# 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 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. UMI Number 9955178 UMI* UMI Microform9955178 Copyright 2000 by Bell & Howell Information and Learning C om pany. All rights re serv ed . This microform edition is protected ag ain st unauthorized copying under Title 17, United S ta te s C ode. Bell & Howell Information and Learning Com pany 300 North Z eeb Road P .O . Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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). 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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, 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. <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) 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( 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 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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] 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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]. 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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) 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ‘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 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -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. 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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. 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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, 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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]. 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 111 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 115 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 116 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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: 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 120 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 121 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. R EFEREN CES [1] N.O.Sadiku, Numerical Techniques in Electromagnetics, Boca Raton, FL: CRC Press, 1992. [2] R.F.Harrington, Field Computation by Moment Methods, New York: Macmillan, 1968. [3] S.Ramo, J.R.Whinnery and T.Van Duzer, Fields and Waves in Communication Electronics, New York: John Wiley, 1994. [4] D.I.Wu and D.C.Chang, A Review of the Electromagnetic Properties and the FullWave Analysis of the Guiding Structures in MMICs, Proc. 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Ltd - Singapore, 1996. 145 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [39] A.E.Ruehli, N.Kulasza and J.Pivnichny, Inductance of Nonstaright Conductors Close to a Ground Return Plane, IEEE Trans MTT, August 1975, pp 706-708. [40] W.T.Weeks, Calculation o f co-efficients of capacitance o f multiconductor tmsmission lines in the presence o f a dielctric interface, IEEE Trans MTT-18, Jan 1970, pp 35-43. [41] Hong You and Mani Soma, Analysis and Simulation o f Multiconductor Transmission Lines for High-Speed Interconnect and Package Design, IEEE Trans on Components, Hybrids and Manufacturing Technology, Vol. 13, No. 4, Dec 1990, pp 839846. [42] Robert W. Jackson and David M. Pozar, Full-wave Analysis o f Microstrip Open-End and Gap Discontinuities, IEEE Trans MTT-33, No.10, Oct 1985, pp 1036-1042. [43] Minoru Maeda, An Analysis o f Gap in Microstrip Transmission Lines, IEEE Trans MTT-20, No.6 , June 1972, pp 390-396. 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[49] Yuh-Sheng Tsuei, Andreas C. Cangellaris, Rigorous Electromagnetic Modeling of Chip-to-Package (First level) Interconnections, IEEE Trans on Components, Hybrids and Manufacturing Technology, Vol. 16, No.8 , Dec 1993, pp 876-883. [50] Tapan K.. Sarkar, Ercument Arvas and Saila Ponnapalli, Electromagnetic Scattering from Dielectric Bodies, IEEE Trans AP-37, No.5, May 1989, pp 673-676. [51] