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Investigation of distributed circuit theory with retardation for analysis of microwave characteristics of three dimensional conductor /dielectric structures

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INVESTIGATION OF DISTRIBUTED CIRCUIT THEORY WITH RETARDATION
FOR ANALYSIS OF MICROWAVE CHARACTERISTICS OF THREE
DIMENSIONAL CONDUCTOR/DIELECTRIC STRUCTURES
by
Ramani Tatikola
Presented to the Graduate and Research Committee
o f Lehigh University
in Candidacy for the degree o f
Doctor of Philosophy
in
Department o f Electrical Engineering and Computer Science
Lehigh University
August, 1999
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Approved and recommended for acceptance as a dissertation in partial fulfillment
o f the requirements for the degree of Doctor o f Philosophy.
°i,
M l
Date (J
Dissertation Advisor
f a * to I f f f
Accepted bate
Committee Members:
‘T
J
PrW. D. Frey
Prof. R. Folk
^ )r H.M.01son
Prof. D. Christodoulides
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A cknow ledgem ents
My sincere thanks to Prof. R.Decker for his guidance throughout my graduate program.
He introduced me to the challenging and exciting field of electronic packaging and
helped me understand and solve efficiently complex problems. I would also like to thank
him for helping me learn the practical needs of the industry and successfully present my
candidature.
I extend my thanks to Dr Olson for helping me expand my knowledge in the field of
applied electromagnetics. His critical reading of the dissertation was of great help in
improving my writing skills. The weekly discussions with Prof.Decker and Dr. Olson
were invaluable to the completion of my dissertation.
I would like to express my deep appreciation for the wonderful patience of my husband
Narsu during the completion of this degree program. I would also like to acknowledge the
help he extended to implement the software design.
Finally, I wish to acknowledge my parents for giving me the opportunity to pursue higher
education, my son Neeraj for his sunny disposition and Mrs. Saroja Acharya for all the
help.
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THESIS OUTLINE
Abstract
1
Chapter 1 Introduction
1.1
Introduction to Complex EM Structures
2
1.2
Electrical Characterization o f Package Parasitics
5
1.2.1
1.2.2
Analog Microwave and Millimeter-wave Packages
5
1.2.1.1 The RF Feed-through Section
7
1.2.1.2 Cavity Resonances and Wall Modes
11
1.2.1.3 Composite Modeling Technique
13
Digital Electronic Packages
17
1.2.2.1 Interconnect Modeling
18
1.2.2.2 Crosstalk and Circuit Impedance
20
1.2.2.3 Propagation Delay Time and Signal Attenuation
1.3
Field Analysis o f Analog and Digital Package Interconnects
22
24
Chapter 2 Comparison of Numerical Techniques
2.1
Numerical Modeling and AnalysisTechniques in EM Field Theory 26
2.2
Widely Used Numerical Techniques in EM Field Analysis
29
2.2.1
Spectral Domain Method
30
2.2.2
Frequency domain Time Difference Technique
35
2.2.3
Transmission Line Matrix Method
39
2.2.4
Finite Element Technique
44
2.2.5
Method o f Moments Technique
46
Limitations o f the Various Numerical Techniques
49
2.3
Chapter 3
3.1
Full wave Analysis using Retarded Potentials
Objectives o f this Research
52
iv
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3.1.1
3.2
Field Modeling using Circuit Concepts
52
Problem Formulation using Retarded Potentials and Circuit Theory 56
3.2.1
Loop Formulation
56
3.2.2
Nodal Formulation
78
3.3
Dielectric Modeling
83
3.4
Summary
91
Chapter 4
4.1
Analysis and Simulation of Equivalent Circuit Network
Description of Equivalent Circuit Parameters
93
4.1.1
Capacitive Segments
93
4.1.2
Inductive Segments
96
4.2
Segmentation Scheme
99
4.3
Modeling with the Retarded Potential Technique
104
Chapter 5 Comparison of Numerical/Simulated Results
5.1
Circuit Analysis Tools
108
5.2
Software Design and Data Files
110
5.3
Comparison o f Results
116
Chapter 6 Conclusions
6.1
Summary
137
6.2
Contributions
139
6.3
Future Work
140
References
142
Appendix I
150
Vita
156
V
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A b stract
The recent progress in the field of high-speed digital circuit technologies and
Monolithic Microwave Integrated Circuits (MMICs) has led to complex microwave
circuit structures that exhibit strong dispersive behavior. Therefore, to model the dynamic
behavior o f electromagnetic properties of complex structures, the use of distributed
circuit models in conjunction with numerical analysis is very promising. The starting
point for the use o f numerical models for field analysis is the generalization of field
theory to circuit theory, to efficiently manage complexity. In this thesis, the equivalence
between KirchofFs current and voltage laws, and Maxwell’s field equations, using
retarded potentials and the method o f moments (Galerkin’s) technique, is established in
the frequency domain. Retarded potentials are directly related to the current and charge
sources and therefore both electric and magnetic fields can be determined [3].
Various numerical methods have been published in the literature that lead to
circuit models o f electromagnetic wave propagation, but a direct evaluation o f equivalent
circuit models and their characteristics using retarded potentials in the frequency domain
and KirchofFs laws has not been implemented previously. This work addresses the issue
and shows the relationship between retarded potentials in field theory and KirchofFs
current and voltage laws resulting in equivalent circuit models. The resulting circuit
models have been evaluated using various analytical formulae available in the literature
and the scattering characteristics of several relevant structures have been obtained using
circuit network analysis and are discussed.
1
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CHAPTER 1
INTRODUCTION
1.1 Introduction to Complex EM Structures
Electromagnetic (EM) phenomena are described and analyzed mathematically by
solving Maxwell’s equations. Field properties such as wave propagation, dispersion,
reflection, scattering, retardation, radiation and many other field concepts are determined
by solving the governing field equations to obtain an analytical expression. Most of the
approaches to solving wave propagation and other field concepts described in many texts
[3,10,12] are limited to ideal cases and cannot analyze complicated electromagnetic
structures encountered in present day microwave and millimeter-wave technology.
However, these ideal cases provide valuable insight in understanding the physics of
complex electromagnetic structures such as electronic packages. Therefore, to design,
analyze and synthesize electromagnetic field properties of complex EM structures, like
electronic packages, it is necessary to understand the concepts o f electronic packaging.
These concepts aid in the use o f numerical methods [Chapter 2] to obtain accurate EM
solutions for complex structures within the specified error margins.
The main function o f an electronic package is to provide physical support,
environmental protection, thermal management and required electrical insulation.
However, in the recent past the electrical characteristics of digital packages (operating at
high clock speeds) as well as analog packages (operating at high frequencies) have
received increased attention due to their deleterious effect on the system performance.
2
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This is because the electronic packaging design has not kept pace with the advances in
the IC technology. Therefore, it is important to analyze the factors contributing to the
increase in package delay that degrade the performance characteristics o f both analog and
digital systems.
In digital electronic systems [Figure 1.1], future trends and advances are to
increase the millions o f instructions executed per second (MIPS) to billions of
instructions executed per second (BIPS). To achieve this high system performance as
well as increased information processing rates, it is necessary to increase the system clock
frequency. The use o f high clock speeds has produced entirely new interconnect lines
supporting complex electromagnetic fields. The interconnection line length and the
electromagnetic characteristics o f these lines are directly proportional to the increase in
package delay. Therefore, it is necessary to obtain the full-wave characteristics of these
lines to accurately predict the system performance.
Analog systems are characterized by the utilization of Monolithic Microwave
Integrated Circuit (MMIC) technology that operates at frequencies in the gigahertz (GHz)
range. Current packaging technology o f GaAs MMICs does not permit operation above
30GHz while the MMICs can operate at frequencies in excess o f 40GHz. The
transmission line structures [Figure 1.3] at these high frequencies are small, complex in
geometry and support several frequency-dependent propagating modes. The major
problems associated with signal propagation at high frequencies are due to the physical
3
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and electrical lengths of interconnect lines and the electromagnetic coupling within the
package. In short, it is necessary to identify and estimate the electrical characteristics of
the package at design level to minimize signal attenuation.
Small Scale
Integrated Circuits
l§si)________
Single
Transistor
Chip
Medium Scale
Integrated Circuits
(MSI)___________
Very Large Scale
Integrated Circuits
(VLSI)__________
Figure 1.1 Developmental Phases of Digital Components
DC Bias
1
Through Via*
wIN I
SO Ohm Mlnoxthp
FaadLine
GlawRIng
5 Mil Quartz
Subatrata
r m
»
Kovarbaaa
Figure 1.2 Analog/MMIC Package
4
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1.2 Electrical Characterization of Package Parasitics
As stated earlier, the electrical characteristics o f an ideal electronic
package must be transparent to the propagating signal. However, the parasitic shunt
capacitance and series inductance (termed package parasitics) o f the package degrade this
transparency. At Millimeter-wave frequencies, the package parasitics that significantly
contribute to signal degradation are from the RF feed-through [Figure 1.4], ground plane
discontinuities and package resonance. In digital systems, fast rise and fall times excite
the parasitic mutual inductances and capacitances leading to cross talk between adjacent
signal lines. The transmission line configurations existing in the feed-through and the
other parts of the MMIC package as shown in figure 1.3 support electromagnetic fields
that do not have closed form solutions and must be analyzed using numerical
electromagnetic analysis techniques. The structural and design requirements for packages
in analog and digital domains are quite different and require separate treatment to identify
and analyze the potential sections of the package contributing towards signal delay. This
detailed discussion is essential to illustrate the level o f complexity involved in the
determination of analog and digital package parasitics.
1.2.1 Analog Microwave and Millimeter-wave Packages
A typical MMIC package as shown in figure 1.2 consists o f several sections, of
which the RF feed-through, cavity and package wall, are crucial to the package design.
The cavity section o f the package is typically designed to permit mounting of one or more
MMIC chips with low-inductance interconnections between the package and chip ground
5
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planes.
The RF feed-through sections of the package provide transition to external
microwave connectors or for circuit board mounting and for routing of additional
interconnections such as bias and control lines. The package wall is designed to provide
structural integrity and environmental protection.
At high frequencies, the choice of base material, substrate thickness, dielectric
constant and fabrication technology strongly determine the package parasitics. The
substrate material introduces significant conductor losses, dielectric losses and radiation
losses that attenuate the propagating waves in the RF feed-through section and other
interconnect lines within the package, thereby increasing package return loss (SI 1). The
cavity section and package wall support hybrid modes that contribute to unwanted
coupling resulting in an increase in insertion loss. However, the design requirements of
one section have a contradicting effect with respect to the design requirements of the
other sections. Therefore, to achieve the basic requirements of a MMIC package it is
important to understand and analyze the factors contributing to the package insertion loss
and return loss in order to provide an optimum package design that is electrically
transparent within the required bandwidth. These factors are discussed in the following
sub-sections.
6
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A. S h ie ld e d M ic ro strip L in e
B. S trip L in e
motal
C .S I o t L in e
D . C o p i a n a r S t r i p L in e
Figure 1.3 Transmission Line Structures at Very High Frequencies
1.2.1.1 The RF Feed-through Section
For low return loss (VSWR), the RF feed-through and transitions at transmission
line junctions need to be carefully designed. As mentioned earlier, a typical RF feedthrough consists of several longitudinally non-uniform transmission line structures with
complex geometries. The feed-through shown in figure 1.4, is a conductor-backed
coplanar waveguide and is designed for a custom MMIC package required to house 5
MMIC chips for X and Ka bands [27]. The main objectives of a good feed-through
design are a. Signal confinement b. Electrical transparency c. Bandwidth and d.
Fabrication as shown in Figure 1.5. The metallization patterns on the dielectric substrate
in a typical RF feed-through have different cross-sections along its length as it passes
from inside o f the package to the outside.
For example, in some custom packages, the feed-through consists o f a section of
microstrip line outside the package, a shielded stripline as it passes through the package
7
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wall and a shielded microstrip line inside the package cavity. These transmission line
structures support hybrid propagating modes with strong dispersion characteristics. The
quasi-TEM mode, which is the lowest order propagating hybrid mode consists of
Transverse Electric (TE) and Transverse Magnetic (TM) components as well as
longitudinal field components. The presence of longitudinal field components in addition
to the transverse components increases the complexity o f field pattern calculation within
the transmission line. Therefore, to compute signal attenuation accurately, it is necessary
to consider the three dimensional properties o f the propagating modes.
Accurate matching o f transmission lines (typically to 50 ohms) is required at all
discontinuities that exist at external launchers or interconnects the feed-through area and
chip(s) and chip-to-chip discontinuities to achieve low return loss. If ribbon or wire
bonding is used for chip interconnects, the span between chips will behave as a high
impedance transmission line above a ground plane introducing series inductance. If the
chips are matched to 50 ohms at input and output, this inductive discontinuity needs to be
compensated in some manner. Therefore, a significant practical factor affecting return
loss is the impedance mismatches at several transitions in the MMIC package. The
impedance mismatches introduced between MMIC chips and interconnections inside the
multi-chip package primarily affect the return loss.
Another area o f concern is the impedance discontinuities resulting from step
height and ground plane mismatches. Height discontinuities are due to the different
8
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heights of the package structure and the thickness of the MMIC chip substrates. These
can cause appreciable impedance mismatches that affect the return loss due to multiple
reflections. The discontinuities in the transmission line at coaxial-to-microstrip line
transitions or between levels of multilayer substrate also significantly affect the return
loss. The predominant factor contributing to signal degradation at these discontinuities is
the conductor loss.
Conductor resistivity affects the magnitude of the fields along the length as well
as the width of the transmission lines. Large conductor cross-sections are required to
achieve low resistivity of the metallized patterns on the substrate. At high frequencies, it
is well known that current distribution in a conductor is non-uniform, with more charge
concentration at the edges. This means that charge concentrations are more at the surface
edges than at the center because o f short relaxation times resulting in frequency
dependent resistive losses also known as skin-effect losses. The conductor skin effect
limits the frequency o f operation and along with surface resistivity increases the
conductor losses. At very high frequencies, the conductor thickness must be taken into
account to accurately predict the electrical parasitics.
The substrates used at high frequencies have large dielectric constants giving rise
to significant dissipative losses. Due to the existence o f several dispersive propagating
modes, transmission line structures encountered in a package require numerical
evaluation techniques to determine the conductor and dielectric losses. The inclusion of
9
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dissipative losses in the three dimensional modeling o f multiconductor transmission lines
considerably increases the complexity of the numerical analysis techniques. However, the
conductor losses and dissipative losses together affect the package return loss and
insertion loss and must be included to predict the system performance. Another area in
the package that significantly contributes towards increased signal attenuation is the
package cavity and the electromagnetic properties of the package cavity are discussed in
the next section.
Package Cover
Signal Conductor
V/
Ground Planes
Figure 1.4 Cross-Section of a typical RF Feed-Through
10
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Properties of a good Feed-through
Electrical
Transparency
Signal
Confinement
i. Negligible EM
radiation
ii. Good isolation
from other
signal paths
Bandwidth
i. Low signal
reflection
i. Good electrical
performance in
the operating
bandwidth
ii. Low attenuation
Fabrication
i. Mechanically
robust
ii. Dimensional
and material
requirements
compatible with
technology
Figure 1.5 Requirements for a Good Feed-Through Design
1.2.1.2 Cavity Resonances and Wall Modes
MMICs are generally enclosed in a package to reduce radiation losses and to
isolate one circuit from another. However enclosing a circuit can have some undesirable
effects, such as, parasitic coupling to resonant modes of the enclosure resulting in power
loss, poor isolation and circuit instabilities. The resonances may manifest themselves as
glitches in the insertion loss characteristics.
11
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Certain signal strengths propagating through the package sometimes excite modes
that exhibit resonance behavior. If the frequency of a given resonance is well outside the
operating band of the package, the resonance may not be troublesome. If this is not the
case and the resonance is strongly excited and means for suppressing the resonance need
to be sought. In some cases, posts or walls are inserted into the package to short the
resonant mode or to shift the resonance frequency. Lossy materials [22] are also used to
reduce resonant mode coupling in regions where the stored energy of resonance is
concentrated. All these solutions to the reduction of resonant mode coupling require
efficient and economical ways to identify resonances.
Since the package wall is made of a substrate with a high dielectric constant, it
guides several dispersive modes. The characteristics of these hybrid propagating
waveguide modes depend on the cross-section and the metallization pattern in the
package wall. Some of the waveguide geometries [Figure 1.6] that can be identified to
model the wall modes are: i. Non-Radiative Dielectric (NRD) waveguide ii. Rectangular
dielectric loaded waveguide and iii. Parallel plate waveguide. The accurate determination
o f electric and magnetic fields in these waveguide geometries is important to estimate the
unwanted coupling to the RF feed-through section of the package and other
interconnections in the package cavity.
However, the evaluation o f the guide wavelength, cut-off heights and widths of
the dielectric layers that excite higher order modes and the determination o f the
12
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propagation constant o f these wall mode is quite complex. Therefore, the propagation
characteristics of the wall modes can be included depending on the level of accuracy
needed and the design requirements o f the package. Since each section of the package is a
complex electromagnetic structure, it is helpful to discuss the composite modeling
technique to summarize the analysis and design of a MMIC package.
A. Rectangular Waveguide
B. Parallel Plate Waveguide
Dielectric
air
air
C. Non-Radiative Dielectric Waveguide
Figure 1.6 Typical Waveguide Geometries in a MMIC Package Wall
1.2.1.3 Composite Modeling Technique
The composite modeling technique illustrates the baseline model for a generic
MMIC package to include the individual characteristics of several sections existing in the
package in order to make a judicious choice in obtaining the required package
characteristics. This model is comprised o f canonical package structures (sub-divisions in
the package) combined into a circuit model o f the overall package. This approach is an
13
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alternative to numerical analysis o f the complete three-dimensional structure as a "black
box". The MMIC package is divided into the following canonical structures:
a) Substrate or base (usually metal or metallized ceramic)
b) RF feed-throughs (generally from 2 to 5)
c) DC feed-throughs (for biasing several chips)
d) Package wall (contains feed-throughs and defines cavity)
e) Lid (usually metal or metallized ceramic)
f) Cavity (the space enclosed by base, wall and lid)
The package can be analyzed electrically either by being considered as a complete
electromagnetic structure or as a composite assembly of the above parts. Electromagnetic
analysis using the composite model approach must consider all transmission pathsand
interactions between the basic package structures listed.
The compositeapproach
to
package modeling offers improved insight into the relationship between package
geometry and the electrical performance, since it becomes relatively simple to assess the
contribution of specific package features to the overall electrical parameters of interest.
The composite package model can be developed at various levels of accuracy or
approximation as needed. The simplest model of a MMIC package is to consider the
performance o f only the RF feed-through, whose characteristics significantly affect the
package return loss. This is termed as the “First Level” [Figure 1.7] of the composite
modeling technique. The first level model is capable of providing reasonably accurate
14
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prediction of package return loss (S i i) and insertion loss (S21) but gives no information
about isolation or possible resonances or interactions. The transmission line structures in
the feed-through section can be modeled with either static, quasi-static or full wave
techniques thereby increasing the accuracy of the numerical model.
