function. The specified signal can be an
75
impulse signal, a step signal or some other type of input signals. These approximations
are called fitting models. The time-domain fitting model involves a superposition of a
series of decaying exponential waves which all turn on at zero time. For interconnects
with finite lengths, all the terms must sum to zero on the far end of the interconnect at
early times to satisfy causality since every wave propagates at a finite velocity. This can
lead to a large number of terms in the macromodel for long interconnects. The accuracy
of the fitting model depends on how many poles are used in (3.68) and whether the set of
poles used in the approximation includes the so-called “dominant poles”.
Zs
+
Zl
Vs
-
Figure 3.3. A simple interconnect model.
A simple interconnect model consists of a transmission line that is driven by a
voltage source with a source impedance, and is terminated in a load (Figure 3.3).
Standard macromodeling techniques sample the circuit response (e.g., admittance matrix
parameters) at a sufficient number of frequency points to properly characterize the
frequency response of the interconnect over the bandwidth of the input waveform. From
these data, we can extract poles and calculate the corresponding residues in order to
express the response in the form of (3.67) and (3.68).
76
3.2.1 Single Moment Matching
3.2.1.1Direct Single Moment Matching
The large number of pole and residue pairs associated with standard
macromodeling techniques will make the simulation very time-intensive. A close
exploration of the poles reveals that although there are large numbers of poles spread
over a broad range of frequencies, only a few of the major poles will dominate in
determining the output waveform. Researchers have proposed that extraction of the
dominant poles and elimination of the excess poles will effectively improve the
simulation efficiency.
The Moment Matching Techniques (MMT) begins with the system transfer
function H(s). H(s) is defined as a ratio of two rational functions:
H ( s) =
P( s)
,
Q( s )
(3.69)
where P(s) and Q(s) are rational functions of s. Partial fraction operations can be applied
to (3.69) to obtain a new form of H(s):
N
rα
,
α =1 s − sα
H ( s ) = hnp + ∑
(3.70)
where rα and sα are the α th pole-residue pair, N is the total number of poles, and h np is
the direct coupling between the input and output signals. The time-domain result is
obtained by applying the inverse Laplace Transform to (3.70), thereby expressing it as
N
h(t ) = hnpδ (t ) + ∑ rα e sα t .
α =1
(3.71)
77
Comparing (3.70) with (3.71), we see that MMTs use a sum of exponential functions to
construct the waveform in the time domain. This is one of the reasons why a large
number of poles are needed to properly model even a simple interconnect.
Next, numerical approximation methods are employed to determine the
coefficients rα and sα . The Asympotic Waveform Evaluation (AWE) method employs a
Pade Approximation at s=0 to model the transfer function [48]. Note that a Pade
Approximation can only approximate a transfer functions with less than 10th order
macromodels. What’s more, as discussed in the previous section, the Pade
Approximation method cannot ensure the stability and passivity of the macromodel.
3.2.1.2 Matrix Rational Approximation
A Matrix Rational Approximation (MRA) method is introduced in [51]. This
method first converts the Telegraphers’ Equations in the matrix-exponential form as
V (ω , l ) 
z V (ω , 0) 
 I (ω , l )  = e  I (ω , 0)  ,




(3.72)
− R(ω ) 
− L(ω ) 
 0
 0
z = A + jω B = 
l + jω 
l.

0 
0 
 −G (ω )
 −C (ω )
(3.73)
For the exponential matrix term e z , a closed-form Pade rational function is
employed as
ez ≈
QN , M ( z )
PN , M ( z )
,
(3.74)
where QN , M ( z ) and PN , M ( z ) are polynomial matrices expressed in terms of closed-form
Pade rational functions defined as
78
( M + N − j )! N !
(− z ) j ,
j = 0 ( M + N )! j !( N − j )!
(3.75)
( M + N − j )! M !
(− z ) j .
j = 0 ( M + N )! j !( M − j )!
(3.76)
N
PN , M ( z ) = ∑
M
QN , M ( z ) = ∑
In order to preserve the passivity of the generated macromodel, M has to be chosen as the
same order as N. By doing so, the exponential matrix in (3.72) can be written as
PN ( z )e A+ sB ≈ QN ( z ),
(3.77)
where
N
PN ( z ) = ∑ pi s i ,
(3.78)
i =0
N
QN ( z ) = ∑ qi s i .
(3.79)
i =0
The MRA method ensures the procedure to obtain the coefficients pi and qi is a passive
operation. Once the coefficients in (3.78) and (3.79) are calculated, (3.72) can be
translated into an ODE and embedded in a circuit simulator with other non-linear
elements.
3.2.2 Multiple Moment Matching
The algorithms of multiple moment matching involve the projection of the system
variables into the Krylov sub-space. Due to page limits, the detailed operation will not be
addressed here. However, a simplified example is given here to compare the Multiple
Moment Matching and Single Moment Matching [50].
79
For example, the Pade Approximation is usually adopted in single Moment
Matching Techniques. The Pade Approximation expands the original function in terms of
a Taylor series or Maclaurin series up to the Mth order as discussed in Section 3.1.3.1.
Obviously, the error bound between the original function and the approximated model is
the next higher order term above the Mth order. Thus, at high frequency points that are
far away from the expansion point, the approximation error is severe, which is illustrated
in Figure 3.4. The solid line in Figure 3.4 shows the original function, while the dashed
line shows the approximation result from the single MMT. The concentric circles show
several approximation model terms with increasing order numbers. Because of the
approximation error of the Taylor series as compared to the original function, the
approximation result deviates from the original function severe after several terms.
f(s)
x
0
Figure 3.4 The single Moment Matching Techniques (MMT) case
s
80
f(s)
x
x
x
0
s
Figure 3.5 The multiple Moment Matching Techniques (MMT) case
The Multiple Moment Matching Techniques approximates the original function at
multiple frequency points as shown in
Figure 3.5. Since multiple expansion points are employed in the approximation, it is
possible to control the behavior of the approximation model so that the model matches
with original function over the entire frequency range.
81
CHAPTER 4
SIMULATION OF LOSSY TRANSMISSION
LINES WITH FREQUENCY INDEPENDENT LINE
PARAMETERS USING SPECIAL FUNCTIONS
In Chapter 3, various techniques for modeling and simulation of transmission lines
were outlined. As discussed in Chapter 3, one important method for transmission line
simulation involves the convolution of the impulse response with the source signal. In
previous work [12,13,15], the authors have derived both the impulse response and a
special case of the unit-step response (i.e., for zero conductance G=0) of lossy
transmission lines with Frequency Independent Line Parameters (FILP). They
demonstrated that these time-domain responses involve Bessel functions of zeroth and
first orders.
In this chapter, we demonstrate that both the impulse response and unit-step
response for lossy transmission lines with FILP can be expressed in terms of Incomplete
Lipshitz-Hankel Integrals (ILHIs). Furthermore, the time-domain responses for the
propagation of complicated signal waveforms (e.g., ramp, exponentially decaying, and
exponentially decaying sine signals) on transmission lines with FILP are derived in terms
of ILHIs. Once these responses are obtained, then the time-intensive convolution
operations that are required for the transient simulation of lossy transmission lines can be
replaced by linear combinations of these various signal responses. In addition, it is no
longer necessary to approximate the propagation function and characteristic admittance
82
using numerical methods such as Pade Approximation or Least-Square approximation for
the simulation of lossy transmission lines with FILP.
In this chapter, we first derive the frequency-domain expressions for lossy
transmission lines with FILP. Then we define the ILHIs and convert the frequencydomain expressions into time-domain expressions involving ILHIs. The expressions
include the far-end voltage and near-end current responses of a single lossy transmission
line with FILP with unit-step signal, ramp signal, and exponential decaying signal inputs.
These expressions are also employed in the next chapter to constitute the time-domain
representations of the transient responses of lossy transmission lines with Frequency
Dependent Line Parameters (FDLP) for various source signal inputs.
The closed form ILHIs expressions are validated by comparing with simulation
results from commercial tools like HSPICE. It is observed that the two types of results
have excellent agreement with each other. Since the closed-form results with ILHIs only
involve rapidly computable special functions, the proposed method is free from
numerical effects like step size control and aliasing.
The organization of this chapter is as follows. First we develop the frequencydomain expressions for the transient simulation of lossy transmission lines with FILP.
Then the ILHIs are introduced and the calculations of the ILHIs are briefly discussed.
Next, the time-domain expressions for lossy transmission lines responses with various
signal source inputs are derived in detail. Finally, comparisons are made between the
results from these closed-forms expressions and commercial simulators like HSPICE.
83
4.1 Frequency-Domain Expressions for the Responses of Lossy
Transmission Lines with FILP
It was shown that for the direct convolution of impulse response method as well as
the recursive convolution methods and the Method of Characteristics (MoC), the key
components for the transient simulation of lossy transmission lines with FILP are the
propagation functions and characteristic impedance. This can be verified by a simple case
of a single lossy transmission line with FILP as shown in Figure 3.3. If a single lossy
transmission line is matched at both the source and the load (i.e., the near- and far-ends),
then there are only two unknown variables i.e., the far-end voltage and the near-end
current. The far-end current and near-end voltage can be derived from the far-end voltage
and near-end current through the V-I characteristics of the source and load impedance.
For a well-terminated transmission line with FILP, the voltage and current on the line at
the location x can be expressed in the frequency domain as
1
E (ω ) exp(−γ x),
2
I ( x , ω ) = V ( x, ω ) / Z 0 ,
V ( x, ω ) =
(4.1)
where
γ = ( R + jω L)(G + jωC ) ,
Z 0 = ( R + jω L) /(G + jωC ) ,
(4.2)
and E (ω ) is the source signal. It is seen from (4.1) that the voltage on the line involves a
positive propagating wave and that the current is related to the voltage by the complex
characteristic impedance. For cases when the transmission line is not well matched, the
84
expressions involving the direct convolution of the impulse response (3.6) and (3.7) are
employed and the results involve both forward and backward propagating waves.
The time-domain solutions for (4.1) can be obtained through the use of Inverse
Fast Fourier Transform (IFFT) techniques or the methods discussed in Chapter 2, i.e.,
numerical convolution, recursive convolution, and various macromodeling methods. Here
a new analytical inverse Fourier Transform method is developed for cases where the
source signal is a unit-step signal, a ramp signal, or an exponential decaying signal.
4.2 Time-Domain Expressions for the Far-End Voltage
Responses of Lossy Transmission Lines with FILP
First, the time-domain expressions for the voltage responses of lossy transmission
lines with FILP are derived. Here we look at three different source excitations.
4.2.1 The Unit-Step Voltage Response
If the source signal is a unit-step signal defined as
0,
e(t ) = u (t ) = 
1,
t<0
,
t≥0
(4.3)
then the frequency-domain expression for the source signal is given as
E (ω ) =
1 1
=
.
s jω
(4.4)
Therefore, based on the definition of the inverse Fourier transform, the time-domain
expression for the voltage at the far-end in (4.1) can be written as
85
v1 (l , t ) =
1
4π
∫
∞
exp  −l ( R + jω L)(G + jωC ) + jωt 
jω
−∞
dω.
(4.5)
In order to carry out the integral in (4.5), the propagation constant is rewritten as
LG + RC   LG − RC 