The next level o f modeling is to include the effect of coupling between the wall
modes and the feed-through and coupling to other interconnections within the package
and is termed the "Second Level" model [Figure 1.8]. These wall to feed-through
coupling circuits can then be integrated with the first level model to provide a composite
model with improved accuracy that is capable of predicting isolation (Si 2 ) effects due to
coupling through the package wall. The package wall can be modeled in detail as
discussed in section 1.2.1.2 to consist o f several types of waveguide configurations for
rigorous characterization. In some cases, simple models can be included if the wall mode
effects are anticipated to induce minimum coupling. However, the choice o f analysis
depends on the package wall geometry and the thickness of the dielectric substrates.
The next level o f approximation is to consider the coupling through the fields
within the package cavity in addition to the effects already modeled and is termed the
"Third Level" mod^L [Figure 1.9].
As discussed earlier, cavity resonances may
significantly affect the insertion loss. It is therefore important to study in detail the cavity
fields in the frequency band o f interest The inclusion of the cavity fields completes the
composite model for a generic package. It is important to note that at each level all the
15
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interconnections between the various canonical structures are included according to their
relative importance. The composite levels determine the accuracy of the overall
numerical model o f the package.
Therefore, to predict and design the electrical characteristics of a MMIC package
requires efficient three-dimensional electromagnetic analysis of transmission line
structures and identification o f package resonances, wall modes and their contribution to
undesired coupling. Additional canonical structures can be included in the composite
model depending on their importance to the package electrical characteristics. The
composite model helps in avoiding complex details o f individual structures that do not
contribute to the required package characteristics. Depending on the accuracy required the
simulation of electromagnetic characteristics could be expedited by including or
removing specific components or canonical parts. The composite model also summarizes
the various design issues that need considerable amount of electromagnetic modeling to
predict accurately the effect of package characteristics in analog systems.
MMIC
CHIP
MMIC
CHIP
FeedThr ough
FeedThr ough
Figure 1.7 Composite Model • Level 1
16
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P a c k a g e W al l
MMI C
CHI P
M M IC
CHI P
FeedThrough
FeedThrough
Figure 1.8 Composite Model - Level 2
P a c k a g e W al l
M M IC
CHI P
M M IC
CHI P
FeedThrough
FeedThrough
Figure 1.9 Composite Model - Level 3
1.2.2 Digital Electronic Packages
The objective in this section is to emphasize the design requirements specific to
digital systems. The primary functions o f a digital electronic package are to provide
power to the semiconductor devices and circuits with a high degree o f stability and to
17
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carry signal from one circuit to another with minimum distortion. In addition, the other
objectives common to both analog and digital packages are to minimize electrical noise,
provide environmental protection and adequate heat dissipation.
The determination o f electrical parasitics such as electrical noise and cross talk
between signal lines is crucial to the design of digital packages. They arise because
electronic components do not have all the properties of idealized circuit elements and can
cause false triggering of the digital logic gates. For example real capacitors, resistors and
even simple wiring can have inductive properties, which may cause significant voltage
drops in power distribution system. These can occur during fast switching transients
when digital circuitry switches from one state to the next. Noise is also generated if
signals on adjacent lines are not completely isolated electrically from one line to another.
Accurate analyses of interconnect lines, component placement and power distributions
are crucial in the design o f a digital package and are discussed in the following sub­
sections.
1.2.2.1 Interconnect Modeling
There are two types of interconnects in digital packages namely logical
interconnects and physical interconnects. The logical interconnects define the function o f
the signal line. The physical inteconnects model and define physical connectivity of the
logic models. The logical interconnects are characterized by short rise and fall times o f
the pulses. The rise time o f a signal is inversely proportional to the frequency of the
18
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important signal components. The fast switching times o f the pulse signals require
equally fast changes in electric current from the power supply, while m aintaining the
voltage within specified limits to all devices being supplied. Meeting this requirement
demands low inductance connections to devices with high capacitance among various
levels in the distribution system.
In digital systems, for an interconnection from one device to another, the
connecting line can be treated as either a transmission line or a capacitive line. The use of
substrates with high dielectric constant introduces (see section 1.2.1.1) significant
conductor and dielectric losses that affect the electrical performance of digital
interconnects. Therefore, electromagnetic analysis must be considered in evaluating the
propagation along interconnect lines [Figure 1.10] to include the dissipative losses. This
means that voltages and currents are replaced by the concepts of electric fields and
magnetic fields respectively. Another key-determining factor for digital signals is the
pulse rise time and the conductor length, which determine the signal conductor treatment
as a transmission line or a capacitive line.
The increase in the density o f circuits per package has increased the number of
interconnections. Therefore, the assumption of instantaneous signal transmission at high
clock speeds under estimates the package delay leading to significant system failures.
Hence, it is necessary to include the finite velocity of propagation in interconnect
modeling to accurately determine the package delay. Another problem that needs careful
19
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modeling is the cross talk between adjacent signal lines and is discussed in the next
section.
Interconnection
IC CHIP
IC CHIP
IC CHIP
J
Figure 1.10 Atypical Interconnection between Digital Components
R A/
n—AAA
/Y m
A / - Length of a section
G A/<
C A/
Figure 1.11 Lumped Circuit Model of a Transmission Line
1.2.2.2 Cross-talk and Circuit Impedance
Cross-talk [Figure 1.12] is the unwanted transfer of energy by the electromagnetic
wave from the source line to other lines called as victim lines. The coupling o f voltage
and current from cross talk can induce spurious signals on the victim lines. Therefore, it
is necessary to keep the cross talk between lines below a threshold value to avoid system
breakdowns.
20
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Among various factors that cause cross talk the most significant are short adjacent
conductor lines, wide lines with less separation and lines with low line impedance.
Different logic families placed in the same package also gives rise to cross talk because of
the mixture of different voltage swings, noise margins and logic levels required. For
example, let us consider the effects due to mixing of Schottky TTL (Transistor Transistor
Logic) and ECL (Emitter Coupled Logic) logic families. The main concern in mixing
these families is coupling from TTL signals to the ECL conductors because TTL swings
are 3volts and ECL family has a logic level of lOOmV. Another factor contributing to
cross talk is the ground return path, when copper planes are used to distribute logic levels.
This problem is also called common mode impedance coupling. The returning signal
causes the rise of ground potential resulting from the DC resistance of the plane.
The cross-talk effects can be greatly reduced by including the noise in the AC
noise budget; confining logic families together and providing return paths for each logic
family. In addition reducing parallel conductors, controlling conductor to conductor
spacing, terminating conductors to reduce reflections that generate more noise and
providing multiple ground planes and power distribution on the circuit board. However,
all these remedies increase the complexity and give rise to several constraints in circuit
board technology. The main challenge for package designers is to make reasonable
tradeoffs at different levels and minimize cross-talk noise to fall within the specified
noise margins.
21
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L12
Lii • Self Inductance
Cii • Self Capacitance
Lij - Mutual Inductance
Cij - Mutual Capacitance
Figure 1.12 Cross Talk between two conductors
Signal Plane
Power Plane
B. Coaxial Line
A. Microstrip Line
.Signal Plane
Signal Plane
Power Plane
C. Wire-Over-Ground
Power Plane
D. Strip Line
Figure 1.13 Transmission Line Cross-sections in a Digital Package
1.2.2.3 Propagation Delay Time and Signal Attenuation
At high-speeds propagation times o f the signal lines are sometimes longer than
the clock cycle time o f the system. Therefore, for efficient functioning o f the systems,
propagation times must be controlled and adjustments may be required in some cases. For
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a capacitive line the propagation time is calculated assuming that the line as well as the
loads connected to it are purely capacitive. The reflections on a short interconnecting line
occur several times during the pulse's rise time, thereby slowing the edge transition times.
However, transmission line interconnects modeling results in a much faster propagation
time. Hence, close attention must be given to the type of modeling used in digital
interconnect modeling to calculate the effect of propagation delay times.
Another significant factor affecting information processing in digital systems is
signal attenuation. Signal attenuation increases the pulse rise time and decreases the
amplitude o f the pulse causing false triggering. The cause of signal attenuation can be
explained by considering the transmitting pulse as the sum of signals of several
frequencies. The high frequency components of the pulse attenuate more rapidly than the
low frequency components. This is due to the resistive losses (skin effect) in the
conductor and the dissipation effect in the dielectric. Resistive losses are directly
proportional to the square root of frequency, while the dielectric losses are directly
proportional to frequency. In the transmission line environment, the resistive losses are
described by more complex equations and all the second order field effects must also be
considered to determine them accurately. Closed form analytical expressions are available
in the literature [18,23] to calculate the signal attenuation due to dielectric losses.
Several tradeoffs must be made in designing the electrical characteristics of a
digital package. This is mainly due to the properties of device characteristics and signal
23
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transmission that have effects opposite to one another. To minimize signal attenuation by
dielectric losses, it is desirable to select materials with low dielectric constant and
conductor configurations that have low loss tangent. However for a given transmission
line geometry, propagation time is directly proportional to the square root o f e r , circuit
impedance is inversely proportional to square root of er and capacitance is directly
proportional to e r . In reality, the main goal is to design and deliver a package that is cost
effective, reliable and reproducible, which means reduce propagation delay and signal
loss, allow high density o f interconnections and minimize electrical
parasitics in
interconnect lines.
1.3 Field Analysis of Analog and Digital Package Interconnects
In summary, the requirements common to both analog package and digital
packages are, to provide the necessary interconnections between the chips within the
package, input/output interconnections and adequate thermal properties. However, the
complexity involved in modeling interconnects is the primary difference in both the
systems. Until recently, lumped elements were sufficient to predict the signal attenuation
in digital systems. But, with the increase in the clock speeds and the density of
interconnections in the digital package it is inevitable to use the distributed circuit
concept to estimate coupling to adjacent signal lines, include retardation and the
propagation delay time. Therefore, both the RF feed-through and the digital interconnect
require three dimensional numerical analysis techniques for electrical characterization
(section 1.2.1.1).
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Several numerical methods are available that analyze the dispersive characteristics
of transmission lines. Therefore, it is very important to investigate the equivalent circuit
models developed using different numerical methods and investigate the field solutions to
compliment experimental data. The numerical field analysis solutions to the transmission
lines shown in figure 1.3 are directly related to the type of Maxwell's field equations
[figure 2.1] used in describing the fields and boundary conditions. This aspect o f the field
equations lays the foundation for the relative advantages of the solution methods. Given
the complexity and fabrication costs involved in designing an electronic package, it is
helpful to use CAD tools in modeling the prototypes. However the performance of a
CAD tool is dependent on the numerical solution used to simulate field behavior and is
the subject matter of chapter 2.
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CHAPTER 2
COMPARISON OF NUMERICAL TECHNIQUES
2.1 Numerical Modeling and Analysis Techniques in EM Field Theory
Several numerical methods are available to model electromagnetic field problems
and are discussed at length in the literature [1]. Discretizing Maxwell's equations [Figure
2.1] or their equivalent form to obtain the desired solution accomplishes numerical
modeling o f complex field equations. The method of discretization and the
approximations used to model the discretized electromagnetic structures lead to the
different numerical techniques. Most of the numerical techniques used to solve
geometrically complex electromagnetic field problems employ Partial Differential
Equations, Integral Equations or Variational Methods.
The similarity between various numerical methods is that, the application
of the technique to the governing field equation results in linear simultaneous algebraic
equations. Therefore, enormous amounts o f matrix manipulation are required to obtain
numerical solutions. The size of the resulting matrices to be solved depends on the type
of the technique, desired accuracy and ease of operation. The main objective of a
numerical technique is to provide a full-wave solution, which in principle has no
frequency limitations. Several factors specific to a numerical method demand particular
type of Maxwell’s equations [Figure2.1] in problem formulation that simplifies the
determination o f EM fields. The static fields neglect both electric displacement current
and time varying magnetic flux. In general, the quasi-static solutions are formulated by
26
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neglecting the displacement currents in the Maxwell's equations. However, several other
approximations also lead to quasi-static definition of field solutions [12].
The goal of numerical methods is to obtain approximate solutions to a field
problem in actual physical environment o f the fields. Therefore, the numerical algorithm
used to solve the field problem must have a unique solution, practical in implementation
and cost effective. Several algorithms are developed to solve a particular field problem
using different numerical techniques. This leads to number o f solutions specific to a class
of problems based on the field equations or class of problems based on the geometry. The
numerical solutions based on the type of Maxwell’s equations can be classified as Static
solutions, Quasi-static solutions, Mixed solutions and Full-wave solutions.
Full-wave techniques are the most accurate solutions to predict the performance
of complicated electromagnetic structures such as those encountered in high-speed and
high- frequency systems. Unfortunately these techniques take too much time for
processing and require very expensive computing facilities. On the other hand, increase in
complexity of electromagnetic structures requires an electromagnetic modeling tool that
can predict distributed effects within realistic time and limited costs. These requirements
lead to challenging developments o f new modeling or simulation techniques that provide
efficient solutions, which are equivalent to full- wave techniques within the required
bandwidth. The choice o f a numerical method [Figure 2.2] for electrical characterization
o f EM structures is directly proportional to the ease of computing the matrix elements
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and the resulting size o f the matrices. The Next section discusses the salient features of
some important numerical methods widely used to analyze electromagnetic structures.
Different C a s e s o f Maxwell's
E quations
Static
Q uasi-Static
Full-Wave
dB
Figure 2.1 Classification of Maxwell's Equations
C h o ic e O f
N um erical M ethod
A n aly sis T ype
T im e /F re q u e n c y D om ain
E a s e o f P ro b le m
F o rm u latio n
G e o m e try o f the
S tru c tu re
F orm o f M axw ell's
E q u atio n
T y p e o f M axw ell's
e q u a tio n u s e d :
S ta tic , Q u a s i­
s ta tic o r Full w a v e
C o m p u tatio n Run
T im e s
Required
B o u n d a ry
C ondition
D isc re tiz a tio n
Figure 2.2 Factors determining the Choice of Field Analysis Technique
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Widely Used Reid
Analysis
Methods
Method Of
Moments
a. Solves
integral and
differential
equations
b. Open and
closed
geometries
Finite Element
Method
Finite
Difference Time
Domain
a. Variational
type solutions
a. Differential
equations
a. Solves wave
propogation
a. Integral
equations
b. Closed
structures
b. Closed
structures
b. Closed
structures
b. Open and
closed
structures
c. Time and
frequency
domains
c. Time domain
c. Time domain
Transmission
Line Matrix
Technique
Spectral
Domain Method
c. Frequency
domain
c. Time and
frequency
rinmains___
Figure 23 Widely Used Numerical Methods in Electromagnetic Analysis
2.2 Widely Used Numerical Techniques in EM field Analysis
The block diagram above [Figure 2.3] shows the most widely used numerical
solution methods and the form o f Maxwell’s equations (differential or integral) used in
the problem formulation of each method. The flow diagram also shows the analysis (i.e.
time or frequency domain) and the type o f problems most efficiently solved by each
individual technique. Therefore, it is clear that there is no single general numerical
technique to analyze an electromagnetic problem. The following sub-sections discuss
briefly the merits and demerits of the numerical methods and the fundamentals of
problem formulation related to each technique. Several modifications are made to the
basic analysis techniques to analyze complex electromagnetic structures.
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2.2.1 Spectral Domain Method
In the spectral domain, Maxwell’s field equations and the boundary conditions are
formulated with spatial harmonic functions. The fundamental advantage of the spectral
domain technique is the transformation of the Green’s function into a relatively simple
algebraic expression. This results in a numerical solution with small matrices with few
calculations. Alternatively, the spectral domain method refers to the application of
Fourier and Hankel integral transformations to the solution of boundary value and initial
value problems. It is categorized as a hybrid technique, due to the large amount of
analytical pre-processing involved to obtain a high degree of computational efficiency.
The spectral domain method has been extensively used to design parameters for
problems such as microstrip lines [Figure 2.4], junctions, and resonators and patch
antennas. In hybrid microwave integrated circuits the conductor thickness is negligible
compared to the length and width of the conductor. Therefore, the assumption that the
conductor thickness is infinitesimal is realistic and greatly simplifies the numerical
analysis. However, in monolithic microwave integrated circuits this is not generally the
case, and we have to consider the finite thickness of the conductor to get accurate results.
In the spectral domain method, finite thickness can be included in the formulation as an
additional layer. However, each additional layer increases the complexity of the spectral
domain solution technique. Loss parameters are usually introduced by perturbation
methods to solutions obtained using the lossless condition.
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Ground Plane
Figure 2.4 Cross-Section of an Open Microstrip Line
Problem Formulation [16]
The natural modes of propagation on a microstrip line [Figure2.4] are the surface
wave modes that are either Transverse Electric (TE) or Transverse Magnetic (TM) modes
with respect to the interface normal. These modes are also called Longitudinal Section
Electric (LSE) and Longitudinal Section Magnetic (LSM) modes [16]. For the dominant
mode on the microstrip line, we can assume that all the field components have a z
dependence o f the form i=d&. The equations describing the LSE and LSM modes are
E = V x y / h( x, y, z) y
LSE modes
(2 .2 . 1. 1)
H = V x\j/t {x, y, z)y
LSM modes
(2 .2 . 1.2 )
In the absence o f the strip, each longitudinal section mode exits from the dielectric by
itself. The presence o f the strip couples the modes. Let us define functions / and g as
follows:
(2.2.1.3)
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(2.2.1.4)
e~m g( y, a) = [eJm~j* y/e( x, y) dx
—
co
where / and g are solutions o f
\A
^ - j + iKkv2 - y 2)
dy‘
iS,
=
0
d2 ,.2
,, \ f 1
= 0
7
+
(
V
r
)
dy
[g\
0 <y<h
(2.2.1.5)
y >h
( 2 .2 . 1.6)
(2.2.1.7)
where y 2 = a 2 + p 1
and k0 is the free space wave number.
The boundary conditions for an infinitely thin conducting strip are:
Et = E. = 0
on the strip
at y = h
-W<x>W
( 2 .2 . 1. 8)
HZ - HZ = J . at v = h
where J x and J. are the components of the current density on the strip. The continuity
of the tangential fields Ex, E, at y = h and x > W requires f ( y , a ) to be continuous
at y = h and to vanish at y = 0. Hence / (y, a) is of the form
f{y,a) =
[.4(a) sin/y
y<h
[/1(a)sin//» e - ply-h)
y>h
(2.2.1.9)
where I2 = tck] - y 2, p 2 = y 2 - k 20 . The continuity of EX, E2 at y = h and fact that
Ex = E, = 0 at y = 0 require g to have the form,
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which makes (1/ k (y))(dg/ dy) continuous at y = h . From this point onwards the factor
e~jPz is suppressed for convenience. The functions J x( x ) , J . ( x ) are defined to be
identically zero for |jcj > W. This will allow us to express the boundary conditions on
H x, H, in the Fourier domain. Let the Fourier transforms of J x(x) and J . ( x ) be
J x(w) = 2 J ( x )
(2 .2 . 1. 11)
y.(w ) = 3 J s(x)
We require the conditions
and
h
:-
h
: =-j
(2 .2 . 1. 12)
x
where the symbol (:) denotes the Fourier transform. We now express H x , H : in terms
o f / and g and obtain
(2.2.1.13)
Kz0^
(2.2.1.14)
dy
and thus
w
/
—
/4(w)------ {p sin/A + / coslh) + j f i B(w)(coslh------- sin lh) = J .
K z0
*p
(2.2.1.15)
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A(w) ^ ( p sinlh + l cosIh) + jw B (w )(coslh— — sin/A) = - J I (2.2.1.16)
kaZ Q
Kp
Now let
A' =
A - ( p sin Ih + l cos Ih)
k oZ o
(2.2.1.17)
B' = B(coslh — — sin Ih)
Kp
then
wA' + j p B' = J .
(2.2.1.18)
P A ' - jw B ' = - J x
Solving the above equation (2.2.1.18) we obtain
„ _ w A ~ PJ X
A’ = — V ~ 2
P l + w2
(2.2.1.19)
5' = -
( 2 . 2 . 1. 20 )
P 2 + w2
In the Fourier domain V • J = -y*u p
becomes - _/(/? J : + w J x ) = jco p so B'
is
proportional to the charge on the conducting strip. The boundary conditions Et = E. =0
on the strip will determine J x, J . . The unknown current densities are determined by
using the Galerkin’s method [see section 2.2.5]. The current densities are expanded using
suitable basis functions. The expansion functions are tested using the same testing
functions resulting in linear algebraic equations.
The solution is obtained by matrix
inversion.
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The main advantage of the technique is that it results in a closed form analytical
solution with a small matrix size and reduces the degree o f the original field problem by
one. The shortcomings o f this technique are the large amount of analytical pre-processing
required for its efficiency. Most of the analytical processing requires solutions to
improper integrals and infinite series with only a moderate rate o f convergence.
Numerous modifications to the spectral domain method are published in the literature
[15] depending on the class of problems being solved. Finally, the simplicity and ease of
obtaining desired design information with the spectral domain method depends on the
assumptions made.
2.2.2 Finite Difference Time Domain (FDTD) Technique
FDTD provides solutions by discretizing Maxwell’s equations in differential form
over finite space and time, thereby reducing them to finite difference equations. For
uniform, homogeneous, isotropic and lossless media Maxwell’s equations governing
wave propagation are
( 2 .2 .2 . 1)
e
( 2 2 2 .2 )
dt
In order to find solutions to the above set of equations FDTD [16] uses the
Leapfrog algorithm to simulate the three-dimensional electromagnetic field information
in reasonable time. The space cell originally used by Yee in the FDTD technique is
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shown below [Figure 2.5]. The systematic calculation of electric field from the magnetic
field and then again the magnetic field from the electric field is called the Leapfrog
algorithm.
Ah
Figure 2.5 Field components in a FDTD unit cell
Problem Formulation for finite difference solutions [10]
Let us apply the finite difference technique to Poisson’s equation in electrostatics
to illustrate the problem formulation. Consider the two-dimensional Poisson’s equation in
rectangular co-ordinates.
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<f>(x,y) is the unknown potential that is to be evaluated. To express equation (2.2.2.3) in
finite differences at any point in the two-dimensional space, it is convenient to use Taylor
series expansion.
f t x + Ax, y) = *(x,y) +
<*(x - Ax, y ) = *(x,y ) - A
ox
x
ox
+ (Ax)2
ox
+ (Ax)2 d- ^ y)
dx
(2.2.2.4)
(2.2.2.5)
Therefore, in the above equations <p(x,y) has been expressed by three terms o f the Taylor
series in the x-direction assuming that it will lead to reasonable accuracy. Adding
equation (2.2.2.4) and equation (2.2.2.5) we can express the second partial derivative of
<f>(x, y) with respect to x in terms o f the values of potential at neighboring points.
d 2t ( x , y ) „ ^(x + Ax, y ) - 2<f>(x, y) + <j>(x - Ax,y)
2
dx2
(Ax)
( 2 .2 .2 .6)
A similar procedure leads to an approximation for the second partial derivative of
<p(x, y) with respect to y in terms o f the values o f potential at neighboring points.
d </>(x,y) <fi(x, y + Ay) - 2<p(x, y ) + ^(x, y - Ay)
dy2 "
(Ay)2
(2 2 2 7)
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Substituting equation (2.2.2.6) and equation (22.2.1) in equation (2.2.2.3) results in the
finite difference equivalent o f the partial differential equation.
</>(x + h,y) + <j>(x - h , y ) + <f>(x, y + h) + <fi(x, y - h ) - 4^(x, y) = - £ - h 2
s
(2.2.2.8)
where Ax = Ay = h. We can improve the accuracy of the solution by keeping the interval
h reasonably small.
For each grid point in the region o f interest, we get equation
(2.2.2.8). Therefore, the finite difference method results in n linear simultaneous
algebraic equations. If the time dependent equation is considered then we have to include
time steps at each grid point. The resulting equations can be solved by iteration, thereby
successively improving the estimate o f the variable at each grid point.
Since the FDTD algorithm directly solves Maxwell’s equations, it has excellent
capacity to solve wave propagation in complex electromagnetic structures. FDTD method
is formulated by setting the tangential electric field components to zero on conducting
surfaces. Appropriate boundary conditions must be enforced on the mesh walls and
source to get an accurate solution with few unstable solutions. Different types of
electromagnetic field excitations are used to simulate the wave propagation depending on
experience, prior knowledge o f the field distribution and desired information. The effects
of finite metalization thickness, conductor losses, dielectric losses and radiation losses on
the circuit parameters can all be taken into account in the FDTD algorithm at the expense
o f computational time.
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Several new modeling techniques and enhancements have been added to the
FDTD technique to increases its capability to provide solutions for a wide range of
problems with minimum errors. For example, structures with high dielectric constants
and dispersion at high frequencies emphasize the need to use modified FDTD algorithms
to model accurate absorbing boundary conditions. The problem formulation in all
modified FDTD techniques is the same as the original FDTD formulation. The current
field values are determined from the past values at the grid point and the surrounding grid
points. Hence, it requires an exorbitant amount o f computational time to handle structural
complexities. Moreover, it has a slow rate o f convergence for electromagnetic structures
where the ratios o f maximum to minimum dimensions are large. This undermines the
simplicity o f the original FDTD with its universal capacity to provide solutions to
complex field problems. Hence, a trade-off must be made between simplicity and the
tolerance limit o f erroneous solutions.
2.2.3 Transmission Line Matrix Technique
The Transmission Line Matrix (TLM) method is a time domain technique based
on the discrete model o f Huygen’s principle [Figure2.6]. The most valuable advantage of
TLM technique is its generality. It is extremely useful in providing efficient solutions to
wave propagation in guided wave structures and modeling arbitrarily shaped
discontinuities in electromagnetic systems. In the three-dimensional TLM technique,
space and time are discretized to obtain a discrete model of the continuous structure.
Then the entire discretized field space is modeled with a network of transmission lines.
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The intersection of the transmission lines that approximate the field at each point in space
is called a node.
Hence, we can say that the TLM technique is a physical model and not a
mathematical one. Impulses are scattered among the nodes and boundaries, in fixed time
step to simulate the electromagnetic field. The response of the resulting circuit is an
impulse response that is equivalent to a transfer function in circuit theory. In other words,
TLM solution can provide an output for any type of excitation.
Problem formulation with TLM method [17]
The basic building block o f a two-dimensional TLM network is a shunt node with
four sections of transmission line o f length V //2 . The lumped circuit model for the two
and three-dimensional transmission line node is shown in Figure2.7. To understand the
steps involved in applying the TLM technique to determine the unknown field
components, let us consider the fields in a rectangular waveguide. Comparing the
relations between voltages and currents in the equivalent circuit with the relations
and electric field component E~ of a
between the magnetic field components
TEm0 mode in a rectangular waveguide, the following equivalencies can be established:
ju = L
e =C
(2.2.3.1)
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For an elementary transmission line in the TLM network, when p r = e r = 1, the
inductance and capacitance per unit length are related by
~
=
ylLC J s eflQ
(2.2.3.2)
where c is the speed o f light.
Hence, if voltage and current waves on each transmission line component travel at
the speed of light, the complete network of intersecting transmission lines represents a
medium of relative permittivity twice that of free space. Therefore, for the equivalent
circuit shown in the figure 2.7, the propagation velocity in the TLM grid is ( c / - J l ) . For a
given TLM grid, the network voltages and currents simulate the unknown electric and
magnetic field components of the propagating modes. This is possible due to the dual
nature of electric and magnetic fields. The TLM circuit network simulates an isotropic
propagating medium only as long as all frequencies are well below the network cut-off
frequency.
There are several different approaches in formulating the TLM technique. The
main difference in the various algorithms is the TLM network node. To name a few we
have i. Symmetrical condensed node ii. Hybrid symmetrical condensed node and iii.
Expanded node. The choice of the TLM network node depends on the class o f problems
being investigated and in the reduction of simulation times. Using the expanded node in
the network to simulate the wave propagation results in a complicated network topology
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compared to FDTD solutions. The use o f condensed nodes leads to asymmetrical
boundaries when viewed from one direction compared to the other. The symmetrical
condensed node network needs additional conditions to suppress instabilities arising from
spurious modes. Hence, the type of the network node used to simulate the
electromagnetic field determines the merits and demerits of the technique. Adding
appropriate transmission line stubs to the network can represent different material
properties.
Time domain techniques have the advantage of providing unique and
stable modeling solutions. However, they are less effective compared with frequency
domain techniques in characterizing the frequency selective components o f the
transmission line structures. Significant errors are introduced by the Fourier transform to
obtain frequency domain information from the TLM technique. This is mainly due to two
reasons: i. Impulse excitation theoretically provides information for an infinite range of
frequencies and ii. The finite truncation o f the impulses in time and finite discretization of
space and time in the TLM network leads to multiple modes at the frequency at which the
transform is performed. To minimize the errors resulting from time domain to frequency
domain conversion, the time domain TLM technique has been modified to the frequency
domain TLM [20] technique. The computational costs in implementing the TLM
technique are comparable to other time domain techniques such as FDTD with a higher
degree o f accuracy. Other properties o f the solutions related to stability, flexibility and
handling irregular geometries are also comparable to the FDTD method.
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D U W TO i
Figure 2.6 Scattering in a 2D TLM Network
^
s___rm
L Al /2
Al / 2
/Y Y L — h. 4
2 C Al
Figure 2.7a Equivalent Circuit for a 2D TLM Node
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I z6
* " /L
ixW .
L Al
12 ^
Al
12
nm
fv
3 C Al
Figure 2.7b Equivalent Circuit for a Scalar 3D TLM Node [19]
2.2.4 Finite Element Method
The Finite Element Method (FEM) is a widely known numerical solution
technique, because o f its implementation in both structural analysis as well as
electromagnetic analysis.
FEM formulations are usually established via a variational
approach or a Galerkin Method of Moment [20] approach. Several different variational
formulations have been proposed in the literature [19] for use with the finite element
method.
In the FEM formulation, the entire domain is divided into finite surface or volume
elements. The elements are usually triangles or quadrilaterals. Triangles are commonly
used because they are easy to adapt to complex shapes. The unknown function, which
may be the scalar potential or a field component is approximated by a polynomial
function. A linear polynomial expansion function is used for the simplest triangular
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element [Figure 2.8] with field values at the vertices of the triangle. Higher order
polynomials are used to approximate fields with a larger number of unknowns in each
element. The polynomial functions used to expand the unknown field information over
the elements must satisfy continuity conditions over the entire domain.
To find the solution to the unknown functions, usually the Rayleigh-Ritz
procedure is applied. This procedure transforms the functional minimization into a
system of linear matrix equations. The use of infinite elements and higher order
polynomials for expansion functions increases the accuracy of the method at the cost of
increased programming effort and increased matrix density. On the other hand,
computational efficiency and memory requirements depend on the choice o f the matrix
manipulation techniques. The finite element method has established itself as the most
popular technique for two-dimensional analysis. Unfortunately, it requires a great amount
of computer time and memory for three-dimensional solutions. This is mainly due to the
requirement of absorbing boundary conditions in analyzing open electromagnetic
structures.
L in e a r
T r ia n g u la r
Prism
R e c ta n g u la r
Cube
Figure 2.8 Finite Elements used in FEM Method
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2.2.5 Method of Moments
The fundamental rule behind the Method of Moments (MOM) technique is the
idea of transforming linear functional equations to linear matrix equations. We can say
that the MOM technique is an error minimizing process with the concept of linear spaces.
EM problems are generally divided into two categories namely: i. Deterministic problems
and ii. Eigenvalue problems. Method of Moments technique can be applied to both
categories o f EM problems to obtain relatively accurate or approximate solutions.
Problem Formulation [2]
The Method of moments is a general procedure for solving linear equations. Consider the
deterministic equation
L(f) =g
(2.2.5.1)
where L is a linear operator, g is a known function and /
is a function to be
determined. Let / be expanded in a series of functions in the domain of L, as
/ =I * ,/,
(2.2.5.2)
n
where anare the constants and /„ are called the expansion or basis functions. For exact
solutions, equation (2.2.5.2) is usually an infinite summation and the /„ form a complete
set o f basis functions. For approximate solutions, equation (2.2.5.2) is usually a finite
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summation. Substituting equation (2.2.5.2) into equation (2.2.5.1), and using the linearity
of L , we get
X a nL(/J = g
(2.2.5.3)
It is assumed that a suitable inner product ( f , g ) has been determined for the problem.
The inner product is defined as
(2.2.5A)
</.g> = | f ( x ) g ( x ) d x
Now define a set of weight functions or testing functions in the range o f L and take the
inner product of equation (2.2.5.3) with each wn . The result is
(2.2.S.5)
m = 1,2,3-- s o on.
This set o f equations can be written in the matrix form as
(2.2.5.6)
[U k]= kJ
where
(wx, Lf n)
(wp Z/,)
(w ,,Z /2> ...
(w2, L f y)
(w2, L f 2) ... (w2,L/„)
i)
(wm, I / n)
u =
(2.2.5.7)
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a,
(2.2.5.8)
\gn] =
(2.2.5.9)
(Wm>g)
If the matrix [/] is nonsingular and it inverse [/]"' exists, then an are given by
(2.2.5.10)
k ] = [ /™ r u m]
and the solution for / is given by equation (2.2.5.2)
Each problem can be solved efficiently with an appropriate set of expansion and
basis functions. The choice depends on the required accuracy, ease of computing the
matrix elements, convergence and the size of the resulting matrix. The various choices for
expansion and testing functions lead to different specialized Method of Moments
technique. Some of them are i. Galerkin's Technique ( wm = f m): the expansion functions
and the testing functions are same ii. Point matching technique (w m = 8 n): the testing
functions are dirac-delta functions iii. Approximate operators: sometimes it is convenient
to use finite difference operators instead of differential operators. The success of applying
the Method o f Moments technique to a particular problem depends entirely on the choice
of the expansion/basis functions.
The Method of Moments technique has been extensively used to-date in analyzing
high frequency analog and digital circuits. Extensive literature [18-19] is available on the
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use o f this technique in time and frequency domains. Several commercially available
two-dimensional and three-dimensional electromagnetic computational packages use the
method of moments technique with the availability of PC-based matrix manipulation
techniques with low run times and good graphic facilities.
2.3 Limitations of the Various Numerical Techniques
The most valuable benefits resulting from the use o f electromagnetic
computational techniques are better performance of complex systems and lower design
expenditure. The choice of the technique and the type of Maxwell’s equations used to
formulate the problem largely determine the accuracy of the desired solution. The trends
in the industry today demand accurate three-dimensional modeling o f the electromagnetic
structures that include all the dynamic effects. This leads to the use of the numerical
techniques with nearly full-wave capabilities, which in principle have no frequency
limitations. The major factors limiting numerical techniques to perform full-wave
analysis are memory requirements, realistic run times, geometry of the structure, problem
formulation for complex structures, to name a few.
The various techniques discussed above have features that are best predicted in
either the time domain or the frequency domain. Therefore, to obtain information in both
domains using a single technique results in some penalties such as increased
programming, run times and spurious solutions to name a few. The governing field
equations determine the suitability o f a numerical technique for a particular problem. For
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example, the applicability o f the MOM for closed or open structures depends on the use
of the governing integral field equation. In the case o f scattering by arbitrarily shaped
objects, the use o f the Electric Field Integral Equation (EFIE) with the method of
moments is applicable to both closed and open bodies. Nevertheless, the Magnetic Field
Integral Equation (MFIE) provides solution to only closed surfaces.
The choice o f commercially available numerical analysis tool for field simulation
greatly depends on the numerical method used to build the characteristics of the field
problems [Figure 2.2]. For example, if radiation problems are to be taken into
consideration with the FDTD technique based CAD tool, we have to take into account the
increase in computer resources and run times to model the problem. This is directly
related to the requirement o f absorbing boundary conditions for the numerical analysis
tools based on Finite difference techniques. In some frequency specific applications, the
spectral domain method is the best choice to predict the performance characteristics but a
great amount o f analytical pre-processing is required to predict full-wave properties of the
problem.
For time domain analysis, modeling with Finite Difference Time Domain
(FDTD) technique and Transmission Line Matrix (TLM) technique (with many
similarities) can be applied to solve electromagnetic wave propagation in threedimensional structures. However, they are more suitable in analyzing closed structures in
comparison to open structures. We can conclude that the use of a particular technique or
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computer aided design tool is entirely dependent on the application and the familiarity of
the user and the physical significance of the mathematical concepts used in the numerical
methods. In addition, computation times are significant in determining the use of a
particular technique. Finally, we can say any field analysis tool can provide solution to
problems encountered at high-speeds and high frequencies within the specified design
requirements at the cost o f large computational expense. However, continued
investigations in the field o f numerical analysis techniques can lead to new modifications
in the existing techniques to solve electromagnetic fields in complex structures in realistic
runtimes.
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CHAPTER 3
FULL WAVE ANALYSIS USING RETARDED POTENTIALS
3.1 Objectives of this Research
The main objective o f the research is to investigate widely used numerical models
in field theory and develop a full-wave field analysis method to analyze complex
electromagnetic structures such as those encountered in the electronic packaging. All the
numerical methods discussed in chapter 2 have features that are more suitable for a
particular geometrical shape and type o f analysis (time or frequency domain). The main
limitations common to all the methods is the loss of simplicity, when applied to complex
geometries that demand full-wave analysis.