γ (ω ) = ( R + jω L)(G + jωC ) = j LC  ω − j
 +
 ,
2 LC   2 LC 

2
2
(4.6)
so that (4.5) can be represented as
(
)
2
exp
j
ω
t
−
jd
ω
− ω p2
1
 t 
v1 (l , t ) =
exp  −  u (t − d ) ∫
dω ,
−∞
ω − jα1
j 4π
 2τ 
∞
(4.7)
where the variables and their properties are defined as
d = l LC ,
τ=
LC
,
LG + RC
ωc =
ω p = ωc2 −
d ∈ℜ,
RG
,
LC
τ ∈ℜ,
(4.9)
ωc ∈ℜ,
(4.10)
1
j
LG − RC ,
=
2
4τ
2 LC
α1 = −
LG + RC
1
=−
,
2τ
2 LC
ω =ω−
j
,
2τ
(4.8)
ω p ∈ », Re(ω p ) = 0,
α1 ∈ℜ,
ω ∈ .
(4.11)
(4.12)
(4.13)
As will be shown later, the properties of these variables will help to determine the proper
closed integration contour.
86
4.2.2 The Exponentially Decaying Signal Voltage Response
An exponential decaying signal is defined as
0,
e(t ) = ep(t ) = 
 exp(− s0t ),
t<0
t≥0
(4.14)
,
where s0 is a positive real value. The frequency-domain expression for the exponential
decaying signal is
E (ω ) = EP(ω ) =
1
−j
=
.
s + s0 ω − js0
(4.15)
Another form of the exponentially decaying function is the exponentially decaying sine
or cosine function that can be expressed in the frequency domain as the sum of a pair of
complex conjugate poles and residues, i.e.,
E (ω ) = EP(ω ) =
u + jv
u − jv
+
,
s + ( s0 + jy ) s + ( s0 − jy )
s0 , y, u , v ∈ ℜ, s0 > 0
(4.16)
In the time domain, ep (t ) can be represented as
ep(t ) = (u + jv) exp [ −( s0 + jy )t ] + (u − jv) exp [ −( s0 − jy )t ] , t ≥ 0.
(4.17)
Then e(t ) can be expressed as
0,
e(t ) = ep(t ) = 
 2exp(− s0t )sin( yt + θ ),
t<0
t≥0
,
(4.18)
where
u
 
θ = arctan   , 0 ≤ θ < π .
v
(4.19)
87
The far-end voltage of the transmission line with FILP for the exponentially
decaying signal input can be represented as
v2 (l , t ) =
1
j 4π
∫
∞
−∞
e−l
( R + jω L )( G + jωC )
ω − js0
e jωt dω
(
)
exp jωt − jd ω 2 − ω p 2
1
 t 
=
exp  −  u (t − d ) ∫
dω,
−∞
ω − jα 2
j 4π
 2τ 
∞
(4.20)
where all variables are defined in the same way as in (4.7) to (4.13) except that α 2 is
defined as
α2 = −
1
+ s0 .
2τ
(4.21)
4.2.3 Ramp Signal Voltage Response
The source signal can also be defined as a ramp signal
 0, t < 0

e(t ) = r (t ) =  1
,
 ∆t t , t ≥ 0
(4.22)
where ∆t is the rise time. Then the frequency-domain expression for the source signal is
given as
E (ω ) =
1
1
=−
.
2
∆ts
∆tω 2
(4.23)
The transient response at the far-end of a lossy transmission line with FILP can then be
represented as
88
(
)
1 ∞ exp −l ( R + jω L)(G + jωC ) jωt
e dω
4π∆t ∫−∞
ω2
exp jωt − jd ω 2 − ω p 2
∞
1
 t 
=−
exp  −  u (t − d ) ∫
e jωt dω.
2
−∞
ω
4π∆t
 2τ 
v3 (l , t ) = −
(
)
(4.24)
4.3 Solutions for the Three Key Integrals for the Voltage
Responses
The integrals in (4.7), (4.20), and (4.24) are key integrals for the solutions of
transient responses for lossy transmission lines with FILP. In order to solve these
integrals, we first have to define the Incomplete Lipshitz-Hankel Integrals (ILHIs) of the
first kind [52] as
ζ
Je0 (a, ζ ) = ∫ exp(− ax) J 0 ( x)dx,
0
(4.25)
where J 0 ( x) is a Bessel function of the first kind. It has been shown [53] that Je0 (a, ζ )
can be expressed in integral form as
e − aζ
euζ
Je0 (a, ζ ) =
du.
2π j ∫Γu ( u − a ) u 2 + 1
(4.26)
In order for the integral representation in (4.26) to hold, the inversion contour, Γu , must
satisfy the conditions that Re (u ± j )e j arg(ζ )  > 0 and Re (u − a )e j arg(ζ )  > 0 . The ILHIs
can be efficiently calculated using algorithms developed in [52] and [53], where two
factorial-Neumann series expansions are derived for the ILHIs and are used together with
a Neumann series expansion in an algorithm that efficiently computes the ILHI Je0 (a, ζ )
89
to a user defined number of significant digits. Thus, one can use ILHIs in the same way
as other special functions like the exponential function e− s0t or Bessel functions in a
programming environment.
Comparing (4.26) with (4.7), (4.20), and (4.24), we find that the integrals in (4.7),
(4.20), and (4.24) look very similar to the expressions in (4.26). In fact, once the ILHI is
defined, then the integrals in (4.7), (4.20), and (4.24) can be represented in terms of
ILHIs by using two methods. In [54], a contour integration method is employed to
express the integral in (4.20) in terms of ILHIs and solve for the transient field
distribution in a rectangular waveguide. In [55], a differential equation method is adopted
to represent the integral in (4.20) in terms of ILHIs and solve the problem of an ultrawide-band electromagnetic pulse propagating through a dispersive media. These two
applications are different from each other in terms of the source waveforms, boundary
conditions, and initial conditions, however, both formulations yield similar results. In this
dissertation, we employ the method and results from [55].
Using the results in [55] (see (38)), we can express the exponentially decaying
signal response v2 (l , t ) as
v2 (l , t ) = exp(−
{
t
1
)u (t − d ) e −α 2t cosh(d α 22 + ω p2 ) +
2τ
2ω p t 2 − d 2
}
 (α d − t α 2 + ω 2 )e a+ζ Je (a , ζ ) + (α d + t α 2 + ω 2 )e a−ζ Je (a , ζ )  ,
p
p
2
0
2
2
0
+
−
 2

where
(4.27)
90
a± =
−α 2t ± td α 22 + ω p2
ω p t 2 − td 2
j G R
−
2 C L
ζ = ωp t2 − d 2 =
,
t2 − d 2 ,
α2 = −
a± ∈ »,
(4.28)
ζ ∈ », Re(ζ ) = 0,
(4.29)
1
+ s0 .
2τ
(4.30)
It is observed that the exponentially decaying signal response v2 (l , t ) in (4.20) is very
similar to the expression for the unit-step response v1 (l , t ) in (4.7). Actually, the unit-step
response is a special case of the exponentially decaying signal response where the
decaying factor vanishes as ω0 = 0 , i.e.,
v1 (l , t ) = exp(−
{
t
1
)u (t − d ) e−α1t cosh(d α12 + ω p2 ) +
2τ
2ω p t 2 − d 2
 (α d − t α + ω )e
 1
2
1
2
p
a+ ζ
Je0 (a+ , ζ ) + (α1d + t α + ω )e
2
1
α1 = −
2
p
1
.
2τ
a− ζ
}
(4.31)
Je0 (a− , ζ )  ,

(4.32)
The expression in (4.27) involves a hyperbolic function cosh(d α 22 + ω p2 ) . In
cases where s0 is a large value, d α 22 + ω p2 may be so large that the hyperbolic function
cosh(d α 22 + ω p2 ) results in numerical overflow errors even though ω p2 is a negative
value as defined in (4.11). In order to avoid these overflow errors, the Complementary
Incomplete Lipshitz-Hankel Integrals (CILHIs) are introduced as
ζ
Je0 (a, δ , ζ ) = ∫ exp(− ax)J 0 ( x)dx,
δ
(4.33)
91
where
 ∞; Re(a ) ≥ 0
.
 −∞; Re(a) < 0
δ =
(4.34)
CILHIs and ILHIs are related by the identity [52]
Je0 (a, ζ ) = − Je0 (a, δ , 0) + Je0 (a, δ , ζ ),
(4.35)
where Je0 (a, δ , 0) is a special case of CILHI in [56] (see (6.611.1)). When Re(a ) ≥ 0 and
a ≠ ± j , then
0
Je0 (a, ∞, 0) = ∫ e − at J 0 (t )dt =
−∞
−1
a2 + 1
,
(4.36)
.
(4.37)
and when Re(a ) < 0 ,
∞
Je0 (a, −∞, 0) = ∫ e − at J 0 (t )dt =
0
1
a2 + 1
After substituting (4.36) and (4.37) into (4.35) and multiplying both sides by an
exponential term, we find that
ea±ζ Je0 (a± , ζ ) =
e a± ζ
a +1
2
±
+ e a±ζ Je0 (a± , δ , ζ ).
(4.38)
Next, (4.38) is substituted into (4.27) and the hyperbolic function is expanded in
exponential form to yield
92
)
(
t
1
)u (t − d )  exp −α 2t + d α 22 + ω p2 +
2τ
2
1
1
exp −α 2t − d α 22 + ω p2 +
2
2ω p t 2 − d 2
v2 (l , t ) = exp(−
)
(

 α 2 d − t α 22 + ω p2


(
(
+ α 2d + t α + ω
2
2
)
2
p
 e a+ζ


+ e a+ζ Je0 (a+ , ∞, ζ ) 
 2

 a+ + 1

)
(4.39)
 ea−ζ
  
a−ζ

+ e Je0 (a− , −∞, ζ )    .
 a2 + 1
 
 −
 
Note that the following identity is valid:
a +1 =
2
±
−α 2 d ± t α 22 + ω p2
ωp t2 − d 2
(4.40)
.
After (4.40) is substituted into (4.39), we obtain the following expression for v2 (l , t )
 t
v2 (l , t ) = − exp  −
 2τ
(
1

 u (t − d ) {u ( −a+ζ ) exp(a+ζ ) +

2ω p t 2 − d 2
)
(
)
}
×  α 2 d − t α 22 + ω p2 e a+ζ J e0 (a+ , δ + , ζ ) + α 2 d + t α 22 + ω p2 ea−ζ Je0 (a− , δ − , ζ )  ,


(4.41)
where u ( −a+ζ ) is a unit-step function with a value of 1 when −a+ζ is positive real and a
value of 0.5 when −a+ζ =0. Thus, the term that leads to the overflow error is absorbed by
the CILHIs and the expression for v2 (l , t ) is well behaved.
The expression in (4.41) is made up of two parts. The first part is an exponentially
decaying term, and the second part is an ILHI involving the Bessel function of the first
kind. Note that this result agrees with equation (5) in [26] except that equation (5) in [26]
93
still has an integral over the Bessel functions. By using (4.27) and (4.41), numerical
integrations are no longer necessary in order to obtain an accurate transient response of
lossy transmission lines with FILP.
Next, the ramp response (4.24) has to be carried out in terms of ILHIs. Comparing
(4.24) with the exponential decaying signal response in (4.20), it is observed that the
integral part in (4.24) can be derived from the integral in (4.20) as shown in [29], i.e.,
(
)
)
(
2
2


exp jωt − jd ω 2 − ω p 2 ∂  ∞ exp jωt − jd ω − ω p

dω = lim
dω  , (4.42)
∫−∞
s0 → 0 ∂s  ∫−∞
ω2
ω − jα 2
0 



∞
where α 2 = −1/ 2τ + s0 . Note that the integral within the limit on the Right Hand Side
(RHS) has been obtained as shown in (4.27). In [29], the derivative and limit operations
are performed on the exponential responses of (4.27) to obtain the ramp response
v3 (l , t ) = −
(
e
−
t
2τ
−
u (t − d ) 
d
e 2τ
−
−
+
ω
ω
t
cosh(
d
)
sinh(
d
)

c
c
2
2τωc
2ζ

t
  −d
a′ a ζ
− tωc ) +2 + +
 (
a+ + 1
  2τ
 d
−d
a′ a ζ 
a′ ζ
d
a′ ζ 
+ tωc ) −2 −  J 0 (ζ ) +  ( + tωc ) 2+ + ( − tωc ) 2−  J1 (ζ ) +
2τ
a− + 1 
a+ + 1 2τ
a− + 1 
 2τ

a′ a 
−d
t
d
− tωc )a+′ ζ + ( + tωc ) 2+ +  ea+ζ Je0 (a+ , ζ ) +
)+(
(d +
a+ + 1 
2τωc
2τ
2τ

 