The goal of this research is to find a
technique that is applicable to a wide range o f problems, easy to formulate, applicable
both in time and frequency domains and obtain full-wave electromagnetic analysis in
realistic times. Therefore, the concepts o f field theory in conjunction with circuit theory
are used to develop numerical models capable o f providing full wave characteristics. The
next section lays the foundation for the relationship between field and circuit concepts
leading to dynamic numerical models.
3.1.1 Field modeling using Circuit Concepts
The formulation o f electromagnetic fields using retarded potentials is established
by Lorentz. We can express the electric field ( £ ) in terms of both, the vector magnetic
potential ( A ) and the scalar electric potential ( a ) known as retarded potentials [12].
Therefore
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( 3 .1 .1 .1 )
For the static case, E is defined in terms of the scalar electric potential. There are several
advantages in using retarded potentials in field analysis, instead o f solving for the field
vectors directly. The main advantage for this thesis is that, they are useful in expressing
the field equations in terms of circuit concepts. Another major advantage is their
relationship to the sources, J and p the current density and charge density, respectively.
Moreover, both the electric field and magnetic field can be derived directly from the
retarded potentials.
For linear media, using the Lorentz gauge the following equations are used to
define the retarded potentials.
V 2A - p s ^ - ^ - = - p J
d t2
d t1
(3.1.1.2)
(3.I .1.3)
e
The definitions of scalar and vector potentials show that only currents contribute to the
vector potential and charges to the scalar potential. However, the electric field E is
related to both currents and charges.
Among other objectives mentioned above, the aim o f this research is to obtain an
equivalent circuit model by applying field concepts and then using circuit theory to
analyze the resulting network. In field theory, Maxwell's equation in quasi-static and
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dynamic form leads to more complex distributed circuit networks as compared to the
static form that leads to lumped circuit elements. Figure 3.1 shows the alternative
definition of circuit elements in field terms that provide physical insight. Circuit theory
explicitly ignores the finite velocity of propagation or retardation o f electromagnetic
waves. Lumped element modeling of transmission lines is the starting point for field
analysis using equivalent circuit models and is described in many texts [3,10] with no
retardation. In this work, the electromagnetic field behavior is expressed using retarded
potentials and the resulting equations are solved by the method of moments [2.2.5] to
obtain an equivalent circuit network. The use of retardation takes into account the finite
time required for wave propagation. Therefore, it is equivalent to a three dimensional
full-wave electromagnetic analysis technique.
The circuit network consists of self-capacitances and mutual-capacitances to
represent the fields due to charges and self-inductances and mutual inductances to
represent fields due to current density. The inductances are in all the three orientations
with mutual coupling among elements along the same direction. This advantage results
mainly because of the use of retarded potentials. It is clear from equation (3.1.1.2), that
rectangular components of A in any one direction have rectangular components of
J as
their sources in that direction. In other words, we can write three separate equations of the
type in equation (3.1.1.2) for each direction. No such relationship exits between field
vectors and current. Moreover, both electric scalar potential and magnetic vector
potentials are continuous functions across any interface.
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To complete the relation between circuit theory and field theory the network
formulation must satisfy KirchofFs laws. The governing field equation is established
using Maxwell's equations to describe the field relations and physical properties of the
system and retarded potentials to describe the electric and magnetic fields. The field
concepts of charge (Q ) and flux (i/s) are used to relate the field behavior to circuit
theory that satisfies KirchofFs laws. The circuit element associated with charge is
capacitance and that with the flux is inductance. Charge and flux in terms of voltage and
current are defined as
Q = CV
(3.1.1.4)
y = Li
(3.1.1.5)
The determination o f the numerical field models using the circuit concepts is
discussed in the following sections. The resulting circuit network is analyzed using circuit
simulators that support frequency domain analysis with delay.
i = dq /dt
Charge
Current
Voltage
Flux
V = d y /d t
Figure 3.1 Relationship of basic circuit elements to field terms
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3.2 Problem Formulation using Retarded Potentials and Circuit Theory
In stead of solving electric and magnetic fields directly, the advantages in using
retarded potentials for determining the characteristics of propagating waves are discussed
in section 3.1.1. We also know that circuit theory offers a very wide range of analysis
techniques to handle complexity. In this work, retarded potentials along with fundamental
postulates for circuit theory are applied to electromagnetic structures to get equivalent
circuit models that can be analyzed by circuit simulators. The loop formulation uses
Kirchoffs Voltage law (KVL) to determine the governing equation and the resulting
equivalent circuit network. The nodal formulation shows the equivalency between the
continuity equation and circuit theory completing the requirements for energy
propagation.
3.2.1 Loop Formulation
Kirchoffs Voltage Law (KVL) states that the algebraic sum of voltages in a loop
is equal to zero. The application of KVL to a mesh or a loop relates the currents and
voltages across each circuit element by Ohm's law. The general expression of KVL for
circuits is
2>„=0
(3.2.1.1)
n
where n is the number o f circuit elements in a single loop or mesh.
In field theory, the relation between current density ( J ) and electric field
intensity ( E ) is necessary to complete the problem specification. Therefore, Ohm’s law
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can be used to define a part or the entire path to relate the flow o f current to the electric
field [1] in a conductor. In the electromagnetic environment, magnetic fields are excluded
in the definition o f Ohm's law, which is
( 3 .2 . 1.2 )
£ =—
a
where cr is the conductivity and J c is the conduction current density. The assumption in
using Ohm's law is that the entire conducting system can be specified by its conductivity.
At any point in the conductor if the distribution o f currents and charges is known
electromagnetic fields can be determined by Maxwell's field equations. However,
currents and charges do not appear explicitly, but the retarded potentials [equations
3.1.1.2 & 3.1.1.3] at any point in a field space are defined in terms of current and charge
densities. Therefore, the electric field is expressed in terms of retarded potentials and the
distribution of unknown currents and charges are determined using the method of
moments technique [1.2.5].
If an external field is impressed on the system, the £ field at any point in a
conductor is the sum o f the applied field ( £ 0) and induced field ( £ ' ) . The induced part
of the £ field is due to the sources (charges and currents) arising from within the
conductor itself. Therefore, the total £ field at any point in the conductor is
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The solution to wave potential equations (3.1.1.2 & 3.1.1.3) result in the following
definitions for retarded potentials
<P(r) =
(3.2.1.4)
4i t e R
(3.2.1.5)
where
(3.2.1.6)
Substituting equation (3.2.1.3) in equation (3.2.1.2), we get
(3.2.1.7)
This is the governing field equation. This governing equation has several advantages. The
fundamental advantage is that, it is a complete equation specifying the fields at every
point and is directly related to the physical properties o f the system. Secondly, it does not
need boundary conditions to uniquely determine the field distribution. The most valuable
advantage is that it relates the field properties to the concepts o f circuit theory through
appropriate analogies resulting in the equivalent circuit network.
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Let us consider a closed conducting path. Integration along the closed path leads
to the integral field equation that defines the potential at any point in the conducting
medium.
j E Q*dl -
• dl - <^<j) • dl - ^
• dl = 0
(3.2.1.8)
Now, the task is to identify and determine the equivalent circuit parameters from their
field properties. Each term in equation 3.2.1.8 is dimensionally equivalent to a voltage.
To relate the field equation to circuit theory, we can say that each is equivalent to a
voltage drop across a circuit element. Therefore, equation 3.2.1.8 is equivalent to the
algebraic sum of voltages in a closed loop satisfying the general form of Kirchoffs
Voltage Law (KVL). In circuit theory, we can write the field equation 3.2.1.8 in terms of
basic circuit elements for a single loop as
di
(3.2.1.9)
where V0 is the voltage due to the applied electric field. The other three terms are
identified as a resistor, capacitor and inductor from their relationship to field concepts as
shown in figure 3.1. Therefore, the equivalent circuit is a series resistor, capacitor and
inductor with a voltage source and is shown in figure 3.2.
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The field equation 3.2.1.8 describes the field behavior at every point along the
closed path. Each field point gives rise to the equivalent circuit described by equation
3.2.1.9 resulting in a network consisting of uniquely defined values for the circuit
elements at that field point. Thus, we have interpreted equation 3.2.1.8 as several
equivalent circuit loops consisting o f capacitors, inductors, and resistors. Each loop in
theory is infinitesimal in extent, infinite in number and coupled to other loops because of
the continuity of electromagnetic fields. In reality the number of loops is finite resulting
in the discretization o f the field equation into a distributed network consisting of self and
mutually coupled terms with delay. The next step is to prove the equivalence between the
field and circuit relations and determine the values for the equivalent circuit elements.
Vn ^ b -
Figure 3.2 Equivalent circuit network for field equation 3.2.1.8
I. Inductive Elements
The contributions from the last term in equation 3.2.1.8 are due to inductances in
the circuit domain. To show the equivalence and derive a formula to compute the
equivalent inductances, the inductive term from the integral field equation is
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where A is the retarded vector potential defined in equation 3.2.1.5 and dl is the
differential length along a closed loop o f wire [Figure 3.3].
p
T
Figure 3.3 A Closed conducting path P and differential current elem ents along the
path
All the terms in equation 3.2.1.8 represent voltages across equivalent circuit
elements. Therefore, expression 3.2.1.10 represents the voltage drop across several
inductive elements that comprise the conductive path. Substituting equation 3.2.1.5 in
expression 3.2.1.10, the voltage across equivalent inductive elements is
(3.2.1.11)
where - jco results from taking the partial time derivative o f A , J ( r ') is the current
density and e~jkR is the retardation in the form of a phase shift. The volume V includes
the entire region that contains current and charge sources.
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Equation 3.2.1.11 is an integral equation with unknown current sources and can
be solved using the Method o f Moments [Section 1.2.5] technique. The unknown current
density J ( r ') in the source region is expanded in terms of known 3D dyadic pulse
functions. The relation between the unknown current density and the expansion functions
is
V
(3.2.1.12)
where N is the total number of volume segments in the entire volume V and pn is
diagonal dyadic with the following definition [9]
(3.2.1.13)
ex, e2 and e2 are unit vectors along x, y and z axes respectively. Pn are 3D pulse
functions defining the current density in an elemental volume. Their properties are
defined as
on all Vv'
(3.2.1.14)
elsewhere
In the expansion, functions [Equation 3.2.1.12] J n are the unknown vector functions and
can be expressed as
j
(3.2.1.15)
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where J nini are the unknown coefficients to be determined.
Substituting equation 3.2.1.12 in the integral equation 3.2.1.1, the voltage across an
equivalent inductor is
v <
-
H£ J- *
e-m
)
p-
P V
(3.2.1.16)
'
Since Pn are 3D dyadic pulse functions with magnitude equal to unity in each volume
segment Av ', they have the same properties as Idem dyad [9]. Therefore
(3-2.1.17)
l-l
Substituting equation 3.2.1.17 in equation 3.2.1.16
yL
\j„ ^ -d V e ,-d l
™
p n-1 i»l V
(3.2.1.18)
^
The contribution o f individual charges to the total current flow in a conductor is
expressed by current density. In equation (3.2.1.18), J ni is the current density in each
volume segment. Let us define a current element as the magnitude of current times the
length over which it extends. The current density can be considered a volume density of
current elements. With this definition the distribution o f current in each volume segment
can be written as [12]
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Therefore J mdS' = I m is the average value o f the current over both dS' and dl' in each
volume segment. The differential length dl in equation 3.2.1.11 is represented as
d l = Y ddlJej
(3.2.1.20)
where e ,, e2, e3 are along x, y and z-axes respectively as previously stated.
Substituting equation 3.2.1.20 and equation 3.2.1.19 in equation 3.2.1.18, we get
(3.2.1.21)
From vector algebra, we know that
e, • e / = 1 for i = j and
e, • kj = 0 for / * j
(3.2.1.22)
Therefore, substituting equation 3.2.1.22 in equation 3.2.1.21, the voltage drop
across the inductive elements can be decomposed into three equations. Each individual
equation is equivalent to the voltage drop across inductive elements in the x, y and zdirections respectively. Another inference from this decomposition is that there is no
coupling between inductive elements in different directions. Coupled inductance exists
only between elements in the same direction, leading to the calculation of three
inductance matrices. Since all three equations differ only in the direction, for simplicity
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let us solve the general equation with no subscripts (denoting the direction) to determine
the inductance element matrix in each direction.
The next step in the method of moments technique is to test the expansion
functions along the entire path P . A field point is chosen in each source region.
Therefore, for N source segments we consider m=l to N test points to weight the results.
The choice of the testing functions determines the accuracy of the solution in addition to
factors affecting other properties o f the solution [see section 1.2.5].
We choose the
testing functions to be same as the expansion functions, which is known as Galerkin’s
method [2]. We can rewrite equation 3.2.1.21 as follows by applying the pulse testing
functions defined in equation 3.2.1.14.
(3.2.1.23)
Equation 3.2.1.23 is the general equation describing the voltage drop across several
segments in the closed conducting path. Therefore, each segment can be considered as an
equivalent circuit element resulting in a network of inductive elements. To equate them to
equivalent inductive circuit elements, the current-voltage relationship across a lumped
inductor L is
(3.2.1.24)
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For the assumed sinusoidal sources, the current I can be expressed as I = /„ e~j a . This
representation provides an efficient solution to include retardation along with the lumped
circuit elements in the complex phasor method. The voltage drop across the inductor in
phasor concept can be expressed as VL = -jco L I n. Using this relation, we can rewrite
equation 3.2.1.23 in matrix notation as
K M
dl
^
n
1-™.] dt
(3.2.1.25)
where
L2
(3.2.1.26)
L^v
-j(d x
-ja li
(3.2.1.27)
dt
_ -M v _
(3.2.1.28)
Ann = — f f— dl' dl
An}} R
Thus, we have determined the inductive circuit elements in relation to the
inductive term [Equation 3.2.1.10] in the general field equation 3.2.1.8. The factor e 'JkR
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in equation 3.2.1.28 represents the shift in phase resulting from finite velocity of wave
propagation or retardation effects. If we consider retardation effects negligible, equation
3.2.1.28 is equivalent to the Neumann’s formula [6] for the inductance of filamentary
circuits. In other words, with negligible phase shift equation 3.2.1.28 is a lumped
inductance circuit and is applicable at low frequencies, when the circuit dimensions are
small compared to wavelength.
However, at high frequencies and in the applications related to electronic
packaging it is necessary to consider the finite effects of wave propagation. Let us
interpret equation 3.2.1.28 in terms of lumped inductances and phase shift. The factor
e~JkR makes it different from the conventional way to compute the inductances using
Neumann’s formula and other inductance formulae [8]. Therefore,
(3.2.1.29)
where
(3.2.1.30)
and (pn, is the phase delay between segments m and n.
Substituting equation 3.2.1.28 and equation 3.2.1.30 in equation 3.2.1.29 provides the
following equation in terms o f phase shift times the lumped inductance.
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Now applying the limit l'n —►0 and /m -> 0 to equation 3.2.1.31 results in
e-jkr'-<M Al'n
;
f —=- = e
, n “ r,'
ft |
A/ A/'
—55— - nK\i
I n
(3.2.1.32)
where
e-i*~ = e -J*rm-i
(3.2.1.33)
Substituting equation 3.2.1.33 in equation 3.2.1.29 leads us to a simplified form of
computing inductance with retardation. The finite propagation velocity is computed in
terms of the distance between the centers of two segments.
[L _]
(3.2.1.34)
This decomposition allows us to use the relations and analytical formulae used to
determine the inductive elements with no retardation, to include the finite propagation
effects. For self inductance rn = r„' and there is no phase shift, but the phase shift affects
all the mutual inductance elements computed by using the formulae with no retardation
[8].
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II. Capacitive Elements
Next, we consider the contributions from the third term o f the field equation
3.2.1.8 to the equivalent network. From the properties of electrostatic fields, we know
that
= 0 . Therefore, consider a small gap in the closed conducting path to
determine the contributions from the scalar potential function to the equivalent network.
It is also well known from electrostatic fields that this term provides the capacitive
elements. Thus the term providing the capacitive elements is
a
rc = |v^«rf/
(3.2.1.35)
b
where Vc is the voltage drop across the equivalent capacitor terminals, <f> is the scalar
potential and dl is the differential length along the path [Figure 3.4]
df
Figure 3.4 A Closed path P with a small gap for capacitive element
In equation 3.2.1.35, the gradient of <j> is the vector sum o f E field variations in
all the three directions. This is because the gradient shows the magnitude and direction of
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the maximum space variation in the function, at any point in space. The scalar product of
the gradient o f <f> and an element o f length di is equal to the change in tf>. Hence, we
have
(3.2.1.36)
d<j>= V<t>*dl
Substituting equation (3.2.1.36) in equation (3.2.1.35)
b
(3.2.1.37)
Vc = jd<t> = tpb -<pa
a
Thus, equation 3.2.1.37 is the potential between any two field points a distance
dl apart in space. The potential at any field point can be arbitrarily fixed and then that
definition of the potential can be used to determine the potentials at different points with
respect to the reference point. Therefore, substituting the definition of scalar potential
[Equation 3.2.1.4] in the equation (3.2.1.37) the potential at any point in space with the
reference at infinity is
(3.2.1.38)
where e = e„ s r . s Q is the permittivity of free space and e r is the relative permittivity of
the medium and is equal to unity for free space.
Hence, the term from equation 3.2.1.8 is reduced to the integral equation 3.2.1.38,
defining the potential at each point with respect to infinity along the closed path P. The
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solution to equation 3.2.1.38 using Method of Moments technique [Section 1.2.5]
completes the relation between concepts and circuit theory. The unknown charge density
p(r') is expanded in terms o f known expansion functions. The expansion or basis
functions are 3D pulse functions, the same as those used in the derivation o f the inductive
elements. Mathematically the unknown charge density is expressed as
V
(3.2.1.39)
where p n are the unknown coefficients to be determined and Pn are the 3D pulse
functions [Equation 3.2.1.14] and N is the number of segments along the conducting path
P.
Substituting equation 3.2.1.39 in equation 3.2.1.38
v
(3.2.1.40)
The properties of 3D pulse functions are defined in equation 3.2.1.14. Using the pulse
function properties in equation 3.2.1.40, we get
(3.2.1.41)
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For an infinite number o f segments along the conducting path, we can rewrite equation
3.2.1.41 as follows
(3.2.1.42)
From field concepts, it is well known that charges in a good conductor distribute
themselves in a thin layer at the surface [10,12]. Therefore, we can approximate the total
charge in an elemental volume as the sum of the charges on all the six surfaces i.e.
6
(3.2.1.43)
Substituting equation 3.2.1.43 in equation 3.2.1.42 and letting the surface charge vary
continuously, we get
(3.2.1.44)
The field term in equation 3.2.1.8 is reduced to equation 3.2.1.44 applying the
Method o f Moments technique. The next step in the procedure is to test the expansion
functions in each segment.