−d
t
d
a′ a 
+ tωc )a+′ ζ + ( − tωc ) 2− −  e a−ζ Je0 (a− , ζ )  ,
)+(
(d −
2τωc
2τ
2τ
a− + 1 

 
where J 0 (ζ ) and J1 (ζ ) are Bessel functions of zeroth and first orders.
(4.43)
94
4.4 Time-Domain Expressions for the Near-End Current
Responses of Lossy Transmission Lines with FILP
The frequency-domain current responses of lossy transmission lines with FILP at
any location x are given in (4.1) and can be expressed as
I ( x , ω ) = V ( x, ω ) / Z 0 =
E (ω ) exp(−γ x)
.
2Z 0
(4.44)
As previously discussed in Chapter 3, the far-end voltage and near-end current are the
two independent variables for solving the lossy transmission line problem. The near-end
current can be approximated by taking the limit of x → 0 in (4.44).
Here the three types of source waveforms that were used for the far-end voltage
response calculations are again used here, i.e., the unit-step signal, the exponential
decaying signal, and the ramp signal.
4.4.1 Unit-Step Signal Current Response
When the unit-step signal is applied to a lossy transmission line with FILP, the
time-domain current response can be derived from (4.44) and (4.4) as
i1 ( x, t ) =
1
4π
=
G
4π
+
C
4π
∫
∞
∫
∞
−∞
−∞
∫
∞
−∞
(
exp − x ( R + jω L)(G + jωC )
jω ( R + jω L) /(G + jωC )
(
exp − x ( R + jω L)(G + jωC )
jω ( R + jω L)(G + jωC )
(
)e
)e
exp − x ( R + jω L)(G + jωC )
( R + jω L)(G + jωC )
jωt
dω
jωt
dω
)e
jωt
d ω.
(4.45)
95
By following a similar procedure as we did in the voltage case, i1 ( x, t ) is divided into two
terms and (4.45) can be written as
i1 ( x, t ) = i1_1 ( x, t ) + i1_ 2 ( x, t ),
i1_1 ( x, t ) =
G exp ( −t / 2τ ) u (t − d )
i1_ 2 ( x, t ) =
j 4π LC
(4.46)
)
(
exp jωt − jd ω 2 − ω p 2
∫−∞ (ω − jα ) ω 2 − ω 2 dω ,
p
1
∞
C exp ( −t / 2τ ) u (t − d )
j 4π L
)
(
exp jωt − jd ω 2 − ω p 2
dω ,
∫−∞
2
ω − ωp2
∞
(4.47)
(4.48)
where all variables are defined in the voltage v1 (l , t ) expression from (4.8) to (4.13).
4.4.2 The Exponentially Decaying Signal Current Response
When the exponential decaying signal is applied to the lossy transmission line
with FILP, the time-domain current response can be derived from (4.44) and (4.15) as
i2 ( x, t ) =
1
4π
(
exp − x ( R + jω L)(G + jωC )
∞
∫ ( jω + s )
−∞
0
)e
jωt
( R + jω L) /(G + jωC )
d ω.
(4.49)
Similarly, i2 ( x, t ) can be represented as a sum of two terms by using partial fractions
i2 ( x, t ) = i2 _1 ( x, t ) + i2 _ 2 ( x, t ),
i2 _1 ( x, t ) =
i2 _ 2 ( x, t ) =
C
4π
∫
∞
( R + jω L)(G + jωC )
−∞
( G − s0C )
4π
(
exp − x ( R + jω L)(G + jωC )
∞
(
(4.50)
)e
jω t
exp − x ( R + jω L)(G + jωC )
∫ ( jω + s )
−∞
0
dω ,
)e
( R + jω L)(G + jωC )
jωt
(4.51)
d ω.
(4.52)
96
Then, the two integrals in (4.51) and (4.52) can be simplified as
i2 _1 ( x, t ) =
i2 _ 2
G exp ( −t / 2τ ) u (t − d )
j 4π LC
)
(
exp jωt − jd ω 2 − ω p 2
∫−∞ (ω − jα ) ω 2 − ω 2 dω,
p
2
∞
( G − s0C ) exp ( −t / 2τ ) u (t − d )
( x, t ) =
j 4π LC
(4.53)
)
(
exp jωt − jd ω 2 − ω p 2
d ω. (4.54)
∫−∞
2
2
ω −ωp
∞
4.4.3 The Ramp Signal Current Response
In cases where the source signal is a ramp signal as defined in (4.22) and (4.23),
the time-domain current response at the location x can be represented as
i3 ( x, t ) = −
=−
(
)
1 ∞ exp − x ( R + jω L)(G + jωC ) + jωt
dω
4π∆t ∫−∞
ω 2 ( R + jω L) /(G + jωC )
1 ∞
4π∆t ∫−∞
(
exp − x ( R + jω L)(G + jωC ) + jωt
ω
2
)
(4.55)
G + jωC
dω.
R + jω L
As will be shown in the next section, the integral in (4.55) can be expressed as a closedform representation in terms of ILHIs.
4.4.4 Solutions for the Key Integrals for the Current Responses
The integrals in (4.47), (4.48), (4.53), (4.54), and (4.55) are the keys to the
solutions of current responses for lossy transmission lines with FILP. Fortunately, these
integrals are either solved in [34], or can be expressed in closed-form ILHIs as in [55].
The first integral to be discussed is [34]
97
)
(
exp jωt − jd ω 2 − ω p 2
d ω = 2π jJ 0 ω p t 2 − d 2 u (t − d ),
∫−∞
2
2
ω − ωp
∞
)
(
(4.56)
where u (t ) is defined as
 0, t < 0

u (t ) = 0.5, t = 0 .
1,
t>0

(4.57)
Thus, (4.48) and (4.54) can be expressed as
i1_ 2 ( x, t ) =
=
=
i2 _ 2 ( x, t ) =
=
=
C exp ( −t / 2τ ) u (t − d )
j 4π L
C exp ( −t / 2τ ) u (t − d )
j 4π L
)
(
exp jωt − jd ω 2 − ω p 2
dω
∫−∞
2
2
ω − ωp
∞
(
)
2π jJ 0 ω p t 2 − d 2 u (t − d )
(4.58)
)
(
C
exp ( −t / 2τ ) J 0 ω p t 2 − d 2 u (t − d ),
2 L
( G − s0C ) exp ( −t / 2τ ) u (t − d )
j 4π LC
( G − s0C ) exp ( −t / 2τ ) u (t − d ) 2π jJ ω
0( p
j 4π LC
( G − s0C ) exp
2 LC
( −t / 2τ ) J 0 (ω p
)
(
exp jωt − jd ω 2 − ω p 2
dω
∫−∞
2
2
ω − ωp
∞
)
t 2 − d 2 u (t − d )
)
t 2 − d 2 u (t − d ).
Next, the following integral expression was derived in [55]
(4.59)
98
)
(
exp jωt − jd ω 2 − ω p 2
u (t − d ) −α t
2
2
∫−∞ (ω − jα ) ω 2 − ω 2 dω = 2π α 2 + ω 2 e 2 sinh(d α1 + ω p ) +
1
1
p
p
∞
{
1
2ω p
(α d − t α 2 + ω 2 )e a+ζ Je (a , ζ )
+
1
1
0
p
t −d 
2
2
(4.60)
}
+ (α1d + t α12 + ω p2 )e a−ζ Je0 (a− , ζ )  .

By substituting (4.60) into (4.47), we obtain
i1_1 ( x, t ) =
=
G exp ( −t / 2τ ) u (t − d )
j 4π LC
)
(
exp jωt − jd ω 2 − ω p 2
∫−∞ (ω − jα ) ω 2 − ω 2 dω
1
p
∞
G exp ( −t / 2τ ) u (t − d )
2π
j 4π LC
α +ω
2
1
2
p
{e
−α1t
sinh(d α12 + ω p2 ) +
1
2ω p
(α d − t α 2 + ω 2 )e a+ζ Je (a , ζ ) +
p
1
1
0
+
t −d 
2
2
}
(α1d + t α12 + ω 2p )ea−ζ Je0 (a− , ζ )  u (t − d )

G exp ( −t / 2τ ) u (t − d ) −α1t
e sinh(d α12 + ω p2 ) +
=
2
2
2 LC −α1 − ω p
(4.61)
{
1
2ω p
(α d − t α 2 + ω 2 )e a+ζ Je (a , ζ ) +
p
+
1
1
0
t −d 
2
2
}
(α1d + t α12 + ω p2 )ea−ζ Je0 (a− , ζ )  .

The expression for i2 _1 ( x, t ) is more complicated than i1_1 ( x, t ) because the
definition for α 2 in (4.54) is α 2 = −1/ 2τ + s0 while α1 = −1/ 2τ . Note that s0 can be a
large real or complex value. When s0 is a small real value, i2 _1 ( x, t ) can be expressed as
99
i2 _1 ( x, t ) =
G exp ( −t / 2τ ) u (t − d )
2 LC −α − ω
2
2
2
p
{e
−α 2 t
sinh(d α 22 + ω p2 ) +
1
2ω p
(α d − t α 2 + ω 2 )e a+ζ Je (a , ζ ) +
+
2
2
0
p
t −d 
2
(4.62)
2
}
(α 2 d + t α 22 + ω p2 )e a−ζ Je0 (a− , ζ )  .

However, when s0 is large, the hyperbolic function sinh(d α 22 + ω p2 ) will result in
numerical overflow errors. Therefore, just like the case for the exponentially decaying
voltage signal response, CILHIs have to be introduced to absorb the exponentially
growing terms in the hyperbolic function. Following similar procedure as were employed
in the voltage response case, the current response can be expressed as
i2 _1 ( x, t ) =
G exp ( −t / 2τ ) u (t − d )
2 LC −α 22 − ω p2
(
{u ( −a ζ ) exp(a ζ ) +
+
+
)
1
2ω p t 2 − d 2
×  α 2 d − t α 22 + ω p2 e a+ζ J e0 (a+ , δ + , ζ ) +

(α d + t
2
)
(4.63)
}
α 22 + ω 2p e a ζ Je0 (a− , δ − , ζ )  .
−

Finally, the integral in (4.55) has to be calculated. It is observed that the
derivative relation in (4.42) also holds for the current response, i.e.,
(
)
(
)
2
2


exp jωt − jd ω 2 − ω p 2 ∂  ∞ exp jωt − jd ω − ω p

dω = lim
dω  , (4.64)
∫−∞
s0 → 0 ∂s  ∫−∞
ω 2 ω 2 − ωp2
0 
(ω − jα 2 ) ω 2 − ω p 2



∞
where α 2 = −1/ 2τ + s0 . By taking the derivative and limit operations on s0 in (4.62), we
obtain the expressions for the ramp signal current response as [29]
100
i3 ( x, t ) = −
=−
(
1 ∞ exp − x ( R + jω L)(G + jωC ) + jωt
4π∆t ∫−∞
ω2
)
G + jωC
dω
R + jω L
u (t − d )  2 C / L
G
2Gt
− 3
+
) sinh(d ωc ) +
(
4∆t  ωc
τωc LC ωc LC
−
 d
Gd
Ge 2τ
a+′ a+ζ
cosh(
d
ω
)
+
(
+
t
ω
)

c
c

2
a+2 + 1
τωc LC
ωcω p LC (t 2 − d 2 )   2τ
t
+(
 d
−d
a′ a ζ 
a′ ζ
d
+ tωc ) −2 −  J 0 (ζ ) − ( + tωc ) 2+ − ( − tωc )
2τ
a− + 1 
a+ + 1 2τ
 2τ
 