As before, using the Galerkin's method the testing or
weighting functions are the same as expansion functions. The inner product of equation
3.2.1.44 with the 3D pulse testing functions results in
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The surface charge density in a segment p sni is the charge per area on an element of
infinitesimal surface. Therefore,
9 ni
P sm
(3.2.1.46)
^ ni
Substituting equation 3.2.1.46 in equation 3.2.1.45, we get
(3.2.1.47)
”
"V S m S'v
The above equation computes the voltage across an equivalent capacitive element.
For i = j the potential difference is computed for self and mutual segments on the same
surface and for i * j the potential difference is computed for segments on different
surfaces. For providing interpretation in relation to circuit theory, let us drop the
subscripts / and j for simplicity. Thus, equation 3.2.1.47 for any single surface or mutual
surfaces can be written in matrix notation as
(3.2.1.48)
where
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cl
cl
K
(3.2.1.49)
h
c.V.
PsX S X
Psi S i
(3.2.1.50)
u =
P sN
P .. =
V
!—
Ak s S
w
£ _
j
(3.2.1.51)
*
The matrix elements defined in equation 3.2.1.51 are called the coefficients of
potential [12]. The coefficients o f potential depend only on the geometry of the segment
and the conductivity of the system. The mathematical properties of coefficients of
potential and the determination o f coefficients of capacitance and their relationship to two
terminal circuit capacitances are discussed in detail in Chapter 4, Section 4.2. Thus, the
third term in equation 3.2.1.8 is equivalent to capacitive elements in circuit theory. The
coefficients o f potential described by equation 3.2.1.51 can be written in terms of non­
retarded coefficients o f potential following the same argument given for inductive
elements as
fe.
(3.2.1.52)
[p - j
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where
[PmJ =
4 x e S I«j
miS,ni...
cJ
(3.2.1.53)
JR
S „ Sm
are the coefficients of potential with no retardation and can be computed using the
analytical formulae derived in the literature [7].
III. Resistive elements
The second term in equation 3.2.1.8 can be identified as resulting form the
internal resistances of the conducting wire. This term contributes to the voltage drop
across resistive elements. Therefore
(3.2.1.54)
where VR is the voltage across a resistive element.
Let us consider a simple resistor consisting of a rectangular bar of conducting
material as shown in the figure 3.5, where I is the length of the conducting wire, cr is the
conductivity o f the material and / is the uniform current flowing through the resistor.
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I = length
■4--------------------- ►
A = C ross sectional area
Figure 3.5 A simple resistor consisting of rectangular bar of conducting material
The current density in the conducting bar of wire is
(3.2.1.55)
|Ji= —
11 A
where A is the area o f cross-section of the wire.
Substituting equation 3.2.1.55 in equation 3.2.1.54
v.-j—
(3.2.1.56)
Dividing the closed path P into N segments and assuming that the current is
uniform in the segment with length dl and using the properties o f dot product [Equation
3.2.1.22], we can rewrite equation 3.2.1.56 as
(3.2.1.57)
-i a ,
a
where /„ is the current in a segment of length /„ and area o f cross-section A.
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The resistance of a segment with varying length [Figure 3.6] and cross-section is defined
as
(3.2.1.58)
Substituting equation 3.2.1.58 in equation 3.2.1.57
•V
(3.2.1.59)
Thus, the equation 3.2.1.59 provides the relationship between the second term in
the equation 3.2.1.8 to the resistive elements in circuit theory.
IV. Applied Voltage Term
The first term in equation 3.2.1.8 is due to the applied electric field. If the applied
electric field is zero, equation 3.2.1.8 consists o f only the circuit elements. The response
o f the resulting equivalent network in circuit theory is called the natural response. In the
presence of the applied field, the term can be identified as an applied voltage producing a
forced response in the circuit domain. Therefore, the potential between the ends of a
conducting wire dl apart is
(3.2.1.60)
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This completes the contributions o f the field terms in equation 3.2.1.8 to the
equivalent circuit elements in the circuit domain. In conclusion, we can say that equation
3.2.1.8 is the algebraic sum o f voltages in a loop contributing to the voltage drops across
the inductance, capacitance, resistance and voltage source in the network. Therefore, we
have proved that the field equation 3.2.1.8 is equivalent KirchofFs Voltage law in a loop
or mesh. The closed conducting path consists of several loops due to the discretization of
the field space and each segment is equivalent to a node in the circuit domain. At each
node, the algebraic sum o f the currents must be equal to zero to satisfy KirchofFs current
law, which is the basis for the nodal formulation and is discussed in the next section.
3.2.2 Nodal Formulation
The second fundamental law in the circuit domain is KirchofPs Current Law
(KCL), which states that the algebraic sum o f all the currents at any node is equal to zero
[Figure 3.6], In the circuit domain the general expression for KCL at a node is
£ /.= 0
(3.2.2.1)
n
where n is the number o f circuit elements connected to each node and /„ is the current
through each element.
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Figure 3.6 Field Model and Circuit Equivalent of KirchofPs Current Law
If a charge q collects at the node, we can write equation 3.2.2.1 to account for this
excess charge as
(3.2.22)
For showing equivalence between field concepts and circuit theory, it is more convenient
to view equation 3.2.2.2 as the KCL at a node.
From the field concept, let us consider a field point (junction) in the conducting
path with the closed surface S as shown in figure 3.7. The distribution o f the currents at
any point in field space is defined by the current density J and the charge distribution.
The currents leaving the junction are equivalent to the currents through an inductive,
capacitive, resistive and applied field. Therefore
(3.2.2.3)
s
Substituting equation 3.2.2.3 in equation 3.2.2.2, we get
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(3 .2 .2 .4 )
which is equivalent to the continuity equation and shows the property of conservation of
charge. The next step is to interpret equation 3.2.2.4 in terms o f circuit elements in the
equivalent system network.
The current density J in equation 3.2.2.4 is the sum of the current density due to
the externally applied field and the induced current density is due to the induced fields.
Therefore
(3.2.2.5)
In addition
(3.2.2.6)
where J c is the conduction current density and J d is the displacement current density.
For linear, isotropic media the definitions o f conduction and displacement current
densities are
(3.2.2.7)
J c =a E
(3 .2 .2 . 8)
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In a d d itio n , th e c o n s titu tiv e r e la t io n s a re
D =s E
( 3 .2 .2 .9 )
B=nH
(3.2.2.10)
where n is the permeability and e is the permittivity of the medium.
Substituting equation 3.2.2.5 to equation 3.2.2.9 into equation 3.2.2.4, we obtain
(3.2.2.11)
We know that the E field can be expressed in terms of scalar and vector potentials and
is defined in equation 3.1.1.1. Substituting the equation 3.1.1.1 in equation 3.2.2.11, we
get
<^y0 • dS + § J C»dS .
,
p
dt
• dS - £<£j— y- + y = 0
p 5/
dt
(3.2.2.12)
Thus, equation 3.2.2.12 is the field equation describing the distribution of the
currents at any point along the conducting path. This equation is the dual of equation
3.2.1.8 and is in the form that relates the field concept to the circuit concept at a node
when the excess charge q is equal to zero. The equivalent circuit equation for the field
equation 3.2.2.12 with q = 0 is
(3.2.2.13)
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‘0
Figure 3.7 Equivalent circuit network for field equation 3.2.2.12
Therefore, the field equation 3.2.2.12 is equivalent to a circuit network consisting
of resistors, capacitors and inductors satisfying equation 3.2.2.13 and the equivalent
circuit is shown in figure 3.7. This equation also shows that the equivalent network
describes the admittance concept in the circuit domain [24]. We can partition equation
3.2.2.12 as contributions from several equivalent circuit elements using the definition of
retarded potentials. Each term in equation 3.2.2.12 provides the interpretation to an
equivalent current in the corresponding circuit element. The first term is the applied
current, the second term is the conductance term, the third term is the capacitance term
and the fourth term is equivalent to the inductance term (actually the reciprocal of the
inductance). This leads to the conclusion that the algebraic sum of currents at a node is
equal to zero.
Thus, we have demonstrated that the general field equation at any point in the
conducting path satisfies the Kirchoffs Current law. In section 3.2.1, we showed that the
segments along the conducting path are equivalent to loops or meshes, with each loop
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consisting o f basic circuit elements in the resulting network. In this section 3.2.2, the
KCL is satisfied at each node formed by the intersection of the basic circuit elements in
the network.
3.3 Dielectric Modeling
In section 3.2, the relationship between field theory and circuit theory has been
demonstrated and equivalent circuit models for the electromagnetic fields in a conductor
are developed. To model dielectrics, we start with the general field equation 3.2.1.7,
which is true for any point in space. For a field point in a dielectric with the applied
electric field ( E0) equal to zero, we have
£ ,( r ) + V # r ) + ^
L
ot
0
(3.3.1)
where Ed is the electric field in the dielectric medium.
In general the current density
(J),
is equal to the sum o f conduction current
density ( J c ), displacement current density ( J D) and polarization current density ( J p).
To model the conductor segment only the conduction current and the displacement
current due to the time varying electric field were included in the theory in section 3.2.1.
To find the equivalent network for a dielectric segment the polarization current must be
included explicitly [14]. The polarization current density in the frequency domain is
defined as
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Jp —
— jca(sr
1'jSgEj
(3.3.2)
where P is the electric polarization [10] and the time dependence term e~ja* has been
suppressed for the sake o f simplicity throughout the thesis document.
Following the same procedure as described in section 3.2.1, let us consider a closed path
in the dielectric to determine the equivalent circuit for a dielectric segment. Therefore, we
have
(3.3.3)
Equation 3.3.3 is the governing field equation in the dielectric region and each term is
equivalent to a voltage drop across an equivalent circuit element. The next step is to
interpret the field equation 3.3.3 in the dielectric in terms of circuit concepts to determine
the equivalent circuit elements.
I. Excess Capacitive elem ents in the Dielectric
The first term in the general equation 3.3.3 is interpreted as the voltage drop
across a capacitance, which is called an excess capacitance to differentiate it from the
capacitive terms due to the charge density p . The term is
(3.3.4)
j
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Substituting equation 3.3.2in equation 3.3.4, we get
• d] = _ • /
*dl
(3-3 -5>
The closed path / consists o f N -dielectricsegments and the current density J P in each
segment can be expressed as
j p = -I7~
As
(3-3.6)
where I p is the current through the segment, As is the cross-section area of the segment
and e, is the unit vector.
Substituting equations 3.3.6 & 3.2.1.20 in equation 3.3.5, we get
Assuming uniform current in each segment, constant cross-section area of the segment
and the properties o f the dot product [Equation 3.2.1.22] in equation 3.3.7, we get
iEd > d l
j
, 1
-J0){sr - l K t f A s ,
(3.3.8)
in each direction.
Therefore, the excess capacitance is defined as
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(3 .3 .9 )
In the time domain
(3.3.10)
Substituting equations 3.3.9 & 3.3.10 in equation 3.3.8, the voltage drop
across each
dielectric segment due to the polarization current density is
(3.3.11)
Therefore, the first term in the field equation 3.3.3 is equivalent to the voltage
drop across the excess capacitance of the segments. The excess capacitance is equivalent
to a parallel plate capacitance with the area of the plate equal to the cross-section area of
the segment and the distance equal to the segment length. The excess capacitances are
defined [Equation 3.3.7] for each dielectric volume segment in all the three directions.
II. Capacitive elements in the Dielectric
We have already shown in section 3.2.1 [Equation 3.2.1.38] that the second term
o f equation 3.3.3 is equal to the voltage across an equivalent capacitance. The electric
polarization P is defined as
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P = D d - s 0Ed
( 3 .3 .1 2 )
where Dd is the electric flux density in the dielectric medium.
The potential at any field point in the dielectric with the reference at infinity is
K =*(»•)= T “ l P , i r l e ‘ d V
47T£0 j.
R
(3.3.13)
where p p is the charge density due to the polarization effects in the dielectric.
The solution to equation 3.3.13 leads to the determination of the equivalent
dielectric capacitive elements. Following the procedure as described in section 3.2.1
[Capacitive Elements] we get
/at m
ma
1 M
«lt nf l«a tl
lal
" m
^
mi
mj
C
S'm
*•
where N is the total number o f dielectric segments, R is given by equation 3.2.1.6 and
q pm is defined as
(3-3.15)
<lpni = P tJmi S 'm
where p spni is the dielectric surface charge density on an element o f infinitesimal surface.
For dielectrics [14], the unknowns are the bound charge and the polarization
currents to determine the electric field. The relative permittivity e r, which is easily
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measurable, accounts for all the polarization effects. The dielectric surface charge density
p sp is expressed in terms of the conductor surface charge density p s as
(3.3.16)
Substituting equation 3.3.16 in equation 3.3.14 and expressing the result in matrix
notation, we get
(3.3.17)
where \Vcm] is same as in equation 3.2.1.49 and [qnp] is same as equation 3.2.1.50 and
[pm« ] = —
L ^
J
^ — W ^ — dS'dS
4 « offr5 . s : sy t
R
(3.3.18)
The difference between the analytical formula for the coefficients of potential on
the dielectric surface and that on the conductor surface [Equation 3.2.1.51] is the factor,
(er - l)/er . The divergence of the electric flux density Dd depends only on the free
charge density [12,14]. Therefore, the capacitive coupling between conductor segments
and the dielectric segments is independent o f the dielectric constant e r.
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III. Inductive elements in the Dielectric
As already stated in section 3.2.1 [Inductive Elements], the contributions from the
last term in equation 3.3.3, are due to the inductances in the circuit domain. The term is
cf— *dl
]dt
(3.3.19)
and A is given by the equation 3.2.1.5.
Substituting equation 3.2.1.5 in the expression 3.3.19, the voltage drop across an
equivalent inductive element is
V, =
1
—
jo ) Li r e J „ ( ? ' ) £ ^
<f f— —
Arc
j}
dV'»dl
(3.3.20)
R
where the current density J p is due to the polarization effects and is the unknown.
Applying the Method o f Moments technique as discussed in section 3.2.1 [Inductive
Elements], we get
(3.3.21)
m-l n*t
r jm
K
Substituting equation 3.2.1.24, we can rewrite equation 3.3.21 in matrix form as
[rj= [£ j
pn
dt
(3.3.22)
where [F^ ] is same as equation 3.2.1.26, [L^ ] is given by equation 3.2.1.28 and
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-Jalpi
dl pn
-ja lp i
(3 .3 .2 3 )
dt
- j CDI p N
where I pn is the current in the dielectric segment.
We have shown in equation 3.3.11 that the electric field in the dielectric gives rise
to an excess capacitance due to the polarization effects. Since the dielectric segmentation
is same along the path /, the current ( 1 ^ ) resulting in the excess capacitance must be
equal to the current ( I pn) in the inductive segment. Therefore, for the dielectric volume
segment the equivalent circuit is an excess capacitance in series with the inductance. The
surfaces o f the dielectric are covered by bound charges, which give rise to the surface
capacitances. The equivalent circuit for a dielectric current between two nodes at the
surface is shown below [Figure 3.8]. The volume currents in the dielectric are represented
by the excess capacitances, the self-inductances and mutually coupled inductances. The
coupling to the other conductors is through the mutual inductances and mutual
capacitances [31]. The difference between conductor modeling and dielectric modeling is
the excess capacitance. The self-capacitances and inductances, mutual capacitances and
inductances are determined from the same analytical formulae.
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Surface Node
Self Inductance
o —1
Excess capacitance
Note: All inductances
are mutually coupled
Figure 3.8 Equivalent circuit between a pair of nodes in the dielectric
3.4 Summary
The formulation o f the field equations governing the electromagnetic wave
propagation using retarded potentials and their connection to circuit theory is shown
using the method o f moments technique in a conductor and dielectric. The choice of
expansion functions is chosen to be same as the testing functions in the implementation
of the method o f moments, which is called the Galerkin’s technique. The advantage of
using the Galerkin’s approach over point matching approach is that it produces better
approximation o f the unknown in the segment.
The physical properties of the structures are governed by Maxwell’s equations
and the vector and scalar potentials are used to describe the electric and magnetic fields.
Ohm’s law in field form along a conducting path is interpreted in terms of circuit
elements that provides the proof for Kirchoffs Voltage Law. Kirchoffs Current Law is
proved at each node of the circuit network by showing its equivalence to continuity
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equation and the governing field equation. The resulting network is made of several
closed circuits called meshes or loops and coupled circuit elements. Each loop has
capacitors, resistors, inductors that are coupled to all the other circuit elements. The
circuit elements represent the charges and currents and the displacement currents are
represented by the retardation. The expressions derived to compute the inductive,
capacitive and resistive elements depend on the geometry of the segments.
Therefore, the solution to the field equations in terms of circuit elements
completes the relation between field theory and circuit theory. Each circuit element
derived has a delay associated with it, which is equivalent to the lumped circuit parameter
multiplied by the delay. Thus, the lumped circuit model of the electromagnetic structure
along with delay for each circuit element is equivalent to a full-wave analysis technique.
The technological advances in electronic packaging have led to transmission lines with
complex geometries such as non-uniform cross-sections, finite thickness and use of
substrates with high dielectric constants. The characteristics of the complex transmission
lines can be evaluated using retarded potentials with substantial mathematical
simplification. This approach shows it is sufficient to know the external characteristics of
the wave propagation in the transmission lines to predict their behavior at high
frequencies and high speeds.
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CHAPTER 4
ANALYSIS AND SIMULATION OF EQUIVALENT CIRCUIT
NETWORK
4.1 Description of Equivalent Circuit Parameters
Each equivalent circuit parameter [Figures 4.2 & 3.8] in the resultant distributed
circuit model with delay is related to the field quantities (charge density, current density
and material properties) as discussed in sections 3.2.1 and 3.3. However, to make use of
the simple and intuitive ideas o f circuit analysis to solve electromagnetic problems, it is
necessary to represent the field distribution as accurately as possible to determine the
equivalent circuit models to get meaningful results. The loop formulation [Section 3.2.1]
allows us to determine the two-terminal capacitances indirectly from the coefficients of
potential. On the other hand, the circuit inductances are computed directly. In this section
the relationship between the equivalent circuit model and the two terminal circuit
elements used in circuit theory is discussed to obtain the circuit file for network analysis
using a circuit simulator.
4.1.1 Capacitive Elements
The equivalent capacitances are determined from the third term of the general
field equation 3.2.1.8. It is well known in electrostatics, that for two-conductor systems
the capacitance is defined as the charge on one conductor divided by the potential
difference between the two. However, for a system with several conductors capacitance is
defined as a matrix consisting o f self-capacitance and the coupling between the
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conductors. This concept is used in the retarded potential circuits, where the segments are
considered as conductors in the system. From the definition of scalar potential [Equation
3.2.1.4], it is evident that the charges qi,q 2,...qM on n segments are linearly related to
the potential at any point in field space on each segment. Using the method of moments
technique and Galerkin’s procedure, the set of linear equations [Equation 3.2.1.48 in
matrix notation] describing the potentials on each segment are
K=
^ 11^1 +
^ 12^2 +
^*1 3 ^ 3 +
""+
V2 =P2xqx + Pn q 2 + Pa q 3 + ....+
Vs
+
^ 2 ^ 2
+
^V 3^3 +
— +
P2Nq s
^m ^s
The coefficients P ^ are known as coefficients of potential. The diagonal terms
Pu Pn
PSN are real and the off-diagonal terms are complex if retardation is included.
All the Pm terms are real if retardation is neglected. In this work, the retarded
coefficients of potential are expressed in terms of the non-retarded coefficients of
potential by equation 3.2.1.51.
The linear equations 4.2.1 can be solved for charges leading to another set of
linear equations relating charges to potential on each segment as follows.
q { =CUV{ +
C,2^2 + C13^3 + ............... + c i
q2 = c 21F, + c ^ y 2 + C23F3 +•.......... + c2NVv
......................................................................................
Qs —CSlVl +
c \ 2 ^ 2 + C V3^3 + ............... +
Cm Vs
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(4.2.2)
where
k . H
(4 .2 .3 )
f J "
The coefficients c ^ a re also known as short circuit capacitances, coefficients of
capacitance, coefficients of induction and coefficients of sub-capacitance in the literature
[3,7,10]. The two terminal capacitances in the equivalent circuit network are determined
from equation 4.2.3 by the following conversion formulae [3].
.V
Cnn = Z Cmn
^ = 1,2,3
(4.2.4)
JV
(4.2.5)
where Cnn are the capacitances from the node to the reference and
are the coupled
capacitances between any two nodes.
To test the validity o f the coefficients of potential and the short circuit
capacitances obtained by any numerical method it is important to know the properties of
these elements.
The general properties o f coefficients of potential and short circuit
capacitances are:
1.All Pm are positive or zero
2.All the cm are positive or zero
3.All
are negative or zero
4.The sum o f all the elements in any row o f matrix [c^, ] is positive or zero.
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The sum of all the elements in [cm ] for any conductor is equivalent to the shortcircuit capacitance o f that conductor with respect to infinity. The efficiency of obtaining
the coefficients of capacitance depends on the matrix inversion algorithm. The GaussElimination and L-U decomposition approaches are commonly used for matrix inversion
in many applications. In this work, L-U decomposition method is used for inverting the
Pnu, matrix [32].
4.1.2 Inductive Elements
The last term in equation 3.2.1.8 contributes to the circuit inductances in the
equivalent network. In general, the inductance is defined for a closed loop or closed wire.
However, the discretization of the governing field equation 3.2.1.8 along the closed
conducting path, using the method of moments leads to the concept of distributed
inductances [8,29] for open volume segments. The mutual inductance M , between any
two conductors with cross-sectional areas Axand A2 and carrying currents /, and / , can
be derived from magnetic energy considerations [29] and is given by
(4.2.2.1)
where M l2is the mutual inductance between a filament carrying a current J tdA,iil the
first conductor and a filament carrying a current J 2dA2in the second conductor. The
mutual inductance o f any two filaments with constant current along their lengths is given
by Neumann’s formula,
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(42.2.2)
Where the distance R is between the two filaments of lengths dlxand dl2 respectively
and n is the permeability o f the medium. If the current density is constant throughout
each conductor, then equation 4.2.2.1 reduces to
(4.2.2.3)
which is independent o f the current in both the conductors and is a function of only their
dimensions. Therefore, using equations 4.2.2.2 and 4.2.2.3 the mutual inductance
between any two conductors having constant cross-sectional areas and uniform (but not
necessarily equal) current densities may be evaluated. These equations are used to
compute the mutual inductance for complex electromagnetic structures (consisting of
conductors and dielectrics) carrying non-uniform current using retarded potential theory.
In this thesis, the conductors and dielectrics are divided into segments carrying uniform
current density, which is sufficient to use the above results.
Grover [6 ] and Hoer & Love [29] have published several working formulas and
tables for computing the inductances o f discretized structures with rectilinear geometry
based on equations 4.2.2.2 and 4.2.2.3. Grover uses the geometric mean distance concept
in determining the distributed inductances. This method assumes that the length is much
greater than the cross-section of the segment. However, Hoer and Love on the other hand
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give exact equations to determine the self-inductance and mutual inductance of
rectangular segments. A general procedure is also described by these authors to determine
the inductance of complicated geometries by dividing them into segments whose
inductance’s can be calculated by analytical formulae. This general procedure is valid for
uniform as well as non-uniform current densities.
Equation 3.2.1.28 derived in section 3.2.1 shows that the inductance formula
obtained is same as the Neumann’s formula for mutual inductance between any two
filaments [Equation 4.2.2.2]. The self-inductance of a segment is determined as a special
case o f mutual inductance between segments, where two identical segments are assumed
to coincide with each other. If the segments are arbitrarily oriented, they are further
divided into filaments to determine the mutual inductance.
The current in all the
segments and filaments is assumed to be uniform in the computation of distributed
inductances for discretized structures. The concepts developed for calculating the
distributed inductances by the aforementioned authors are used in [8 ] to obtain closed
form analytical formulae for computing the self and mutual inductances of volume
segments. These formulae [Appendix II] are used in this work for calculating the
distributed inductances in the equivalent circuit model.
The exact formulas developed in [29] and the other closed form formulae used in
this work for inductance calculations are all dependent on the geometry of the segments.
Several simplified formulae are obtained by further development o f the concepts
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presented in [29] for segments on the same axis and segments parallel to each other.
These simplified formulae are given in [6 ] and have been used in the computations
presented in this thesis work. It is important to mention that the computation of the
segment inductances does not require any restrictions on the size or spacing of the
conductors.
4.2 Segmentation Scheme
For any given electromagnetic structure, the solutions to the governing field
equations using a numerical method involves discretization/segmentation in time, space
or both depending on the properties o f the solution techniques and the desired results. The
effect of field segmentation on the solution is directly related to the accuracy of the
numerical analysis technique. Hence, it is very important to know the details of
segmentation to assess and choose a numerical method. In addition, it also helps us make
judicious choices in analyzing a problem under several restrictions such as, geometry of
the structure, efficiency and boundary conditions, to name a few. In this work, the
accuracy of the resultant circuit network is largely dependent on the segmentation scheme
[Figure 4.1]. The segmentation of the conductor and dielectric bodies is described in this
section and the implementation of the technique and effects of segmentation are discussed
at length in section 4.3 and chapter 5. The coupling between the conductors and the
charge crowding at the edges o f the o f the conductor and transmission line discontinuities
are accurately interpreted by using finer/smaller segments [Section 4.2]. Each segment
represents a small section o f the entire geometry under consideration. Therefore, they are
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also termed as partial capacitances and partial inductances in the works published by
other researchers using retarded potentials in the time domain [3,8].
The segmentation o f the electromagnetic structure for full-wave analysis using
retarded potentials is parallel to the work by A.Ruehli [3] and the theory o f coupled
networks used by Ramo, Whinnery and Van duzer [7]. The electrical and magnetic fields
in the electromagnetic structure are expressed as self-capacitances, coupled capacitances,
self-inductances and coupled inductances in all the three directions respectively, in the
resultant equivalent circuit model. We can add resistive elements to the equivalent
network to model the effects o f finite conductivity. In this work, the discretization leads
to subdivisions of the structure, which are called segments. The segmentation of a
conductor or a dielectric surface is shown in Figure 4.1 below. At high frequencies, the
entire electromagnetic structure is capable of storing and absorbing electromagnetic
energy. Therefore, the entire volume consisting of conductors and dielectrics is divided
into segments. For rectilinear geometries, each conductor with finite thickness and
dielectric structure is modeled as six surfaces with common nodes at the edges and at
comers.
Capacitive Call
Full/ Center
Call
Node
Figure 4.1 Segmentation of a Conductor/Dielectric Surface
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The discretized structure consists o f two types of segments, surface segments and
volume segments. The surface segments represent the charge distribution (free and bound
charge densities) at each field point. The size o f the segments is directly proportional to
the concentration of the charge and is dependent on their location and the type of
expansion functions used in the method o f moment solution. In this work 3D pulse
functions are used as expansion and weighting functions resulting in the Galerkin’s
technique that leads to better averaging of the unknowns over the entire segment.
The volume segments model the current densities in conductors and dielectrics
and are used to determine the equivalent inductive circuit elements. As stated earlier, the
geometries analyzed in this thesis are rectilinear, therefore the surface segments are
rectilinear in shape and the volume segments consist of square or rectangular crosssections. In general, the center o f each surface segment corresponds to a node in the
equivalent circuit network. However, to take into account the physical dimensions of the
structure, nodes are placed along the boundaries of conductor and dielectric bodies,
leading to unequal segments at the edges. Another reason for unequal segmentation is to
model more accurately the charge crowding at the edges of the conductors.
In figure 4.1, the segments in the center are called full segments, those on the
sides half segments and the comer ones are known as quarter segments. The shunt
elements in the equivalent network [Figure 4.2] are the self-capacitances at each node
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corresponding to the respective surface segment representing the charge distribution on
the segment. The series elements between any two nodes are the resistances, self­
inductances and mutual capacitances. The mutual coupling between all the inductances
are not shown in Figure 4.2 for the sake of simplicity.
Conductor/Surface Node
Self Inductance
m
Self Capacitance
Mutual Capacitance
_
Note: All inductances are
mutually coupled and all
mutual capacitances are
not shown
Figure 4.2 Equivalent Circuit for a node on the Conductor Surface
The evaluation o f circuit elements in conductors and dielectrics differs with
respect to both, material and field properties, and the determination of the equivalent
network for a conductor and a dielectric is derived in chapter 3. A typical circuit node for
a conductor is shown in figure 4.2 and for a dielectric in figure 3.8. All the surfaces of the
dielectrics are laid out with surface cells, which contribute to the bound charge. The
volume cells contribute to the polarization currents in the dielectric in all the three
directions. In addition, the volume cell in the dielectric is considered as a parallel plate
capacitor to model the contributions from the dielectric constant to the charge density.
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This is called the excess capacitance as shown in figure 3.8 for the dielectric node
[Section 3.2.3]. The volume segments are coupled to all the inductive segments on the
conductors as well as the dielectrics. Similarly, the surface segments are coupled to all the
other segments both, on the conductors and on the dielectrics. Therefore, the continuity of
charge density and current density at all conductor-dielectric interfaces is accounted for in
the equivalent circuit network.
The equivalent circuit network at each node as shown in figures 4.2,4.3 & 3.8 are
quasi-static lumped models with respect to field theory, if retardation is neglected.
However, in this work each quasi-static lumped element has a delay, therefore the circuit
simulation o f the equivalent network results in a full-wave solution for infinite number of
segments. Therefore, to get a meaningful engineering solution a compromise must be
made between the number o f segments required and the computational run times. We
have shown in chapter 5 that few segments judiciously placed to appropriately represent
the field distribution does provide dynamic as well realistic run time solutions. We must
point out that, in modeling a problem with the retarded potential theory it is necessary to
relate the equivalent circuit parameters in accordance with field concepts. Some of the
circuit parameters obtained in the equivalent network are in excess and must be
discarded. This could be due to many reasons, such as software coding, numerical error
introduced in the network analysis program or numerical errors due to matrix inversion
programs and the analytical formulae used.
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Nodes
X-Directe
Cells
Y-Directed Cells
Figure 4.3 Three-Dimensional view of Conductor/Dielectric Segment Inductances. All the
inductances are mutually coupled, but are not shown for simplicity.
4.3 Modeling with the Retarded Potential Technique
The numerical errors involved in modeling the conductors and dielectrics with
retarded potential technique are discussed in this section. There are several modeling
issues that lead to numerical errors that are not intuitive. Some of the main issues that
must be addressed are modeling of the ground plane, alignment of segment width along
the strip width and ground plane width and modeling of the dielectric bound charge to
name a few. Let us consider two types o f microstrip configurations to obtain results using
the theory presented in chapter 3. These examples enable us to identify and reduce the
numerical errors that manifest themselves in the resultant circuit file due to the issues
mentioned above.
First, a microstrip air-line with strip dimensions 0.0001cm (thickness) X 0.025cm
(width) X 1.0cm (length) and ground plane dimensions 0.0001cm X 1.0cm X 1.0cm
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separated by a distance 0.25cm was used as a test problem to obtain the scattering
parameters [Figure 5.1]. The propagation delay of the air-line obtained from circuit
simulations was compared with the calculated values as a function of frequency [Figures
5.5-5.7]. A Few important issues with reference to ground plane and strip conductor
modeling were identified that affected the prediction of the propagation delay. In
addition, the issue o f reference node in distributed circuit modeling needs special mention
and must be located at the correct position. At this juncture, it is appropriate to repeat
that, it is important to relate the field behavior and the equivalent circuit elements to
obtain realistic results.
For instance, most o f the conductors used in high frequency integrated circuit
applications are electrically thin and have high conductivity; hence the electric field
inside the conductor is negligible. Therefore, all the nodes on the strip edges must be
merged and all capacitive couplings inside the conductor must be removed in the
resultant circuit file. We cannot liberally assign a bunch of nodes on the ground plane as
ground nodes. The ground plane serves as the return path for inductive currents.
Therefore, the distributed inductances on the ground plane must be modeled accurately
to include the all the couplings to the other conductors. Since the bottom surface o f the
ground plane does not contribute to both, the electric and magnetic fields o f the structure,
it is numerically and computationally advantageous to model the ground plane as a single
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surface with finite thickness. This improves the condition number o f the coefficient of
potential matrices and the partial inductance matrices.
Another important observation made regarding the modeling o f finite conductors
with high conductivity is the segment inductances along the merged edges. It is extremely
important to know that the circuit analyzer has the capability to correctly handle mutual
coupling between two branches that are merged together. In Retarded potential theory the
computed circuit elements are dependent on the geometry of the segments. Two inductive
branches of the same segment dimensions and parallel to another are strongly coupled
with mutual inductance value almost equal to the self-inductance o f the individual
branches. If the nodes are merged as in the case of edge nodes along the strip conductor,
the branch inductances are parallel in circuit theory. But according to the field concept the
inductance of the merged nodes is that o f a segment with twice the width of the original
two segments. Therefore, if the circuit analyzer does not take into account accurately the
coupled inductance between any two branches whose nodes are merged, this could lead to
a few degrees error ( about 30 percent) in estimating the propagation delay.
Another area o f concern is the modeling of the interior fields in a dielectric. If the
substrate thickness is comparable to the strip width it is unnecessary to have several
layers to represent the polarization currents in the dielectric medium. Each additional
dielectric layer increases considerably the size of the circuit file. In reality, most of the
network analyzers have limitations on the length of the circuit files thereby limiting the
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analysis o f complex structures. Therefore, judicious choice must be made in modeling the
fields inside a dielectric. In this work, the equivalent circuit node for a dielectric as shown
in figure 3.8 introduces internal nodes leading to an increase in circuit analysis time. The
charge density on the dielectric is modeled as bound charge and the current density is
modeled as polarization current. However, if the dielectric is modeled entirely by surface
currents and charge densities, it would lead to a solution that is both computationally and
numerically less expensive. This approach is presented in [34] and is discussed in chapter
5 and chapter 6 as future work.
In three-dimensional modeling the finite width of the ground plane must be
cautiously segmented to avoid overlap with conductor segments and enough segments
must be used to represent the gaussian charge distribution. The ground plane modeling
also affects the resultant inductance computations. Additional discussions on the issues
mentioned above are presented in the next chapter [Chapter 5].
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CHAPTER 5
COMPARISON OF NUMERICAL/SIMULATED RESULTS
5.1 Circuit Analysis Tools
Research in the area o f computational electromagnetics is on the rise with the
availability o f better computing facilities and the need for accurate and efficient 3D
numerical analysis techniques. Microwave CAD tools perform both circuit analysis and
circuit synthesis to provide the characteristics of transmission line devices and structures.
This information is crucial to the design process in a number o f ways as emphasized in
chapter 1. Field modeling in the electromagnetic circuit simulator can be based on any of
the numerical techniques discussed in chapter 2. Therefore, the choice o f commercially
available microwave CAD tools is greatly dependent on the relative merits of the
numerical technique that performs the field analysis [Figure 2.2] to determine the
equivalent circuit.
We have shown in chapter 3 that the dynamic retarded potentials provide an
equivalent circuit that is potentially able to predict the full-wave properties of
electromagnetic structures. This method requires an efficient circuit simulator that
includes the effects o f retardation to obtain the wave-properties and various field
interactions within the structure. In general, circuit simulators provide results in either
time or frequency domains. To model the dispersion effects efficiently at high-speeds and
high frequencies, it is much more convenient to use frequency domain methods and tools.
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The use o f frequency domain methods in the implementation o f a retarded potential
technique eliminates the need for a history mechanism. The requirement for a history
mechanism is costly as its demands on computation time and memory requirements are
extensive. On the other hand, the use of frequency domain retarded potential theory can
provide solutions within reasonable run times. Simulated results obtained for a microstrip
line [Figure 5.1] in this work, had run times that were in the range from a few seconds to
15min. The simulation times are dependent on the geometry of the structure, the
segmentation of the geometry, the inclusion of various inductive and capacitive couplings
and length o f the circuit file. The use of many of the commercially available circuit
simulators for the equivalent circuits developed in this thesis is limited mainly due to the
following reasons
a.
Most circuit simulators do not support lumped elements with delay built into
them
b.
Many simulators have limitations on the number of mutual inductors in the
circuit file
c.
The use of delay elements provided in the simulators increases the number of
nodes thereby increasing the computational times considerably
d.
Lastly, almost all the simulators have limits on the length o f the circuit file and
are primarily customized to the structures that can be modeled by the numerical
techniques used by the schematic editor.
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To overcome the problems mentioned above in the circuit analysis tools, we have
used a microwave circuit analyzer developed in-house. This circuit analyzer supports
lumped circuit elements with delay which simplifies the creation of the circuit file and
does not require additional node numbers. The delay can be specified as a distance in
centimeters or a time in picoseconds. The circuit files generated for simulation in this
work had delay between the centers o f segments expressed as a distance in centimeters,
which translates to a delay between any two equivalent circuit elements. The default units
for capacitances are in pico-farads, resistance in ohms, inductance in nano-henries and
frequency in gigahertz. The output can be expressed as two-port S-parameters, Zparameters or Y-parameters depending on the required form.
5.2 Software Design and Data Files
The software code for implementing the retarded potential technique described in
chapter 3 was developed in C language. The software is designed to handle threedimensional conductor and dielectric structures with rectangular cross-sections. The
subdivision of the structures into segments as described in section 4.1 can be done either
automatically or manually. The automatic mode sub-divides the structure according to the
frequency o f analysis with the length o f each full segment equal to X /15, which has been
obtained by considering the numerical accuracy of the method [See Appendix II].
However, the use o f this mode for the software design leads to increased computation
times for capacitance and inductance matrices and lengthy circuit files that require long
simulation times. We have compared the values of segment capacitances and inductances
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with those published for microstrip multi-conductor lines and discovered that it is
sufficient to get reasonable results with segment lengths larger than X /15, which leads to
realistic run times [See Tables 1 & 2]. Therefore, segmentation in the software is
specified manually in the input circuit file along with the description o f the structure
dimensions.