−G
a−′ ζ 
(
)
J
ζ
 + 
 1
2

a− + 1 
  ωcω p LC
+(

−d
t
)+(
− tωc )a+′ ζ
(d +
2τωc
2τ


d
a′ a 
C/L d
G
d
( + tωc ) +
(
)
ω
+ tωc ) 2+ +  −
+
t
c 

2τ
a+ + 1 ωcω p 2τ
2τωc3ω p LC 2τ



G
t
(d −
)+
e a+ζ Je0 (a+ , ζ ) + 

2
2
 ω ω LC 
2
τω
t −d
c
c
p

e
(
− 2tτ
−d
d
a′ a 
C/L d
+ tωc )a−′ ζ + ( − tωc ) 2− −  +
( − tωc ) −
2τ
2τ
a− + 1 ωcω p 2τ
G
2τωc3ω p
(4.65)
 e− 2tτ

d
( − tωc ) 
e a−ζ Je0 (a− , ζ )  ,
 t2 − d 2
LC 2τ


where all variables are defined in (4.8) to (4.13).
4.5 Numerical Validations of the Voltage and Current
Responses
The complicated expressions in sections 4.1 to 4.4 can be validated by some
examples for given RLGC parameters and the input waveforms.
101
W
t
εr=4.4
T
Figure 4.1 The cross-section of a microstrip line with W=5µm, t=2µm, T=10µm,
l=5cm, and εr=4.4.
Here an example is given to model the transient response of a line with FILP for
various source waveforms. We define a 5cm lossy microstrip line with the dimensions
shown in Figure 4.1 and employ the FILP R = 1720.3Ω / m , L = 485.37 nH / m ,
C = 107.13 pF / m , and G = 0 .
4.5.1 Definitions for the Source Signals
Next, we define some source waveforms, i.e., a triangle source signal and a
trapezoidal source signal. The triangle source signal waveform is shown in Figure 4.2.
v in (t )
1
0
∆t
2 ∆t
t
Figure 4.2 Time-domain triangle input
102
v in (t )
1
∆t
0
T + ∆t
T + 2∆t
t
Figure 4.3 Time-domain trapezoidal input
The time- and frequency-domain expressions for the triangle source signal can be
represented as
vin (t ) =
Vin (ω ) = −
1
2
1
t − (t − ∆t ) + (t − 2∆t ),
∆t
∆t
∆t
1
2
1
+
exp(− jω∆t ) −
exp(−2 jω∆t ).
2
2
∆tω
∆tω
∆tω 2
(4.66)
(4.67)
The trapezoidal source signal waveform is shown in Figure 4.3. The time-domain
and frequency-domain expressions for the trapezoidal input have the forms
vin (t ) =
Vin (ω ) = −
1
1
1
1
t − (t − ∆t ) − (t − ∆t − T ) + (t − 2∆t − T ),
∆t
∆t
∆t
∆t
(4.68)
1
exp(− jω∆t ) exp(− jω∆t − jωT ) exp(−2 jω∆t − jωT )
+
+
−
. (4.69)
2
∆tω
∆tω 2
∆tω 2
∆tω 2
Note that in the time domain, the triangle and trapezoidal inputs are linear
combinations of ramp signals with various time delays. Since a lossy interconnect is a
Time Invariant (TI) system, the triangle and trapezoidal responses are linear
combinations of ramp responses. Fortunately, the ramp response of a lossy interconnect
has been previously represented in terms of (4.27) and (4.41). Therefore, by using the
103
expressions in (4.27) and (4.41), it is then straightforward to obtain the time-domain
triangle and trapezoidal responses.
4.5.2 Validation of the Voltage Responses
We first calculate the transient responses of the line using (4.27), (4.31), and
(4.43). Note that the results in (4.31) model the unit-step signal voltage response of the
line, the results in (4.43) model the ramp signal voltage response of the line, and the
results in (4.27) model the exponentially decaying signal response of the line. These
responses are also simulated using HSPICE where the W-element constant RLGC lossy
transmission line model, as discussed in Chapter 3, is employed. In order to make the two
results comparable, we configure the SPICE simulation scheme as shown in Figure 4.4.
In order to reduce the effects of the mismatch at the far end of the line in the HSPICE
simulations, we attach a very long transmission line T8 (i.e. 1m) with the same line
parameters to the line under test T7. We also only check the transient result at the far end
of the 5cm line prior to the moment when the wave bounces back from the long 1m line.
A short triangle pulse with a rise time of 10ps is input into the above transmission line
with matched loads of R1=R2=70 Ω at both ends. The voltage at the far end of line T7,
i.e., V3, is then observed.
The results of the unit-step voltage response from the closed-form model in (4.31)
are displayed as the solid line and the HSPICE simulation results using the frequencyindependent lossy line model are displayed as the dotted line in Figure 4.5. Note the
excellent agreement between the two results.
104
Next, the Triangle Impulse Responses (TIR) from the two methods are obtained
and compared in Figure 4.6. According to the superposition principle, the TIR can be
obtained through the combination of the ramp signal responses as described in (4.66) and
(4.67). Here we set the slope of the ramp signal to 1/ ∆t = 1e11 . It is observed in Figure
4.6 that ILHI results agree with HSPICE W-element result fairly well. However, at the
peak of the TIR, the HSPICE W-element result has a round peak while the ILHI result
has a sharp peak. This is because the numerical approximation approach employed in the
HSPICE W-element method causes a loss of the high frequency components.
Next, the exponentially decaying signal responses of the line are calculated and
compared as shown in Figure 4.7. Here we assume that the pole in (4.30) is 1e9 in the
frequency domain, which corresponds to a time constant of 1ns in the time domain. Also
the expressions involving CILHIs (4.41) are employed. Once again, excellent agreement
is observed for the two results.
Finally, the exponentially decaying sine signal voltage responses are calculated
using both the closed-form CILHI expressions in (4.41) and HSPICE W-element with
FILP. In the closed-form CILHI method, a pair of complex conjugate poles and residues
are employed as
poles = 3.456357423117727e11 ± 2.096223914009156e12,
residues = 2.061598165353282e10 ± 1.267057502262814e11.
By using the method described in Section 4.2.2 , the poles and residues combine in the
time domain are normalized and form a decaying sine function
105
vin = 2exp ( −t 3.456357423117727e11) sin ( t 3.336243977419973e11) .
The output voltage results are shown in Figure 4.8. It is observed that the two methods
agree with each other very well.
As has been demonstrated, the results from the two methods agree very well with
each other for all four different signal inputs. Furthermore, the closed-form ILHI method
is free from numerical issues like high-frequency truncation, simulation step size control
in numerical integrations, or aliasing as in IFFT techniques. As an example, an enlarged
part of Figure 4.7, between 0.35ns and 0.38ns, is shown in Figure 4.9. Note that the
application of numerical approximations in HSPICE causes the truncation of the highfrequency components. In SPICE-type simulators, the Local Truncation Error (LTE) and
Global Truncation Error (GTE) control the transient simulation in terms of step size and
precision. As long as the high-frequency component truncation errors are less than a
default or pre-defined LTE and GTE, the current step simulation result is accepted as a
correct one. Otherwise, a smaller step size is chosen to reduce the truncation error. This
effect yields the rounded results in Figure 4.9. To get better results, a smaller simulation
step has to be applied in HSPICE at the cost of simulation time and memory. In contrast,
the closed-form expressions involving ILHIs and CILHIs don’t have any algorithms for
controlling the simulation step size. However, at each simulation step, since the
simulation result is expressed in closed form, the precision of the simulation is controlled
totally by the calculation of ILHIs and CILHIs. The work in [52] and [53] employed
contour integration techniques to derive the Bessel series expansions for the ILHIs and
CILHIs. The algorithms in [52] can be used to choose the expansion that provides the
106
optimal efficiency for a user’s specified number of significant digits of accuracy. Thus,
the error in the closed-form expressions is determined by the round-off error in the
calculations of the ILHIs and CILHIs. It has been observed that the close-form
expressions involving ILHIs and CILHIs produce very accurate time-domain simulation
results as shown in Figure 4.5 to Figure 4.9.
R2
V5
Iin
T7
LOSSY
V3
T8
R1
LOSSY
Vout
LEN = 0.05
LEN = 1
Figure 4.4The SPICE simulation scheme for the transmission line with FILP
Figure 4.5 Comparison of the unit-step responses of the FILP between HSPICE and the
closed-form ILHI results.
107
Figure 4.6 Comparison of the TIR of the FILP lossy line between HSPICE and the
closed-form ILHI results
Figure 4.7 Comparison of the exponential decaying response of the frequencyindependent lossy line between HSPICE and the closed-form ILHI results
108
Figure 4.8 Comparison of the exponential decaying sine response of the frequencyindependent lossy line between HSPICE and the closed-form ILHI results
Figure 4.9 An enlarged part of Figure 4.7 showing the numerical issues in the HSPICE
results
109
4.5.3 Validation of the Current Responses
The near-end current responses, as discussed in Section 4.4 , are now checked
against SPICE simulation results. Given the example shown in Figure 4.1, we calculated
the near-end, unit-step signal current response based on equations (4.47) (4.58), and
(4.61), we also found the near-end, exponentially decaying signal current responses based
on equations (4.50), (4.59), (4.62), and (4.63), and the near-end, ramp signal current
responses based on (4.65). The SPICE simulation scheme in Figure 4.4 is also employed
to generate comparable simulation results and the near-end current is denote as Iin in
Figure 4.4.
First, the near-end unit-step signal current response Iin from the closed-form ILHI
expressions and HSPICE W-element are plotted in Figure 4.10. It is observed in Figure
4.10 that the two results agree with each other very well.
110
Figure 4.10 Comparison of the unit-step signal current response of the frequencyindependent lossy line between HSPICE and the closed-form ILHI results
Figure 4.11 Comparison of ramp signal current response of the frequency-independent
lossy line between HSPICE and the closed-form ILHI results
111
Next, the near-end, ramp signal current results from the closed-form ILHI
expressions and HSPICE W-element are plotted in Figure 4.11. Here the slope of the
ramp signal is set to 1/ ∆t = 1e11. Again, the two results agree with each other very well.
Finally, the near-end exponentially decaying signal current response results from
the closed-form ILHI expression and HSPICE W-element are plotted as Figure 4.12.
Note that the three cases that are labeled as CASE A, CASE B, and CASE C in Figure
4.12 correspond to different poles in the exponentially decaying signal, i.e., 1e6, 1e8, and
1e9, respectively. Since all three cases are the simulation results for the near-end current
response, it is hard to distinguish them from each other. Thus, CASE B and CASE C, i.e.,
the exponentially decaying signal with poles of 1e8 and 1e9, are artificially delayed with
1ns and 2ns, respectively, from CASE A for a better view.
Since both ILHIs and CILHIs are introduced in the exponentially decaying signal
response, both ILHIs and CILHIs are used in this example. In Figure 4.12, CASE A with
pole of 1e6 is calculated with the ILHI expression in (4.62), while CASE B and C with
poles 1e8 and 1e9 are calculated with the CILHI expression in (4.63). It is observed that
for all three cases, the closed-form expressions involving ILHIs and CILHIs agree very
well with HSPICE simulation results.
112
Figure 4.12 Comparison of the exponential decaying signal current responses of the
frequency-independent lossy line between HSPICE and the closed-form ILHI results for
three cases
In conclusion, we have successfully derived closed-form expressions involving
ILHIs and CILHIs for the transient voltage and current responses for the simulation of
lossy transmission lines with FILP. Several forms of source signals, i.e., the unit-step, the
ramp, and exponentially decaying signals, are used as the inputs to lossy transmission
lines with FILP. These closed-form expressions are checked against the commercial
simulation tool of HSPICE with the W-element model. Examples are given to show that
113
the closed-form results agree very well with the HSPICE W-element model. Furthermore,
the closed-form results are free from aliasing and numerical truncations.
In the next chapter, it will be demonstrated that these closed-form expressions
involving ILHIs and CILHIs can also be used to generate a Dispersive Hybrid Phase-Pole
Macromodel (DHPPM) for the simulation of lossy transmission lines with FrequencyDependent Line Parameters (FDLP).
114
CHAPTER 5
MODELING AND SIMULATION OF LOSSY
TRANSMISSION LINES WITH FREQUENCY
DEPENDENT LINE PARAMETERS USING DISPERSIVE
HYBRID PHASE-POLE MACROMODELS
In Chapter 4, several closed-form expressions involving Incomplete LipshitzHankel Integrals (ILHIs) and Complementary Incomplete Lipshitz-Hankel Integrals
(CILHIs) are used in the simulation of lossy transmission lines with Frequency
Independent Line Parameters (FILP). The simulation results are compared with
commercial simulation tools like HSPICE. However, since frequency-dependent RLGC
parameters must be employed to correctly model skin effects and dielectric losses for
high-performance interconnects, the simulations of lossy transmission lines with
Frequency Dependent Line Parameters (FDLP) are more desirable for electronic system
designs. As discussed in Chapter 2, lossy transmission lines with FDLP have to be first
properly modeled before simulation.
In this chapter, we first study the behaviors of various lossy interconnects that are
characterized by FDLP. We then develop a frequency-domain Dispersive Hybrid PhasePole Macromodel (DHPPM) for such lines, which consists of a constant RLGC
propagation function multiplied by a residue series. The basic idea is to first extract the
dominant physical phenomenology by using a propagation function in the frequency
domain that is modeled by Frequency Independent Line Parameters (FILP). A rational
115
function approximation is then used to account for the remaining effects of FDLP lines. It
is desired that macromodels for lossy transmission lines with FDLP not only have high
accuracy and good efficiency, but also satisfy the stability, causality, and passivity
requirements. Next, the properties of the DHPPM, i.e., stability, causality, and passivity,
are discussed and the DHPPM method is proved to be a stable and causal macromodel.
The passivity of the DHPPM can only be checked after macromodeling.
By using a partial fraction expansion and analytically evaluating the required
inverse Fourier transform integrals, the time-domain DHPPM can be decomposed as a
sum of canonical transient responses for lines with FILP for various excitations (e.g.,
trapezoidal and unit step). These canonical transient responses are then expressed
analytically as closed-form expressions involving ILHIs and CILHIs of the first kind and
Bessel functions. The closed-form expressions for these canonical responses were
previously validated by comparing with simulation results from commercial tools like
HSPICE in Chapter 4.
Next, the DHPPM method is extended from single interconnect structures to
coupled interconnect structures to perform transient simulations for various input
waveforms such as trapezoidal and triangle source signals. Comparisons between the
DHPPM results and the results produced by commercial simulation tools like HSPICE
and a numerical Inverse Fast Fourier Transform (IFFT) show that the DHPPM results are
very accurate.
116
5.1 Analysis of the Properties of Lossy Transmission Lines
with FDLP
Modeling lossy interconnects with Frequency Dependent Line Parameters (FDLP)
has become more important as electronic system clock frequencies approach multiple
GHz, with rise and fall times shrinking to less than 0.1ns. For such cases, harmonic
components with frequencies up to 100GHz need to be taken into account in a signal
integrity analysis [26]. In a typical interconnect modeling and simulation tool, the
interconnects are often modeled by transmission lines with RLGC parameters, i.e., perunit-length resistance, inductance, conductance and capacitance matrices, respectively.
Note that all of these matrices can involve FDLP in high-speed interconnect applications.
At frequencies above a few gigahertz, the frequency-dependent effects of the RLGC
parameters are very important. For example, the resistance of on-chip, lossy interconnects
can be thousands of ohms per cm due to the small cross sections of the interconnects.
Another trend is the increasing lengths of the lossy interconnects for both on-chip and
off-chip structures, which leads to more delay, dispersion, and decay on the
interconnects. While transmission line RLGC parameters are typically characterized in
the frequency domain, it has been both a desire and a challenge to properly model
interconnects in the time domain for practical system designs, where a large number of
interconnects are used to connect nonlinear devices like transistors.
As discussed in Chapter 2, one of the reasons for the frequency-dependent
behaviors of RLGC parameter is the skin effect, which dictates that the resistance of the
117
transmission line increases at the rate of the square root of the frequency at high
frequencies. The skin effect results from the crowding of the electrical current at the
surface of the transmission line, which also leads to a decreasing inductance of the
transmission line with respect to frequency.
For on-chip interconnects, the signal propagation modes on the interconnects have
changed from the previous “slow wave mode” into the “quasi-TEM mode” or the “skineffect mode”[57]. Even though frequency-dependent R(f) and L(f) and constant C (i.e.,
R(f)L(f)C) equivalent networks have been the dominant models in recent years, substrate
losses and the corresponding frequency-dependent G(f) parameters cannot be neglected
in many applications [26,57,58].
For off-chip interconnects, the high-frequency
properties of the lossy materials (e.g. FR-4) make the inclusion of frequency-dependent
G(f) parameters an important factor in signal integrity analysis. What’s more, recent
studies [32] have revealed the close inter-relationships between the RLGC parameters. It
has been shown that R(f) and L(f), as well as G(f) and C(f), are related through the
Hilbert Transform, and the correct relationships between these pairs of parameters is
required to satisfied the causality principle. These relationships are called the KramerKronig conditions [31]. Thus, mathematical operations such as interpolation of the FDLP,
may produce uncausal results if such operations cannot be shown to meet the causality
requirements. Based on these constraints, one can conclude that only lossy transmission
lines with FDLP in a tabular format that satisfy the causality requirements will produce
accurate simulation results.
118
Numerous algorithms have been developed to model lossy transmission line
structures that are characterized by FDLP [12,13,15-18,36,59-62]. In the earliest
Simulation Program with Integrated Circuits Emphasis (SPICE) simulation tools, lossy
transmission lines were sectioned into many pieces along the length of the interconnect
and each piece was represented by a FILP model, i.e. frequency-independent RLGC
circuit elements. This method obviously suffers from the large number of required circuit
nodes and the correspondingly low simulation efficiency. In order to overcome these
problems, the impulse response of the uniform transmission line with FILP is first
represented analytically in terms of a modified Bessel function in [13]. Then a
convolution algorithm is applied to find the transient response for various source
waveform excitations. The convolution method doesn’t require the sub-sectioning of the
transmission line and avoids introducing extra nodes in the transmission line model in a
SPICE-type simulator. Unfortunately, the convolution method is time consuming because
at each simulation time step, the responses at previous time steps and the source
waveform have to be recalled, thereby leading to a computational complexity of O(t 2 ) ,
where t is the simulation time. In addition, this procedure is limited to lossy transmission
lines with FILP and cannot be applied to transmission lines with FDLP because a closedform representation doesn’t exist for the impulse response for transmission lines with
FDLP. Thus, macromodels are usually developed for the impulse response prior to
applying convolutions [12,13,15-18,36,59-62].
As an example, two transient responses for a transmission line with FDLP are
shown in Figure 5.1. A triangular source with 10ps rise and fall times is input into a 5cm
119
lossy Multiple Chip Module (MCM) line with the FDLP listed in Table 1 in [29]. An
Inverse Fast Fourier Transform (IFFT) method is first applied to the transmission line
with the FDLP model to calculate the transient responses, i.e., the solid line in Figure 5.1.
Then the R (ω ) and L (ω ) line parameters at 2GHz in Table 1 in [29] are used together
with constant C and G line parameters to form a transmission line for the FILP model.
The transient response of this FILP model is plotted as the dashed line in Figure 5.1. The
solid line shows the response for the FDLP model shown in Table 1 in [29] and the
dashed line is the response for a FILP model where the parameter at 2 GHz are
employed. Since the two responses are completely different in terms of delays,
dispersion, and decay, FDLP models are necessary if one desires accurate time-domain
lossy transmission line simulations.
Figure 5.1 The time-domain output waveforms obtained for a MCM line for a 10ps
triangle impulse input.
120
The Method of Characteristics (MoC) [14], which features the extraction of linepropagation delay from the propagation function and rational function approximation of
the remaining residue series, was first introduced to efficiently model lossless
transmission lines where only constant LC parameters exist. Later, the MoC was
successfully extended to frequency-dependent lossy line simulations [15,32,59]. In [15], a
difference approximation followed by indirect numerical integration is used to generate
state space models for a delayless propagation function and the characteristic impedance
in order to simulate lossy transmission lines with FDLP. This method is very successful
at low frequencies. However, at high frequencies, where the line parameters change a lot,
numerical effects, e.g. ripples at late time, are observed in the simulation of FDLP lines.
These effects are assumed to be associated with the limited capability of the
approximation to model the rapidly decaying delayless propagation function at high
frequencies. In [32,60], the MoC method is discussed in details and a lumped-element
model involving a lossless transmission line is suggested to model transmission lines with
FDLP over a wide range of frequencies. The idea of extracting the propagation delay was
even adopted in the classical macromodel method, e.g., in [61].
In all MoC implementations, the extraction of line-propagation delay is the first
step in the method. There are several advantages associated with extracting the
propagation delay. First, the extracted frequency-domain phase delay can be easily
converted to a pure delay term in the time domain. Since all electromagnetic waves
propagate on interconnects at a finite speed, the time between the excitation of the signal
at the source end and the appearance of the signal at the load end is called the time of
121
flight. Compared with classical macromodels, where rational function approximations are
directly applied to the propagation function, MoC-based methods need far fewer poles
and residues because a large number of poles and residues are used in classical
macromodels to cancel each other in order to realize a zero output state at the far end of
the line before the time of flight. On the other hand, in MoC-based methods, all the poles
and residues are used to model the line response after the time of flight. The second
advantage of the MoC method is that the causality requirement is guaranteed. Causality is
realized in MoC methods by enforcing the output at the far end of the line to be zero
before the time of flight.
MoC-based methods work well for lossless lines and low-loss lines. However, for
high-loss lines, a large number of poles and residues are still needed to model the rapid
decay and dispersion of the propagation function at high frequencies. Therefore, in this
dissertation, an improvement over the classical MoC is suggested for modeling
propagation functions for high-loss lines more efficiently. The organization of this
chapter is as follows. We first briefly discuss the propagation function for interconnects
with FDLP. Next, we develop a frequency-domain Dispersive Hybrid Phase Pole
Macromodel (DHPPM) for lossy lines. An inverse Fourier transform is then applied in
order to find the time-domain DHPPM. Contour integration techniques are used to obtain
closed-form representations for the required time-domain canonical integrals. Finally the
time-domain DHPPM is used to simulate the transient responses of lossy FDLP
interconnects. Different source waveforms, such as triangle and trapezoidal source, are
122
used as inputs. Moreover, the DHPPM model results are compared with those produced
by commercial simulation tools such as the latest version of HSPICE.
5.2 Development of a Frequency-Domain DHPPM for
Frequency-Dependent Lossy Interconnects
The frequency-domain interconnect port voltages and currents should satisfy
Telegrapher’s Equations, i.e.,
∂V (ω , z )
= − R(ω ) I ( z ) − jω L(ω ) I ( z ),
∂z
(5.1)
∂I (ω , z )
= −G (ω )V ( z ) − jωC (ω )V ( z ),
∂z
(5.2)
where R(ω ), L(ω ) , G (ω ) , and C (ω ) are the frequency-dependent per-unit-length
resistance, inductance, conductance, and capacitance matrices, V (ω , z ) and I (ω , z ) are
the voltage and current on the line at the location z , and ω is the angular frequency. At
the location of the far end of the line where z = l , (5.1) and (5.2) can be rewritten in
matrix-exponential form as
V (ω , l ) 
V (ω , 0) 
 I (ω , l )  = exp(−γ (ω )l )  I (ω , 0)  ,