It is important to take into account both field behavior and circuit analyzer
limitations in specifying the sub-division of structures. Since most of the conductors ca
be considered as infinitely thin for many applications and those published in literature,
the element calculation software is designed to handle two layers o f sub-division along
the x-direction for thin conductors and more for electrically thick conductors. The same
sub-division scheme is implemented for dielectrics. For a test problem [Figure 5.1] the
conductor width is lOmils, substrate height is lOmils and the thickness o f both conductors
(strip & ground plane) is 0.04mils. Therefore, the conductors and the dielectric were sub­
divided into two layers along the x-direction. This subdivision helps in reducing the
computation time to evaluate the 3D capacitances and inductances and significantly
reduces the length o f the circuit file, thereby providing the solution in reasonable time.
The conductors are treated as immersed in free space to determine the static
charge distribution. However, one has to remember that the dynamic behavior of the
entire structure is obtained by considering the delay between different circuit elements
computed from the static field behavior [Chapter 3]. The dielectric is also treated as
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immersed in free-space, but in conjunction with a total charge concept [14]. Therefore, all
dielectric surfaces represent capacitive cells that determine the bound charge. This
treatment o f conductors and dielectrics leads to the total charge on the conductor and
dielectric interfaces. All capacitive couplings between segments on the conductor and the
dielectric are through free space as described in references [37] and [40]. The software
was designed to provide circuit files for microstrip multi-conductor lines with a single
dielectric layer. It can be easily extended to include multiple dielectric layers. The
generality of the code is limited only in generating the circuit files and not in field
analysis of a microstrip/stripline geometry with multiple dielectric layers.
For accurate modeling with the retarded potential technique, nodes at the
conductor-dielectric interface or dielectric-dielectric interface should coincide. The code
is written to subdivide a 3D rectilinear object with equal segmentation on each plane (i.e.
x, y and z planes). In the code developed the bottom surface of the dielectric consists of
the same number of segments as the top surface of the ground plane and hence the same
number of nodes that coincide. For the strip, the nodes on the bottom surface do not
necessarily coincide with those on the top surface of the dielectric. The strip needs to be
finely divided to represent the charge distribution accurately, however the dielectric and
the ground planes with far greater widths than the strip width need not be divided finely.
Therefore, to compromise between the requirements for accurate field modeling and to
obtain simulation results for practical run times, the mismatched nodes on the common
surface between strip and the dielectric are merged using interpolation. To effectively use
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interpolation the ratio o f segment widths along the dielectric and strip must not be greater
than unity. In our experience, to even maintain a ratio of unity, leads to significantly
longer computation times in generating the circuit file and to lengthy circuit files that will
increase the simulation times considerably.
The input file consists of the conductor and dielectric information with the last
conductor being the ground plane. Global parameters are defined by a structure named
“conductor list” that consists of the number o f conductors and dielectrics. Each
conductor and dielectric body is defined in rectangular coordinates with the origin
defined for the entire structure. The material properties for each conductor and dielectric
body are defined as input variables for each of them respectively. Each conductor and
dielectric structure consists o f six surfaces, with capacitive segments defining the nodes
for the equivalent circuit network. Since the inductances represent branch currents, two
nodes define them. Therefore, the size of the inductance matrices along any given
direction is less than that the capacitance matrix for any single 3D (conductor/dielectric)
structure [Figure 5.1].
The time required to compute the inductance matrices is determined by the size of
the matrix and the number o f filaments specified in the input to determine the mutual
inductances between the segments. The elements in the inductance matrices are computed
using subroutines for self-inductances and mutual inductances that are parallel and
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arbitrarily oriented [Appendix I]. Formulae for the inductance computations are published
in references [6 ], [8] and [29]. The capacitance matrix is determined by inversion of the
co-efficient of potential matrix and the time required to compute all the co-efficient of
potential is governed by the size of the potential matrix. The larger the number of
segments for a given structure, the longer it takes to compute the capacitance matrix. LU
decomposition method is used to invert the matrices as required . The orientation of any
two capacitive segments for rectangular geometries is either parallel or perpendicular to
one another. The co-efficient of potentials for these two configurations are determined
from the analytical formulae [Appendix I] published in reference 7. The self terms for the
co-efficient of potential along the diagonal are determined from the formula for the
segments oriented parallel to each other.
The inverse o f the [/*] matrix is the same as the short-circuit capacitance matrix.
The circuit file is built by converting the short-circuit capacitance matrix into two
terminal capacitances. Since the matrix is symmetrical the number of equivalent circuit
capacitances ( N Cap) in the circuit file is equal to
(5.2„
where N p is the total number o f surface segments. Moreover, each segment at the edges
have common nodes representing the same potential. Therefore, the segment capacitances
having common nodes can be summed to get the equivalent capacitance. This process
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further reduces the number o f capacitances in the circuit file, in turn reducing the length
of the circuit file. The number o f equivalent self and mutual inductances along the zdirection in the circuit file is equal to
(5.2.2)
where N u is the total number of volume segments along z-direction. Similarly the
number o f equivalent self and mutual inductances along y and x-directions can be
determined using equation 5.2.2. The length of a circuit file for an airline with the number
of strip segments 2x3x10 and the number of segments for the ground plane is 2x7x10 is
about 4000 lines for a 1cm long strip and the other dimensions defined in figure 5.3. The
circuit file is obtained by ignoring coupled capacitances less than 10"4 p F . Since values
for coupled capacitances less that those specified do not significantly affect the simulated
results and reduce both the circuit file length and simulation times. The next section
discusses the simulated results for a microstrip line, airline and an open-end microstrip
line.
9urf-x2yz
T
surf-xlyx
surf'Xyzl
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Ny > 4 S eg m en ts along width
Nz = 5 S eg m en ts along length
Nx = 2 S eg m en ts along th ick n ess
Figure 5.1 A 3D conductor/dielectric structure and segmentation of a surface in the
software design
2L m __
/tv
*
r tr \
r tn
jn \
im
z
__
im
N = 2 * 3 * 5 =30
= 2 * 4 * 4 = 32
Figure 5.2 A section of inductances along y and z-directions for the segmentation shown
in 5.1
5.3 Comparison of Results
The objective o f the theory presented in chapter 3 is to accurately determine the
characteristics o f complex circuit structures such as MMIC chips and interconnect lines in
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packages by describing the field behavior in terms o f an equivalent circuit network for
microwave and millimeter wave circuits. For the equivalent network determined using
static circuit elements without delay, the solution is equivalent to results obtained by
other quasi-static methods. Several results are published in the literature for multi­
conductor transmission lines using quasi-static two-dimensional analysis techniques [15,
32, and 35] for multi-conductor transmission lines for infinitesimally thin and infinitely
long conductors. However, the quasi-static field analysis using the theory developed in
chapter 3 without delay is simple in formulation with reasonable run times and includes
the finite thickness of the conductors, which is required in most practical applications.
The method includes the effects of finite conductor length and dielectric/ground plane
width in predicting the transmission line characteristics. The transmission line parameters
for conductors with no ground planes or remote ground planes can also be determined.
Since the analytical formulae used to calculate the equivalent static capacitances
and inductances have been tested against several published results [6-8,31,62], the
validity of these analytical
formulae to
predict three-dimensional quasi-static
transmission line behavior is given. We have observed that the accurate determination of
the mutual inductances between segments is extremely important to get meaningful
results. There are several analytical formulae in the literature to compute mutual
inductances, therefore depending on the size o f the segments and their relative position
appropriate formulae must be used to compute the mutual inductances. The filament
method described in [8] leads to significantly low mutual inductance values for thin and
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wide segments. The formulae given in [29] provide accurate mutual inductance values for
thin and wide rectilinear segments that are parallel or perpendicular to each other. To test
the software code developed and the limitations of the analytical formulae, we have
generated data for several quasi-static problems. Tables 1 & 3 shows the comparison of
2D published data on transmission line capacitances with values computed using the
given formulae. The published free-space line capacitances are obtained with the
assumption of infinite line length and ground plane (2D) and negligible conductor and
ground plane thicknesses. The capacitance values computed in this work are about
± 10 percent, compared to the published data [Tables I & 3] depending on the geometry
o f the transmission line system and segmentation. Table 1 shows the variation of the
computed results with segmentation and the type of segments (i.e. uniform or nonuniform). The results are 3D capacitance values and the discrepancies are due to the
inclusion of finite length, width and thickness for the strip and ground plane conductors
and numerical errors due to matrix inversion algorithms.
Table 2 & 3 compares the published data on inductances with those computed
using the above formulae. Table 2 shows the affect of segmentation on the computed
inductance values using the closed form analytical formulae given in Appendix I. The
computed inductances are 5 to 15 percent higher than the published data. The higher
values for inductances are due to finite length, width and thickness o f the conductors. For
straight conductors with uniform cross-section and current flow along the length only, it
is sufficient to consider one segment depending on the relative size o f the ground plane
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with total number o f conductors. But to determine the effective three-dimensional static
inductances for a system o f multi-conductor geometries, we have used the coupled
inductor theory and network topology [33]. This approach determines the resultant
inductances directly from the geometry of the system and shows the effect of
segmentation on the inductance computations.
The published data (method of moments approach) calculates inductance matrix from
the free-space capacitance matrix, which requires matrix inversion to determine the
inductance matrix and leads to a lower bound on the inductance values. In this work, the
inductances are computed directly using the analytical formulae [Appendix I] leading to
an upper bound on the inductances. In the retarded potential theory the inductance
segmentation is dependent on the capacitive segmentation. The uniform segmentation
leads to lower values for the capacitances, which affects the accuracy of the calculated
propagation delay. The size o f the inductive segments is dependent on the geometry of
capacitive segments. To achieve accurate capacitance values, unequal segments must be
used to model the charge crowding at conductor edges. But, this leads to increase in
programming logic, computer memory and run times. Therefore, depending on the
application and required accuracy a decision must be made on the type of segments
required. The advantages o f using the static retarded potentials in comparison to other
techniques with unequal segmentation are:
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a. Three-dimensional transmission line capacitances and inductances can be
computed that account for the finite width and thickness o f conductors as well as
ground and power planes.
b. Static retarded potentials calculate easily inductance and capacitance values for
interconnect lines with remote or no ground planes and for irregular goemetries
such as L-shaped conductor [39] and interconnect lines in IC packages.
c. The computed capacitance and inductance values are determined from the
geometry o f the segments and do not need boundary conditions to find the overall
transmission line characteristics.
For testing the microwave characteristics, a microstrip line [Figure 5.3] and an
open-end microstrip line [Figure 5.4] configuration have been used as test problems. The
microstrip line structure was simulated to obtain scattering parameters as an airline (with
dielectric removed) and as a conventional microstrip line (dielectric present) with 2x3x10
segments along the thickness, width and length of the strip and 2x7x10 segments along
the thickness, width and length o f the dielectric and ground plane. The charge distribution
along the microstrip conductor follows a gaussian curve, therefore to represent the charge
distribution accurately it is essential to segment the ground plane and the dielectric with
more cells at the strip/dielectric interface. However, a large number o f segments leads to
a lengthy circuit file that takes about 15-25min or more per simulation at each frequency.
On the other hand, the software as designed has no capability for unequal segmentation
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along the width of any structure. Therefore, to approximately model the charge behavior
for a gaussian distribution the width o f the dielectric was reduced to 0.175cm with the
same segmentation as described above. The results presented show the end effects due to
the finite width, length and thickness o f the ground plane as well as the conductors.
Figure 5.5 shows a difference o f about 33 percent o f the propagation delay of an
air line (dielectric removed) compared to velocity of light. The main reason for the
discrepancy in the simulated values is a result of not considering the mutual inductances.
This is due to an unfortunate inconsistency in the network analysis program in handling
the mutual inductances. However, figures 5.6 & 5.7 show that the simulated values for
|S 111and |S21| compare well the ideal values. Therefore, we can infer from these results
that the mutual inductances have negligible effect on the transmission co-efficient and
reflection co-efficient upto 7.5 GHz. For an air-line (dielectric removed) structure the
characteristic impedance computed from empirical formulae [32, 69] is equal to 125
ohms. The characteristic impedance calculated from the simulated values is 118 ohms for
line lengths 1.0cm and 0.2cm. The simulated value for the characteristic impedance is
about 6 percent lower than the empirical data and shows that the contribution of mutual
inductances to the total line inductance is about 36 percent. Other inaccuracies that may
manifest in the simulated data are due to the matrix inversion algorithms at various levels
and the network analysis program that could introduce numerical errors.
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The same arguments apply for the microstrip line (with dielectric) results
presented in figures 5.10-5.11. The characteristic impedance obtained from the simulated
results for the microstrip line [figure 5.4] is 41 ohms and is about 18 percent lower than
the expected value o f 50 ohms. In addition, to not considering the mutual inductances,
the discrepancies in the microstrip line characteristics may be the result of equivalent
capacitance values obtained using the total charge, which is equivalent to the sum of the
free charge and bound charge on the conductor/dielectric interface [36,47].
In this work, the total charge concept is used to represent the charge on the
conductors. This procedure involves large matrices and increased programming effort. It
is presented in [34] that the total charge concept used to determine static capacitances
leads to inaccurate values for the case of high-dielectric materials at high frequencies that
includes the alumina substrate. An alternate solution is presented in [34], which solves for
the free charge distribution on conductors directly. In this method the conductors and
dielectrics are treated as closed bodies as if immersed in free space. Then the integral
equations are solved to obtain the coefficients of potential for all the surfaces in the entire
system. This method is said to lead well conditioned and smaller size matrices. In our
work, this approach will greatly reduce the number of nodes and modify the dielectric
node leading to reasonable size circuit files. Therefore, to overcome some of the
inaccuracies it is necessary to modify the designed software code. The segment
capacitances for thin conductors can obtained with better precision if the strip conductors
are modeled as consisting of just two-surfaces separated by the finite thickness and the
122
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ground plane with only one surface. This will lead to smaller matrix size and less
computation times for both, generating the circuit file and scattering parameters.
The second test problem is the open-end microstrip line [Figure 5.4] with
following dimensions : Ls = 1.8cm, W = 0.6cm, Wg = 3.0cm, h = 3cm, L = 3cm and
e r = 2.55. The number of segments are 2x3x6 along the strip thickness, width and length
respectively and the number o f segments along the dielectric and ground plane are
2x7x10. This disagreement can again be due to all the numerical and software design
issues discussed above for the microstrip line configuration. Since the substrate height for
this problem is 3cm, more segments must be used along the dielectric thickness.
However, this leads to increase in the nodes and the length o f the circuit file significantly
limiting our circuit analysis capability. Therefore, in conclusion the results simulated for
the test structures show that the equivalent network parameters are sensitive to
segmentation and the mutual inductances.
Normalization o f the parameters leads to
results within the acceptable error margins.
123
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vv
w , = G round P la n e Width
w
h
= Length of th e structure
= Strip width
= S u b strate height
Figure 5.3 Geometry of a microstrip line for test problem
S r = 2.55
= Ground Plane Width
= Length of the structure
w = Strip width
h = Substrate height
Ls = Strip Length
Figure 5.4 Geometry of an open*end microstrip line for test problem
124
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Table 1 - Comparison of capacitance values with published data
Dimensions of the Multi­
conductor Transmission Lines
(Units - cm)
1
2
3
# of conductors =
W/H = 1.0
W=0.025cm
L=lcm
T=0.0001cm
ground plane
Width = 1cm
£r = 1.0
1
# of conductors = 2
W/H = 1.0
S/H = 1.5
T=0.0001cm
S = 1.5cm
L = 8 cm
Ground plane
width = 7.5 cm
£r = 1.0
# of conductors = 2
W/H = 1.0
S/H = 2.0
T=0.0001cm
S = 3.0cm
L = 10cm
ground plane
width = 8 .0 cm
£ r = 1.0
Published Data using
Method of Moments [41,69]
(Units - pF/cm)
C „ = 0-268
Ref [69]
C u = 2AU
C 12 = -0.262
Ref [41]
Data computed using Retarded Potentials
[Chap 3] (Units -pF/cm)
# of segments on the
conductors
condl -2x3x7
ground - 2x7x7
uniform segments
condl-2x3x7
ground - 2x7x7
uniform segments
condl - 2x5x20
ground - 2x7x20
uniform segments
C , 2 = -0.237
Ref [41]
Cu = 0.252
Cu = 0.255
Cu =0.261
condl =cond2 - 2x3x8
ground - 2 x 8 x8
uniform segments
Cn = 2.341
condl=cond2 - 2x5x12
ground - 2x9x12
non-uniform segments
C tl = 2.358
condl=cond2 - 2x5x16
ground-2x12x16
non-uniform segments
Cu = 2.618
Results
C 12 -0-252
C t2 = -0-256
Cu = 2-361
C 12 = -0-257
condl=cond2 - 2x7x16
ground-2x12x16
non-uniform segments
C12= -0.259
condl=cond2 - 3x3x6
ground-2x6x16
uniform segments
Cu = 2.968
C 12= -0-235
condl=cond2 - 3x5x10
ground- 2 x6 x 10
uniform segments
C„= 2.971
C,2= -0.236
condl =cond2 - 3x7x20
ground- 2 x 12 x2 0
uniform segments
Cn = 2.977
C„= 2.362
C l2 = -0.237
Note: 2x3x8 is equivalent to 2 segments along the thickness, 3 along the width and 8
along the length o f the conductor
125
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Table 2 - Comparison of inductance values with published data
Dimensions of the Multi­
conductor Transmission Lines
(Units - cm)
1
tt of conductors = 2
W/H = 1.0
S/H = 1.5
T=0.0001cm
S = 1.5cm
L = 8 cm
Ground plane
width = 7.5 cm
3
# of conductors = 2
W/H = 1.0
S/H = 2.0
1 =0 .0 0 0 1cm
S = 3.0cm
L = 10cm
ground plane
width = 8 .0 cm
<N
2
# of conductors =
W/H = 1.0
W=0.025cm
T=0.0001cm
L=lcm
ground plane
Width = 1cm
II
I
Published Data using
Method of Moments [41,69].