G (ω ) 
L(ω ) 
 0
 0
γ (ω ) = 
.
+ jω 

0 
0 
 R(ω )
C (ω )
(5.3)
We start with the single FDLP case, where
γ (ω ) = α + j β = ( R(ω ) + jω L(ω ))(G (ω ) + jωC (ω )).
(5.4)
123
Since the exponential term in (5.3) doesn’t have a closed-form time-domain expression
even for the single FDLP line case, it is hard to find a time-domain relationship between
the source and load voltages or currents, especially when the source signal V (ω , 0) or
I (ω , 0) is a the complicated waveform like a trapezoidal signal.
Macromodeling techniques are widely used to properly model the propagation
function [12,13,15-18,36,59-62]. The general goal of macromodeling is to replace the
complex electromagnetic model with a reduced-order model, while maintaining the
characteristics of the system. Classical reduced-order models (e.g. UACAPRE [45,46]),
can be applied to the system transfer function or admittance parameters, and are written
in the form
M
RSclassical = ∑
α =1
Rα
+ Q ≈ exp [ −lγ ( jω )]
jω − sα
(5.5)
where Q represents the non-pole components in the frequency domain, Rα and sα are the
poles and residues of the system, l is the length of the line, and exp [ −lγ (ω )] is the
complex propagation function defined in (5.3).
In [17,22], a Hybrid Phase Pole Macromodel (HPPM) was introduced to provide a
reduced order model for complex interconnect structures. The form of the HPPM is
similar to the classical macromodel except that a propagation delay factor τ is included
with the system poles and residues. By factoring out the propagation delay, the HPPM
significantly reduces the order of the macromodel and also guarantees the causality of the
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macromodel. In practice, the propagation delay is best captured by using a lossless line
propagation factor with parameters L(∞), C (∞) , i.e.
exp [ −lγ (ω ) ] ≈ exp  − jlω L(∞)C (∞)  RS HPPM ,
(5.6)
where
M
RS HPPM = ∑
α =1
Rα
+ Q.
jω − sα
(5.7)
In the above expression, the extracted propagation term models most of the effects of the
propagation delay on low-loss transmission lines, so the pole and residue pairs mainly
have to account for the system poles and residues.
In order to better handle moderate to high-loss interconnects, we propose a
DHPPM with the following expression in the frequency domain [62],
exp [ −lγ (ω ) ] ≈ exp  −l