Units - nH/cm
Ref [69]
Z,u = 4.276
L x2 = 0.529
Data computed using Retarded
Potentials [Chapter 3]. Units-nH/cm
# of segments on the
conductors
Results
condl= lxl
ground = lxl
6.590
condl= 1x 10
ground = 1x 10
6.582
condl = 3x1
ground = 3x1
5.06
condl = 3x5
ground = 3x5
4.982
condl = 3x1
ground = 7x1
4.408
condl= 3x5
ground = 7x5
4.402
condl = lxl
cond2 = lxl
ground = lxl
L u =4.55
L n = 0.596
Ref [41]
condl= 1x5
cond2 = 1x5
ground = 1x5
L u = 4.280
Ln = 0.387
L u =4.49
Ln = 0.596
condl= lxl
cond2 = lxl
ground = lxl
Lu =4.586
condl=1x5
cond2 = 1x5
ground = 1x5
L n =4.587
L n = 0.420
Ref [41]
1,2 = 0.419
Note: l x l is equivalent to 1 segment the width and 1 along the length o f the conductor
126
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Table 3 - Comparison of additional inductance and capacitance values with
published data
Dimensions of the Multi­
conductor Transmission Lines
(Units - cm)
I
# of conductors = 6
last conductor is the
ground
T = 0.00127cm
L = 3.81cm
W = 0.00508cm
Spacing (S) = 0.0116cm
Published Data using
Method of Moments [ 8,40,47].
Units - inductance nH/cm
capacitance pF/cm
£ „ = 15.85
£,3=8.73
= 10.69
£ ,2
£ ,4
=7.11
Data computed using Retarded
Potentials [Chapter 3].
Units - inductance nH/cm
capacitance pF/cm
£,, = 14.7
£ 23
= 9 .6 8
£ 24 = 7.45
1,25 = 5.29
= 10.1
£,3=8.15 £ u =6.54
1,5 = 5.16
1,32=15.0
£ 12
£ 22 =
£ ,5
= 4.65
13.8
£23
£ 24 = 6.89
= 9 .0 6
£ 25 = 4.76
Ref [8]
£ 35
£ 44=
2
# of conductors = 2
W/H = 3.0
S/H = 1.0
S = 2.0cm
L = 20cm
= 5.52
1 2 .2 2
£45
£33
= 12.7
£34
= 7 .6 8
t"-'
Ul
II
O
Z,33= 13.84 L m = 8.31
= 6 .2
£ 4 4 = 11.1
£45
= 5.51
Lj5 =9.44
L,5 =8.35
C u = C 22 = 0.609
C „= C 22 = 0.619
Cl2 = -0.077
C, 2 = -0.086
C„ = C 33 = 0.325
C „= C 33 = 0.331
C, 2 = C 23 = -0.054
C, 2= C 23 = -0.058
C 13 = -0.008
C,3 = -0.009
C jj =0.335
C 22 = 0.346
Ref [47]
3
# of conductors = 3
W/H = 1.0
S/H = 1.5
L/W = 10
six segments along length
and three along the width
T=0.1cm
S = 1 .0cm
L= 10cm
Ground plane
width = 7.0 cm
Ref [40]
127
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Ground Plane Width = 1,0cm
-25
Ideal behavior
propagation delay (in Degrees)
-50
-75
-100
-125
-150
-175
-200
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Frequency (In GHz)
Figure S.S Comparison of propagation delay for an airiine 1cm long
(7 segm ents along the length and w idth) with ideal behavior
128
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15
1
0.9
■Ground P lane Width(Zc 114 ohm s) = 1.0cm
0.8
■Ideal behavior (Zc 125 ohm s)
0.7
0.6
IllSl
0.5
0.4
0.3
Frequency in GHz
Figure 5.6 Variation of reflection co-efficient for an airline 1cm long
with ground plane width (7 segm ents along the length and width)
129
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0 .9
0.8
0.7
I 821 |
0.6
0.5
0.4
0.3
• Ground P lane width (Zc 118 ohm s) = 1 .Ocm
0.2
■Ideal behavior (Zc 125 ohm s)
0.1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Frequency (in GHz)
Figure 5.7 Variation of the transmission co<«fflcient for an airline 1cm
long with 7 segm ents along the length and width
130
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15
0.9
Length Of Airline (118 O hm s)= 1.0 cm
0.8
Length of Airline (120 ohm s) = 0.2 cm
0.7
ideal behavior (125 ohm s) = lengths:
0.2cm and 1.0cm
0.6
I us I
0.5
0.4
0.3
0.2
0.1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Frequency (In GHz)
Figure 5.8 Comparison of reflection coefficient for airlines with
different lengths (7 segm ents along the length of the line) and
groundpiane width 0.175cm
131
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15
1.1
0.9
0.8
0.7
I tz s l
0.6
0.5
0.4
0.3
Length Of Airline (118 ohm s)= 1.0 cm
0-2
Length of Airline (120 ohm s)= 0.2 cm
°-1
- e — ideal behavior (125 ohm s)= lengths: 1cm & 0.2cm
‘
>
3
4
5
6
7
8
9
10
11
12
13
14
Frequency (In GHz)
Figure 5.9 Comparison of transm ission coefficient for airlines with
different lengths (7 segm ents along the length) and ground plane
width 0.175cm
132
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15
Line Length (41 Ohms) = 1,0cm
-25
Empirical formula forSO ohm line = 1.0cm
Propagation delay (in Degrees)
-50
-75
-100
-125
-150
-175
-200
-225
1
2
4
3
5
6
7
Frequency (in GHz)
Figure 5.10 Propagation delay Vs frequency for 50ohm microstrip line
( 10mil alumina substrate) for different line lengths
133
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Line Length = 1.0cm , Zo = 41ohm s
0.8
Line length = 0.2cm , Zo=41ohm s
I S11 I
0.6
0.4
0.2
1
2
3
4
5
6
7
Frequency (In GHz)
Figure 5.11 Comparison of reflection coefficient of a 50 ohm
microstrip line ( 10mil alumina su b strate) for different line
lengths.The ideal |S11| for both the line lengths is approximately
equal to zero
134
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0.9
|S 2 1 |
0.8
0.7
Strip Length (41 Ohms) = 1.0 cm
0.6
Strip Length (41 Ohms) = 0.2cm
0.5
1
2
3
4
5
6
7
Frequency (In GHz)
Figure 5.12 Comparison of transmission coefficient for a 50 ohm
microstrip line (10mil alumina substrate) for different line lengths.
The ideal values for |S21| with frequency for both the line lengths is
equal to unity
135
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0.9
0.8
0.7
0.6
0.4
0.3
Ground Plane Dim. = 3cm x 3 cm.
Strip length = 1.8cm
Strip width = 0.6cm
Substrate height - 3cm
0.2
0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50
Frequency (in GHz)
Figure 5.13 Variation of |S11| with frequency for an open end
microstrip line ( dielectric constant = 2.55) as shown in figure 5.4. The
ideal behavior is unity.
136
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CHAPTER 6
CONCLUSIONS
6.1 Summary
The formulation o f the field equations, governing the electromagnetic wave
propagation using retarded potentials in frequency domain and their connection to circuit
theory is demonstrated using the method of moments (Galerkin's) technique in a
conductor and a dielectric [Chapter 3]. The main task involved in implementing the
method of moments technique is the choice of expansion and testing functions [2.2.5].
The work presented in this thesis uses 3D pulse functions, also known as the subsectional
bases [2 ] as expansion and testing functions in determining the unknown charge and
current distributions. Since retarded potentials are directly related to the current and
charge distributions, both electric and magnetic fields can be derived.
The determination of the unknown current and charge densities and using the
relation between field and circuit concepts [3.1.1] leads to a resultant equivalent circuit.
The resulting network consists of several closed circuits called meshes or loops and
coupled circuit elements (coupled capacitances and inductances). In other words, each
loop has capacitors, inductors that are mutually coupled to each other in the entire
equivalent circuit. The lumped circuit elements represent the charges and currents, and
the finite propagation time is included as a delay between each circuit element.
137
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The segmentation of conductor/dielectric structure is not intuitive and we have
discovered several modeling techniques not reported in literature that could lead to
accurate determination o f the scattering parameters. There are several analytical formulae
available in the literature to compute the segment self and mutual inductances as well as
capacitances. All the formulae are strongly dependent on the segment geometry and the
relative positions o f the segments with one another. Hence, the choice o f the appropriate
formula is important to obtain meaningful results. In addition, we have shown that the
mutual inductances contribute significantly towards obtaining accurate solutions.
Therefore, using a network analysis program that can analyze circuits with large
inductance matrices is inevitable in using this approach.
The scattering parameters for a microstrip line and an open-end line are obtained
using a network analysis program developed in-house. The static results compare well the
other published data and are presented in chapter 5. Due to an unfortunate inconsistency
in the network analysis program we could not include the mutual inductances in the
circuit files. Therefore, the data presented in chapter 5 shows the contributions o f only the
self inductances, self capacitances and mutual capacitances with delay. We have shown
that the 3D static characteristics o f complex transmission lines can be evaluated using
retarded potentials with relatively simple mathematical equations in comparison to other
numerical approaches. As stated earlier, to obtain full wave solutions it is crucial to have
access to a network analysis program that can analyze large inductance matrices with
delay.
138
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In conclusion, to improve the accuracy o f the results using the approach presented
in this work the solution must be obtained in two steps. The first step is to model the
problem in accordance with the field concepts to accurately represent charge and current
densities. The results obtained at this level can be termed as microwave analysis and
consist of large inductance and capacitance matrices. The second step is to condense
these large matrices depending on the required system interactions and appropriate delays
to obtain a circuit file for full-wave analysis. This part may be termed as network
analysis. This two step approach will lead to accurate results, reduced simulation run
times and the possibility o f using a wider range of network analysis programs.
6.2 Contributions
Equivalent circuits for conductors and dielectrics have been obtained directly
using retarded potentials in the frequency domain and Kirchoffs laws [Chapter 3].
Recently, Garrett et.tal [11] have shown that, by evaluating the phase at the centroid of
each segment/cell improves the accuracy o f the partial element equivalent circuit [13]
approach. We have presented an efficient method, which is parallel to Ruehli's [13]
approach, to compute the segment capacitances and inductances with delay to include
retardation effects, without making any initial assumptions on the distribution of current
and charge density on each segment. To our knowledge this method has not been
implemented previously. This method (provided an appropriate network analysis program
is available) can avoid the costly history mechanism required for full wave analysis using
the partial
element equivalent circuit method
[7,8]. Static three dimensional
139
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multiconductor inductances are obtained directly from the exact geometry o f the system.
The partial inductance matrices are reduced to the circuit inductance matrices by using
coupled inductor theory and network topology [33].
6.3 Future Work
We have observed that the numerical accuracy of the results presented in this
work can be significantly improved with alternate capacitance extraction methods. In
reference [34], a simple and efficient method is presented to compute the capacitance of
conductors in the presence o f dielectric materials with low as well as high dielectric
constants. The authors claim that the free charge distribution can be computed in a
straightforward manner that is computationally less expensive. The other advantage we
would obtain in using this method is that, it would replace the equivalent dielectric node
[Figure 3.8] with the equivalent surface nodes thereby reducing the number of nodes.
This will lead to reduced circuit file lengths and simulation run times.
Another approach is to use the dual of the governing field equation 3.2.17 to
obtain the equivalent circuit. The concept of duality is used to obtain solutions with
simplified numerical formulations in field theory, and a more intuitive approach to solve
problems in circuit theory. In circuit theory, the dual of loop equations for a given
network is the nodal equations. Therefore, duality in circuit theory is related to the
network topology and the dual quantities current and voltage are realistic. In field theory,
duality is related to the mathematical form of the governing equations. The duals of
140
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electric fields are magnetic fields. The quantities in the dual equation need not be
realistic. As stated earlier, the objective o f dual relationships is to aid in the visualization
of field behavior and simplify the evaluation of the equivalent circuit elements. Using the
dual approach, we can determine the circuit capacitances directly, however inductances
will be obtained indirectly by matrix inversion.
The governing equation 3.2.1.7 led to the loop formulation, where the
capacitances are determined indirectly and the inductances directly. It is well known that
large potential matrices require long runtimes, and to get accurate results one must use
large number of segments. The condition number of co-efficient o f potential matrix is
poor with respect to the numerical accuracy obtained by the inversion algorithms used.
This is mainly due to the fact that we require unequal segments to represent the charge
distribution, which leads to element values in the potential matrix with large ratios
between the smallest and largest number. In contrast, the solution to the dual of the
governing equation 3.2.17, will lead to direct computation of capacitances. This method
will lead to indirect calculation of segment inductances in each direction. The inverse
inductance matrices may have better condition number because the segment sizes are
larger than the capacitive segments and will lead to fewer numerical inaccuracies.
Moreover, for thin conductors it is practical to neglect the mutual inductances along the
thickness o f the conductor and similarly, depending on the geometry o f the conductors
one may need to compute few inductance calculations resulting in reduced computation
times.
141
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APPENDIX I
1.1 CAPACITANCE FORMULAE
The relative positions o f any two segments in general is: a. the segments are parallel to
each other or b. the segments are perpendicular to each other. The following analytical
formulae for these general segment orientations have been developed by Dr. A.Ruehli
and are published in reference [7].
a. Parallel Segments
(al)
const
where
const =
(a2)
4* efafbS'S,
Al = —
ak ln(at + p)
(a3)
(a4)
(a5)
A4 = bnC a k tan ' 1
(a6 )
pC
and
P=4ai+bi+e
(a7)
150
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f a, f b, sa, sb,
T h e v a ria b le s
C,
atJ and by a re
s h o w n in f ig u r e 1 .1 .
b. Perpendicular Segments
p„= ——
y y y ( - i y +,,,+*+,[ 5 i + 5 2 + 5 3 - f i 4 - 5 5 - 5 6 ]
const w ^ w
(bo
where
const = -----------------4n e f j esasb
51 =
(b2 )
(*1 El?c, ln(5m + p )
\ 2
52 =
(b3)
6
'a l
(b4)
ba In(c, + p )
53 = akbn c, ln(at + p ) - ^
-p
(b5)
54 = — tan ' 1
6
(b6)
151
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(b7)
m
and
(b9)
A
2
(b 10)
The definitions of p and ak are given by equations (a7) and (a8 ) respectively. The
variables f a, f c, s a, s b, cir a:j and btJare shown in figure 1.2 .
1.2 INDUCTANCE FORMULAE
The inductance formulae are also dependent on the geometry of the segments for both
self and mutual inductance computations [8 ].
a. Self-inductance for rectangular segm ents
Lpti = ^ [ b l + b2 + b3 + b4 + b5 + b 6 + b l + b& + b9]
where
/
length o f rectangular segment
w
width o f the rectangular segment
( 1)
152
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_ t _ thickness o f the rec ta n gular segment
w
width o f the rectangular segment
(2)
61 = —— <ln
24k
b2 =
24k v
{ln(v + A2) - ^ 6}+ T T -(a * ~ A 2)
60k
! f / K+
63 = — <jln
241
j
\
K2
----- tan
6v
/ V
^uA4
J
/
f
\
An
1
KV
u + ----------tan
65 = — A6 - - t a n *1
4v 6 6
4 6v
\ A * J
V
\ vA*J
66 = - J - r {ln(a + 4 ) - A, }+ - ^ y (A, - A, )
20v
24v
67 = — ^ - ( l - A 2 +A a - A , ) + — (A} - A 4)
60v k
2
1 20v 3
47
]- 4 + J
V“ J V I
68 = — J - I n f 1+ ^
24v v
60v
-A<
-/! [ )+
and
2
II
<2 + u 2J
II
r
+ V
2
+K
2 V /2
153
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b. Self-inductance for infinitely thin rectangular segments ( v < 0.01)
—
3 lnj^ + (u2 + i j ‘ l + i r + «~‘ +3u ln> -
6K
i
where u and v are given by ( 1) & (2 )
c. Mutual inductance between any two arbitrarily oriented segments filament
approach
where i and j are segments, r and s are the number of filaments in segments i and j
respectively. The closed form solution for the filament inductance is
where
Si = 1+ P
S 2 = 1+ P - V
ii= P ~v
and
g4 = p
The normalizations used are
154
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where (xr - xt ), (yr - y s) and Dz are the respective differences in the filament
coordinates and lr and ls are the lengths o f the filaments. The accuracy of the mutual
inductance calculated between any two segments using the above formula depends on the
relative position o f the segments.
155
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VITA
Ramani Chintalapuri Tatikoia
Born
July 26, 1966
Secunderabad, India
Parents
Chintalapuri Sathyanarayana
Chintalapuri Naveena
High School ST.Philomena Girls High School, Secunderaba, India
St.Francis College, Secunderabad, India
Higher Education & Projects
PhD
Electrical Engineering (08/92 -09/99). Lehigh University, Bethlehem, P A .
(G.P.A. - 3.63/4.0)
•
Multichip MMIC Package for X and Ka bands - sponsored by JPL, Pasadena, CA.
•
Analysis of Particle Impurities in steel using optic principles.-. Department o f Physics
Pennsylvania State University, State College, PA - 16801. (G.P.A. - 3.67/4.0)
•
Electromagnetic analysis o f Radar Meteorological Data
M.S
•
Applied Electromagnetics, March'90. IIT- Kanpur, INDIA. (C.P.I. - 8.07/10)
A Generalized Scattering Matrix Representation of Slot Radiators Excited by a NonRadiative Dielectric Waveguide - Masters Thesis
B.E
Electronics & Communication Eng., June'88 . Osmania University - Hyderabad,
INDIA.
•
Analysis of Waveguide Fed Aperture Couplers at X-band - sponsored by
Department of Electronics and Radar Engineering, Bangalore, INDIA.
•
Designed, Fabricated and tested multiple Microstrip Discontinuities and Components
and studied and compared their performance - Senior Year Projects
156
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•
Designed a 8x 8 Planar Microstrip Array at X-band and Feed Networks for Arrays sponsored by Council of Scientific and Industrial Research, Bangalore, INDIA.
Experience
RF Engineer Bell Atlantic NYNEX Mobile, Plymouth Meeting, PA (02/96 - 12/96)
Research Assistant
• Lehigh University, Department o f Electrical Engineering (07/92 - 07/95)
•
Communications Space Sciences Lab. (CSSL) Department o f Electrical Engineering.
Pennsylvania State University, State College, PA. (08/91 - 02/92)
Teaching Assistant
• Lehigh University, Department of Electrical Engineering ( Fall’9 4 , Spring ’95)
•
Lehigh University, Department o f Electrical Engineering (Summer ’94)
Offered course ECE 108 (Signals and Systems)
•
Department o f Physics, Pennsylvania State University, State College, PA.
(Spring’92)
•
Department o f Electronics and Communication Engineering, Osmania University,
Hyderabad, INDIA. (04/90 - 07/91)
Offered the following courses - Signal Analysis and Transforming Techniques, Basic
Circuit Analysis, Electronics Circuits Lab (Supervised)
•
Department o f Electrical Engineering, Kanpur, INDIA. (09/88 - 02/90)
Supervised and Assisted the lab for following courses - Electronic Circuits,
Microwave Circuits
Publications
• "Multichip MMIC Package for X and Ka Bands" - IEEE Trans on Components,
Packaging and Manufacturing Technology - Part B, Vol. 20 Feb '97, pp 27-33.
•
"Multichip MMIC Package for X and Ka Bands" - MMIC Symposium 1994
•
"Microstrip to Waveguide Large Aperture Couplers" - Electronics letters U.K, Vol
24, No. 15, July 1988, pp 913-914.
157
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