[ R(0) + jω L(∞)][G (0) + jωC (∞)]  RS DHPPM ,
M
RS DHPPM = ∑
α =1
Rα
+ Q,
jω − sα
(5.8)
(5.9)
where R (0), G (0), L(∞ ), and C (∞) are the RLGC parameters at dc and the highest
frequency values, respectively. The poles and residues in the DHPPM are obtained by the
Vector Fitting algorithm [47,63,64]. In the HPPM and other MoC based methods, it has
been shown that extraction of a lossless propagation function can properly model the
propagation delay of the signal and satisfy the causality requirements. In the DHPPM in
(5.8), we also include the DC resistance and conductance in the extracted propagation
125
function so that part of the loss and dispersion in the propagation function is covered by
the exponential term in (5.8).
Note that R and L parameters, and G and C, represent the real and imaginary parts
of the serial line impedance as Z (ω ) = R (ω ) + jω L(ω ) and the parallel line admittance as
Y (ω ) = G (ω ) + jωC (ω ) , respectively. As discussed in [32], for a real causal system, the
real and imaginary parts of the series impedance and parallel admittance should be
related by the Kramer-Kronig conditions through the Hilbert Transform. The parameters
R (0) and L(∞) , as well as G (0) and C (∞) , constitute an inherently causal model.
Therefore the DHPPM in (5.8) satisfies the immittance consistency requirement.
Furthermore, poles and residues are used to capture the physical phenomenology that is
not accurately modeled by the constant RLGC model.
The stability of a system is determined by its pole locations. A stable system only
has poles that are located on the Left Hand Side (LHS) of the complex plane. In the
DHPPM, the Vector Fitting algorithm is employed to approximated the residue series in
(5.8). The extracted constant RLGC part of the DHPPM is a rapidly decaying and stable
term. Thus, the stability of the Vector Fitting algorithm determines the stability of the
DHPPM. In the realization of (5.9) in the Vector Fitting algorithm, a routine to check the
properties of the poles sα has been implemented to ensure that all the poles sα are
located on LHS of the complex plane. Therefore, the stability of the DHPPM is
guaranteed.
126
Another important issue is the macromodel passivity. As discussed in [32], MoC
based methods cannot theoretically prove passivity because the rational approximation
procedure used in the DHPPM or other MoC based methods is realized by least-square
based algorithms, e.g., UACAPRE [45,46] or Vector Fitting [47,63,64]. However,
passivity can be checked at each frequency point after the rational approximation, e.g.,
the Vector Fitting algorithm as in [63]. In the DHPPM procedure, by carefully selecting
the macromodel order and controlling the residue series, we didn’t encounter any
passivity problems during our tests.
By extracting the dominant physical phenomenology from the propagation
function in (5.8), we hope to reduce the macromodel order M to a minimum level so that
the simulation efficiency can be improved. As is demonstrated in the next section, the
DHPPM requires fewer terms than the HPPM and other MoC based methods for lossy
interconnects
5.3 Validation of the Frequency-Domain DHPPM
The DHPPM offers an improvement over the HPPM and other MoC techniques
because the propagation function that is extracted in the DHPPM in (5.8) and (5.9)
includes the DC resistance and conductance to approximate part of the loss. Therefore,
one would expect the DHPPM to work better than MoC-based techniques on lossy
transmission lines. Extraction of a constant R (0) L(∞)G (0)C (∞) model should work very
well for interconnects that exhibits cross-sectional dimensions that are approximately one
skin depth or smaller in size. In such cases, it is safe to assume that the currents are
127
distributed evenly inside the interconnect, thereby allowing a frequency-independent,
lossy transmission line model with constant RLGC parameters to be used up to a few
GHz. The frequency-dependent behaviors of the R and L parameters for three different
lines are shown in Figure 5.2 in order to show the skin effects, i.e., a typical on-chip
microstrip line with a 1.84 µ m by 1.23 µ m cross section [26], a typical MCM line, and a
typical PCB line [5]. It is shown in Figure 5.2(a) that the R and L parameters for the
MCM and PCB lines are constant for frequencies below 0.1 GHz, while the R and L
parameters for the on-chip line are constant for frequencies below 10 GHz. These
changes impact the modeling of the interconnect, but can be easily captured by the
residue series in the DHPPM.
(a)
128
(b)
Figure 5.2 The frequency-dependent behaviors of the (a) R and (b) L parameters for a
typical on-chip, MCM, and PCB lossy line.
To test the DHPPM, we first employed a typical on-chip microstrip interconnect
with a 1.84 µ m by 1.23 µ m cross section, height of 0.76 µ m above the ground plane,
length of 1cm, and a material with dielectric constant of 3.898. First, a two-dimensional
Method of Moments (MoM) based modeling tool was employed to obtain the frequencydependent R and L parameters up to 100GHz as well as the constant G and C parameters
[30]. The propagation function was then calculated at the sample frequency points.
The residue series functions RS in (5.5), (5.7), and (5.9) are the target functions
that are approximated by pole and residue pairs. It is desired that the residue series data
are well-behaved from low frequency to high frequency so that the macromodels
129
accurately approximate the propagation function in terms of both the magnitude and
phase at all frequency points. The magnitudes and phases of the residue series data in the
two macromodels of (5.7) and (5.9) for the on-chip lossy interconnect structure are
plotted in Figure 5.3 together with the magnitude and phase of the original propagation
function. It is observed in Figure 5.3 that the magnitude of the original propagation
function is close to one at DC and starts to decay exponentially as frequency increases,
while the phase of the original propagation function starts out at zero at low frequencies
and then the phase starts to vary rapidly at high frequencies as the electrical length of the
interconnect increases. After extracting a pure time delay, the HPPM residues series
phase exhibits much less variation than the original propagation function, especially at
lower and high frequencies. The reason that the phase is close to zero at these points is
because the pure time delay term correctly models the phases at these two extremes.
However, the extraction of the pure time delay term in the HPPM doesn’t affect the
magnitude of the residue series data so the magnitude of HPPM residue series data
overlaps with the magnitude of the original propagation function.
In the DHPPM, the constant R (0) L(∞)G (0)C (∞) model is extracted from the
propagation function. In Figure 5.3(b), the phase of residues series in the DHPPM has
even less variation than either the original propagation function or the HPPM residue
series. This is because the constant R (0) L(∞)G (0)C (∞) term not only models the phases
at low and high frequencies correctly, but it also captures part of the dispersion effects
between the low and high frequencies. Furthermore, it is observed in Figure 5.3(a) that
the magnitude of the DHPPM residue series data is close to one from DC up to 10GHz.
130
This means that the extract constant RLGC propagation function closely model the
original propagation function up to 10GHz. Therefore, the residue series in the DHPPM
is better behaved than the original propagation function and the HPPM.
The three macromodels defined in (5.5)-(5.9), with model orders of ten, were then
used to model the on-chip propagation function. The results for the magnitudes and
phases of the propagation function are shown in Figure 5.4(a) and (b), respectively. In
order to better show the accuracies of the various macromodels, we also plot the absolute
magnitude error for the different models in Figure 5.4(c).
(a)
131
(b)
Figure 5.3 The magnitudes and phases of in the propagation function, the residue series
data of HPPM and DHPPM for the on-chip lossy interconnect
For the classical macromodel, the Vector Fitting algorithm is directly employed
to approximate the propagation function with poles and residues as defined in (5.5). As
shown in Figure 5.4, a tenth-order classical macromodel does a reasonable job of
approximating the propagation function at low frequencies, but it leads to relatively
larger errors at high frequencies. Therefore, a classical macromodel requires more than
10 terms to accurately model this interconnect. It is found that when the macromodel
orders are 18 or higher, the classical macromodel are comparable with the DHPPM and
HPPM. The magnitude and absolute errors of the classical macromodel with 18 poles and
residues pairs are compared with other macromodels and shown in Figure 5.5.
132
For the HPPM, extraction of the phase delay term is first performed as in (5.6).
Then, the Vector Fitting algorithm is applied to approximate the residue series as in (5.7).
Figure 5.4 shows that the HPPM with 10 terms yields approximately the same accuracy
at low frequencies as the classical macromodel, but it has better accuracy at higher
frequencies.
For the DHPPM, a constant R (0) L(∞)G (0)C (∞) model is first extracted from the
propagation function as in (5.8). Then, the Vector Fitting algorithm is employed to
approximate the residue series as in (5.9). It is observed in Figure 5.4(c) that the DHPPM
with a model order of 10 approximates the on-chip propagation function very well, i.e., it
provides a 10 times and 100 times improvement in the absolute error when compared
with HPPM and classical macromodels with the same orders. We also plot a DHPPM
with order 2 in Figure 5.4(c) and find that this DHPPM provides approximately the same
accuracy as the tenth order HPPM.
The magnitude and phase of the extracted constant R (0) L(∞)G (0)C (∞)
propagation function is also plotted in Figure 5.4. We find that the constant
R (0) L(∞)G (0)C (∞) model approximates the propagation function so well at low
frequencies that the absolute error approaches zero as shown in Figure 5.4(c). However,
at frequencies above a few GHz, the constant R (0) L(∞)G (0)C (∞) model significantly
deviates from the propagation function. Comparing Figure 5.4 with Figure 5.2, one can
conclude that the residue series in (5.9) compensates the constant R (0) L(∞)G (0)C (∞)
model at high frequencies in the DHPPM so that the DHPPM is well behaved at both low
133
and high frequencies. This also explains the large differences between the transient
responses for the transmission lines with FDLP and FILP that are shown in Figure 5.1.
The three macromodels in (5.5) to (5.9) are also applied to a MCM line as
described in Table 1 of [29]. One of the notable differences between the MCM line and
on-chip line is that the MCM line has a much larger conductor cross-section and length.
Thus, the MCM line has a much lower resistance and the skin effects affects the wave
propagation at much lower frequencies than for the on-chip line in the first test (See
Figure 5.2). Furthermore, extraction of the constant R (0) L(∞)G (0)C (∞) model from the
propagation function will not reduce the DHPPM order as effectively as for the on-chip
line case.
The results for the MCM line case are shown in Figure 5.6. It is observed in
Figure 5.6(c) that the DHPPM absolute magnitude error is once again smaller than for the
HPPM or classical macromodel when a tenth order approximation is applied for all three
macromodels. However, this time a DHPPM with order 2 is not as good as the HPPM
with order 10. Also, the constant R (0) L(∞)G (0)C (∞) model starts to deviate from the
propagation function at frequencies as low as 0.2 GHz.
The Root-Mean-Square (RMS) error for the on-chip and MCM lines are listed in
Table 5.1. It is observed in Table 5.1 that the DHPPM with 10 poles has the minimum
RMS errors when compared with other macromodeling methods. The RMS errors for the
classical macromodel are the largest when compared with DHPPM and HPPM. As the
134
cross-section dimension and length decrease from the MCM case to the on-chip case, the
DHPPM shows its advantage over HPPM in terms of the RMS error.
DHPPM with
10 poles
DHPPM with
2 poles
HPPM with 10
poles
Classical Macromodel
with 10 poles
On-Chip
Case
0.001654
0.02109
0.005565
0.03751
MCM
Case
0.003106
0.03001
0.01540
0.2994
Table 5.1 The RMS errors of different macromodels for on-chip and MCM cases
These tests clearly illustrate that the simulation error can be greatly reduced if one
first extracts the dominant physical phenomenology from the problem. However, the
physical phenomenology is different in different kinds of interconnects. In long low-loss
lines, the HPPM can approximate the propagation function efficiently, while in short
high-loss, on-chip lines, where the cross sectional dimension, are comparable to the skin
depth, the DHPPM can significantly reduce the macromodel order. The DHPPM can also
be applied to long low-loss lines. However, the DHPPM doesn’t have as much of an
advantage over the HPPM in terms of reduced macromodel order as it does for on-chip
lines. Furthermore, the DHPPM has a more complicated time-domain expression as
illustrated in the next section.
135
(a)
136
(b)
137
(c)
Figure 5.4 A comparison between the propagation function for an on-chip interconnect
modeled by different macromodels in terms of (a) magnitude, (b) phase, and (c) absolute
error
138
(a)
(b)
Figure 5.5 A comparison between the propagation function for an on-chip interconnect
modeled by different macromodels in terms of (a) magnitude, and (b)absolute error
139
(a)
140
(b)
141
(c)
Figure 5.6 A comparison between the propagation function for an MCM interconnect
modeled by different macromodels in terms of (a) magnitude, (b) phase, and (c) absolute
error
5.4 Development of a Time-Domain DHPPM for FrequencyDependent Lossy Interconnects
Now that we have shown that the frequency-domain DHPPM has advantages over
classical macromodels and the HPPM, we now develop an efficient time-domain
142
DHPPM. Here we use triangular impulse and trapezoidal input waveform as examples to
illustrate the conversion of the frequency-domain DHPPM to a time-domain DHPPM.
5.4.1 Expression for the Triangular Impulse Response
When a triangular impulse is used as the input into the lossy interconnect, the
frequency-domain output is expressed as the product of the triangular waveform
spectrum and the propagation function,
4 sin 2 (ω∆t / 2)
Vout (ω ) =
exp ( − jω∆t ) exp [ −lγ ( jω )]
ω 2 ∆t
exp [ −lγ ( jω ) ]
=−
1 − 2 exp ( − jω∆t ) + exp ( −2 jω∆t )  .
ω 2 ∆t
(5.10)
Here we assume that the line is terminated with a matched load, and Vout is the
frequency-domain output signal response. In order to find the time-domain representation
for this propagating signal, we substitute (5.8) and (5.9) into (5.10) and perform an
Inverse Fourier Transform, i.e.,
vout (t ) = vout1 (t ) + vout 2 (t )
(5.11)
[ R(0) + sL(∞)][G (0) + sC (∞)]}
∫−∞ ∑
∆tω 2 ( jω − sα )
α =1
×4sin 2 (ω∆t / 2) exp  jω ( t − ∆t )  dω ,
vout1 (t ) =
1
2π
∞ M
{
Rα exp −l
{
}
1 ∞ Q exp −l [ R(0) + sL(∞) ][G (0) + sC (∞) ]
2π ∫−∞
∆tω 2
×4sin 2 (ω∆t / 2) exp  jω ( t − ∆t )  dω.
vout 2 (t ) =
(5.12)
(5.13)
143
According to Euler’s Theorem and the properties of the Fourier Transform, (5.12) can be
simplified as
vout1 (t ) =
Rα M
∑ [ 2φ1 (t − ∆t ) − φ1 (t ) − φ1 (t − 2∆t )],
j 2π∆t α =1
M
φ1 (t ) = ∑ ∫
α =1
∞
(
exp jωt − jtd ω 2 − jω / τ − ωc 2
−∞
ω (ω − jsα )
2
where td = l L(∞)C (∞) represents the time delay, τ =
ωc =
) dω ,
(5.14)
(5.15)
L(∞)C (∞)
, and
L(∞)G (0) + R(0)C (∞)
R (0)G (0)
. Likewise, vout 2 (t ) can be written as
L(∞)C (∞)
vout 2 (t ) =
φ2 (t ) = ∫
Q M
∑ [ 2φ2 (t − ∆t ) − φ2 (t ) − φ2 (t − 2∆t )],
2π∆t α =1
∞
(
exp jωt − jtd ω 2 − jω / τ − ωc 2
−∞
ω
2
) d ω.
(5.16)
(5.17)
The integration in (5.15) cannot be carried out unless we perform a partial fraction
expansion on the denominator,
1
1
j
1
= 2 +
− 2
.
2
ω (ω − jsα ) sα ω sα ω sα (ω − jsα )
2
(5.18)
By using this partial fraction expansion, we expand the integral in (5.15) into three
integrals
 j

1
1
φ2 (t ) + 2 φ3 (t ) − 2 φ4 (t ) ,
sα
sα
α =1  sα

M
φ1 (t ) = ∑ 
(5.19)
144
φ3 (t ) = ∫
(
exp jωt − jtd ω 2 − jω / τ − ωc 2
∞
ω
−∞
φ4 (t ) = ∫
∞
(
) dω,
exp jωt − jtd ω 2 − jω / τ − ωc 2
−∞
ω − jsα
) d ω.
(5.20)
(5.21)
5.4.2 Expression for the Trapezoidal Response
The time-domain and frequency-domain expressions for the trapezoidal input
have the forms as defined in Chapter 4, i.e.,
vin (t ) =
Vin (ω ) = −
1
1
1
1
t − (t − ∆t ) − (t − ∆t − T ) + (t − 2∆t − T ),
∆t
∆t
∆t
∆t
(5.22)
1
exp(− jω∆t ) exp(− jω∆t − jωT ) exp(−2 jω∆t − jωT )
+
+
−
. (5.23)
2
∆tω
∆tω 2
∆tω 2
∆tω 2
Note that in the time domain, the trapezoidal input is a linear combination of ramp signals
with various time delays. Since a lossy interconnect is a Time Invariant (TI) system, the
trapezoidal response is a linear combination of ramp responses. Fortunately, the ramp
response of a lossy interconnect has been previously represented in terms of φ1 (t ) and
φ2 (t ) in (5.15) and (5.17). Therefore, by solving the conical integrals in (5.15) and (5.17),
it is then straightforward to obtain the time-domain trapezoidal response.
The integrals for φ2 (t ) , φ3 (t ) , and φ4 (t ) in (5.17), (5.20), and (5.21) have been
thoroughly studied in Chapter 4. For example, φ2 (t ) models the transient response of a
ramp signal for a lossy transmission line with FILP and is given in (4.43). Likewise,
φ3 (t ) models the unit-step response of a lossy transmission line with FILP and is given in
145
(4.31). Finally, φ4 (t ) models the exponentially decaying signal response for a lossy
transmission line with FILP and is given in (4.27) and (4.41). By combining the above
results, one can obtain closed-form expressions for the transient response of lossy
transmission lines with FDLP.
5.5 The Time-Domain DHPPM for Coupled Lines
For coupled lines, a modal analysis has to be adopted to analyze the wave
propagation on the lines. It is well known that there are two propagation modes on a
coupled line system, i.e., an even mode and an odd mode. First, let us suppose that the
[ R ] , [ L ] , [G ] , and [C ] matrices for a two-line system are expressed as
R
[ R ] =  R11

21
G
[G ] = G11
 21
R12 
,
R22 
G12 
,
G22 
L
L12 
,
 21 L22 
−C 
C
[C ] =  −C11 C 12  ,
22 
 21
[ L ] =  L11
(5.24)
where we assume that every element in the matrices are frequency dependent. Then, the
even- and odd-mode propagation constants are defined as
γ 1 = ( R11 + R12 + jω ( L11 + L12 ))(G11 + G12 + jω (C11 − C12 )),
γ 2 = ( R11 − R12 + jω ( L11 − L12 ))(G11 − G12 + jω (C11 + C12 )),
(5.25)
and the even- and odd-mode impedances Z1 and Z 2 are defined as
Z1 = ( R11 + R12 + jω ( L11 + L12 )) /(G11 + G12 + jω (C11 − C12 )),
Z 2 = ( R11 − R12 + jω ( L11 − L12 )) /(G11 − G12 + jω (C11 + C12 )).
(5.26)
146
The characteristic impedance matrix of this two-line system is
[ ZC ] =
1  Z1 + Z 2
2  Z1 − Z 2
Z1 − Z 2 
.
Z1 + Z 2 
(5.27)
The port voltages and currents are given as the summation of these two modes. Using the
modal analysis technique, the incident and reflected amplitudes of the two modes, i.e., A,
B, C, and D should satisfy the following equations
ΨX = Φ,
(5.28)
where
Z
Z
Z
Z


1 + G1
1 + G1
1 − G1
1 − G1


Z1
Z2
Z1
Z2


ZG 2
ZG 2
ZG 2
ZG 2


1+
−1 −
1−
−1 +


Z1
Z2
Z1
Z2
,
Ψ=


Z L1
Z L1
Z L1
Z L1
) exp(−γ 1l ) (1 −
) exp(−γ 2l ) (1 +
) exp(γ 1l ) (1 +
) exp(γ 2l ) 
 (1 −
Z1
Z2
Z1
Z2




Z L2
Z
Z
Z
) exp(−γ 1l ) (−1 + L 2 ) exp(−γ 2l ) (1 + L 2 ) exp(γ 1l ) (−1 − L 2 ) exp(γ 2l ) 
 (1 −
Z1
Z2
Z1
Z2


 A
VIN 
B
 0 


, Φ =  ,
X=
C 
 0 
 
 
D
 0 
(5.29)
VIN
is the source signal that is applied on the active port and ZGi and Z Li represent the
generator and load impedances on the two lines (i=1,2). The far-end active line voltage is
V1 (l ) = A exp(−γ 1l ) + B exp(−γ 2l ) + C exp(γ 1l ) + D exp(γ 2l ),
and the far-end passive line voltage is
(5.30)
147
V2 (l ) = A exp(−γ 1l ) − B exp(−γ 2l ) + C exp(γ 1l ) − D exp(γ 2l ).
(5.31)
If we let ZGi = Z Li = ( Z12 + Z 22 ) / 2 and assume that there are no reflections from either end,
then the far-end voltages on the active and passive lines satisfy
V1,2 (l , ω )
E1 (ω )
where p1 = 1/(1 +
=

1 1
1
 exp(−γ 1l ) ± exp(−γ 2l )  ,
2  p1
p2

(5.32)
ZG
Z
) and p2 = 1/(1 + G ) . The “+” sign is for the active line voltage
Z1
Z2
V1 (l , ω ) and the “-” sign is for the passive line voltage V2 (l , ω ) . The near-end voltage on
the passive line satisfies

V2 (0, ω ) 1  1
1   1
1
=  −  1 −  exp(−2γ 1l ) + exp(−2γ 2l )   .
E1 (ω )
2  p1 p2    p1
p2

(5.33)
The DHPPM in (5.8) and (5.9) will be used to model the propagation function
exp [ −lγ ( s )] . Note that the even and odd mode impedances in (5.26) are also frequencydependent. In order to address the frequency-dependent impedance, it is necessary to
modify the DHPPM as

ZG 
 1 +
 exp(−γ 1,2l ) ≈ exp  −l
 Z1,2 
[ R(0) + sL(∞)][G (0) + sC (∞)]  RSDHPPM ,
(5.34)
M
Rα
+ Q.
α =1 s − sα
RS DHPPM = ∑
(5.35)
Then, by following a similar procedure as in the single line case in the previous section,
the time-domain expression for the coupled line signal response can also be obtained.
148
As a summary, the general algorithm flow chart for the simulation of lossy transmission
line with FDLP is shown in Figure 5.7. The operation of the DHPPM simulator is made
up of two parts, i.e., the frequency-domain operation and the time-domain operation. In
the frequency domain, tabular RLGC data are first input into the transient simulator.
Then, a constant RLGC parameter is extracted from the propagation function. Next, the
Vector Fitting algorithm is employed to approximate the resulting residue series in terms
of pole and residue pair. The number of pole and residue pairs is gradually increased until
the approximation deviation reaches a pre-defined accuracy. In the time domain, the
expressions involving the constant RLGC term and the poles are expressed in terms of
ILHIs and Bessel function. Then, the residues series are summed up to obtain a timedomain result at each time step. In the next section, the DHPPM closed-form results are
compared with commercial simulation tools for validation purposes.
149
Start
Read Tabular
RLGC Data
Simulation
Finished?
Vector Fitting
Algorithm
Increase
Model
Order
No
Pole and
Residue
Extraction
Converged?
Time
Domain
End
Yes
Frequency
Domain
Sum Residue
Series
Yes
No
Increase
Simulation
Time
Calculate ILHIs for
Pole/Residue Pairs
Figure 5.7 The general DHPPM algorithm flow chart for the simulation of lossy
transmission line with FDLP.
5.6 Application of the Closed-Form Results to the Simulation
of Transmission Lines with FDLP
Two examples are now given to demonstrate the capabilities of the DHPPM
simulator. First, a microstrip line over a substrate with frequency-dependent R and L
parameters, as listed in Table 5.2, is shown as an example. The substrate has a dielectric
constant of 4.4. This line also has a frequency-independent capacitance C=107.13pF/m
and no conductance G=0. A trapezoidal input with 10ps rise and fall times, and 0.1ns
duration is input into a 5cm long line. First an Inverse Fast Fourier Transform (IFFT)
method is used to calculate the output waveform at the far end of the line, which is shown
150
as the dash-dotted line in Figure 5.8. Then a HSPICE W-element model with frequencydependent tabular RLGC parameters is used to model this interconnect and the result is
shown as the dashed line in Figure 5.8. Finally, the DHPPM simulator is used to calculate
the far-end voltage of the lossy line and the result is shown as the solid line in Figure 5.8.
It is observed that the three results agree fairly well with each other in Figure 5.8.
However, the HSPICE results differ slightly from those produced by the other methods.
All three methods have captured the frequency-dependent losses, the dispersion effects,
and the propagation delay on the line fairly accurately. However, the simulation
efficiencies of the three methods are different. In the IFFT method, the simulation time is
545.60s, while the simulation time for HSPICE is 0.972s. For the DHPPM method, the
simulation time is 0.93s.
The effects of the rise time of the trapezoidal input are also studied. A trapezoidal
input with a 100ps rise time and 1ns duration time is applied to the FDLP line. All three
methods are employed to solve the transient simulation problem and the results are
shown in Figure 5.9. The longer rise time has effectively reduced the bandwidth of the
signal. Therefore, fewer numerical artifacts are observed in the IFFT and HSPICE
simulation results and the three methods are in excellent agreement as shown in Figure
5.9. This time the simulation time in the DHPPM method is 0.69s, while the simulation
times in the HSPICE and IFFT methods are 0.98s and 560.24s, respectively.
Next, a triangle impulse with a rise time of 10ps is input into the lossy microstrip
interconnect. The far-end voltage waveforms are simulated using the DHPPM method
with model orders of 3 and 4, the IFFT method with 220 and 223 sampling points and the
151
W-element frequency-dependent tabular model in HSPICE, and the five results are
shown in Figure 5.10. It is shown that as the sampling points in IFFT method increases
from 220 to 223 , the two IFFT results converge very well and almost overlap each other.
Thus, the IFFT result can be treated as the correct result in comparisons with the other
methods. Similar agreement among the three results is observed in Figure 5.10 as was
seen in Figure 5.8. In order to test the convergence of the DHPPM, different macromodel
orders of 3 and 4 were used to calculate the transient responses of the line. As seen in
Figure 5.10, as the macromodel order increases from 3 to 4, the DHPPM method agrees
with the IFFT results very well. However, HSPICE does not correctly produce the peak
response. In addition, a closer look at the early- and late-time behaviors of the three
results reveals that the IFFT method suffers from the Gibbs phenomenon as shown in
Figure 5.11. Although the spikes due to the Gibbs phenomenon are small, it suggests the
IFFT method in nature violates the transmission line causality requirements. Also, the
HSPICE result has late time oscillations. On the other hand, the DHPPM method is free
from these numerical artifacts. In HSPICE, the simulation step size is a key factor in
determining the transient simulation accuracy and efficiency. Two different simulation
steps are adopted for the on-chip lossy interconnect transient simulation case and the
results are shown in Figure 5.12. In Figure 5.12, the HSPICE result 1 is obtained by
using a simulation step size of 1ps, while the HSPICE result 2 is obtained by using a
simulation step size of 0.2ps. These two HSPICE results are compared with the IFFT
results and the DHPPM with 10 poles results from Figure 5.10. It is observed in Figure
5.12 that the HSPICE result 2 with the smaller step size agrees the best with IFFT,
152
especially at the peak value. However, the better performance in the HSPICE result is
obtained at the cost of simulation time and efficiency. The simulation time for the
DHPPM with 10 poles is 0.67s and the simulation time for the IFFT is 540.78s. The
simulation time for the HSPICE result 1 with the 1ps step size is 0.67s, while the
simulation time for the HSPICE result 2 with the 0.2ps step size is 2.34s.
The second example is a MCM coupled line problem. The FDLP R11, R12, L11,
and L12 are shown in Table 5.3. G11 and G12 are assumed to be zero, C11 is 50.115
pF/m, C12 is 13.664 pF/m, and the length of the transmission line is 10 cm. The input
signal is a triangle impulse with 20ps rise and fall times. The Far End Active (FEA) and
Far End Passive (FEP) line voltages are plotted in Figure 5.13 and Figure 5.14,
respectively. In order to make sure that there are a sufficient number of frequencydomain sampling points to make the IFFT method converge, two sampling schemes are
applied in this example. In the IFFT result 1, 220 sampling points are used, and the
simulation time is 620.11s. When the number of the sampling points in the IFFT result 2
is increased to 223 , the simulation time is 8906.29s. To obtain the DHPPM result which
employs 4 poles, the simulation time is only 0.82s. Note that the three results agree with
each other fairly well. However, the IFFT results do exhibit frequency-domain truncation
errors and the Gibbs phenomenon in the early-time responses in Figure 5.13 and Figure
5.14. Furthermore, because of the lack of the L parameters at some of the GHz frequency
points, numerical interpolation has to be employed on the L parameter before the IFFT
procedure. As indicated in [32], an interpolation of the L parameters may violate the
Kramers-Kronig conditions between the R and L parameters. Thus, non-causal IFFT
153
results are observed in Figure 5.13 and Figure 5.14. However, as can be seen in Figure
5.13 and Figure 5.14, the DHPPM results are free from this problem because the
causality requirement is enforced in the method.
f(Hz)
1e1
1e3
1e4
1e5
1e6
1e7
1e8
4e8
1e9
2e9
4e9
5e9
7e9
1e10
R(Ω/m)
1720.3
1720.3
1720.3
1720.3
1720.3
1720.3
1720.5
1724.2
1744.5
1811.2
2017.4
2132.9
2361.3
2683.2
L(nH/m)
529.48
529.48
529.48
529.48
529.48
529.48
529.48
529.39
528.94
527.48
523.33
521.05
517.24
512.91
F(Hz)
1.4e10
2e10
3e10
4e10
5e10
6e10
8e10
1e11
2e11
5e11
1e12
2e12
5e12
1e13
R(Ω/m)
3079.9
3612.4
4342.2
4923.7
5409.7
5831.6
6545.2
7135.1
10090
15960
22560
31910
50450
71350
L(nH/m)
508.84
504.59
500.09
497.29
495.39
494.02
492.14
490.89
487.50
486.10
485.60
485.50
485.40
485.37
Table 5.2 Frequency-dependent R and L parameters for a microstrip line calculated using
UAPDSE [30] where W=5µm, t=2µm, T=10µm, l=5cm, and εr=4.4.
f(Hz)
R11(Ω/m)
1e1
49.9
1e6
49.901
1e7
49.938
1e8
53.615
4e8
101.99
1e9
157.08
2e9
226.19
4e9
309.13
8e9
434.29
1.6e10 612.23
3.2e10 862.56
L11(nH/m)
560.91
560.91
560.83
546.17
523.14
523
522
R12(Ω/m)
1.2155e1.2152e-4
1.1859e-2
4.3366e-1
9.0e-1
2.7
3.8
5.36
7.55
10.65
15
L12(nH/m)
111.69
111.69
111.69
111.94
112.30
112.30
112.30
Table 5.3 Frequency-dependent line parameters for the MCM coupled transmission lines
as listed in [29].
154
Figure 5.8 The time-domain output waveforms obtained from different models for a 10ps
trapezoidal inputs
155
Figure 5.9 The time-domain output waveforms obtained from different models for a
100ps trapezoidal input
156
Figure 5.10 The comparison of the DHPPM results, HSPICE results, and IFFT results for
a 10ps triangle impulse input
157
Figure 5.11 The comparison of the DHPPM results, IFFT results, and HSPICE results for
a 10ps triangle impulse input
158
Figure 5.12 The comparison of two HSPICE results, IFFT results, and DHPPM with 10
poles results
159
Figure 5.13 The Far End Active (FEA) line voltage comparisons of the DHPPM results
and the IFFT results for a 20ps triangle impulse on a coupled line
160
Figure 5.14 The Far End Passive (FEP) line voltage comparison of the DHPPM results
and the IFFT results for a 20ps triangle impulse on a coupled line
161
CHAPTER 6
CONCLUSION AND FUTURE WORK
In this dissertation, a new transient simulation method for FDLP interconnects
that is based on closed-form results involving ILHI and CILHI has been developed. We
have successfully applied the above method to both transmission lines with FILP and
FDLP.
First, the behaviors of lossy interconnect modeled by transmission lines with FILP
and FDLP are studied. Plots of the propagation functions for various interconnect
structures, such as on-chip lines, MCM lines, and PCB lines, reveals that frequencydependent effects have to be included in the models of lossy interconnect structures for
accurate time-domain simulations. The inter-dependent relationships between the FDLP
are also examined. It is found that the frequency-dependent R and L, as well as G and C
parameters form a Hilbert transform pair, i.e., the frequency dependent R parameters will
determines the frequency dependent L parameters and vice versa.
Some general modeling and simulation algorithms are overviewed for the
modeling and simulation of lossy transmission lines. These methods can be divided into
the Model Order Reduction (MOR) methods and macromodeling techniques. In the
macromodeling methods, the Method of Characteristics (MoC) and its variations,
especially the W-element in HSPICE, are discussed. It is observed that the MoC-based
algorithms can efficiently reduce the macromodel order by pre-extraction of a
propagation delay term from the propagation function. The macromodel employed in the
162
MoC algorithms can be proved to yield a stable and causal system. However, the
passivity of the MoC-based methods cannot be proved.
Then, contour integration theory is successfully applied to solve some canonical
integrals associated with the time-domain propagation functions on a lossy transmission
lines with FILP. By introducing the Incomplete Lipschitz-Hankel Integrals (ILHIs) and
Complementary Incomplete Lipschitz-Hankel Integrals (CILHIs), the transient responses
of lossy interconnects with FILP can be represented as closed-form expressions involving
ILHIs and CILHIs. These ILHIs and CILHIs can be calculated using pre-defined series
expansions with high precision. Furthermore, the expressions involving ILHIs and
CILHIs are free from common numerical effects encountered in other algorithms such as
aliasing, numerical truncation error, violation of causality, etc. Several source signal
waveforms, such as the unit-step signal, the ramp signal, and the exponentially decaying
signal are used as inputs to lossy transmission lines with FILP. Both far-end voltage and
near-end current expressions are successfully derived. Finally, comparisons are made
between the closed-form expressions involving ILHIs and CILHIs and commercial
simulation tools such HSPICE W-element. Excellent agreements are observed between
the two methods.
Next, it was demonstrated that the time-domain propagation function on a lossy
transmission line with FDLP can be expressed as a sum of these canonical integrals. The
DHPPM was also applied to coupled FDLP lines to simulate the crosstalk and coupling
issues. Various source waveforms were used to test the DHPPM simulator and
comparisons were made with commercial simulators like HSPICE and other numerical
163
methods like IFFT. Good agreement was observed between the three methods. It should
be noted that the DHPPM method not only produces efficient simulations, but also
produces results that are free from various numerical artifacts like the Gibbs phenomenon
and truncation errors.
The DHPPM for lossy transmission line with FDLP is an extension of the MoC.
However, due to the complex time-domain expressions involving ILHIs and CILHIs, it is
hard to find a recursive convolution relationship for the DHPPM for lossy transmission
lines with FDLP. In regard to the future work, a recursive convolution relationship is
desired to further improve the transient simulation efficiency. The current DHPPM
simulation tool is based on a stand-alone code that allows for the implementation of the
ILHI calculations. In order to generalize this algorithm, it is desirable to import the
DHPPM and the required ILHI and CILHI into a SPICE-type simulator.
164
APPENDIX A GLOSSARY OF TERMS
FDLP:
Frequency Dependent Line Parameters
FILP:
Frequency Independent Line Parameters
VectFit:
Vector Fitting algorithm
ILHI:
Incomplete Lipshitz-Hankel Integrals
CILHI:
Complementary Incomplete Lipshitz-Hankel Integrals
GMoC:
Generalized Method of Characteristics
MOR:
Model Order Reduction
HPPM:
Hybrid Phase-Pole Macromodel
DHPPM:
Dispersive Hybrid Phase-Pole Macromodel
IFFT:
Inverse Fast Fourier Transform
RMS:
Root-Mean-Square
MCM:
Multiple Chip Module
PCB:
Printed Circuit Board
165
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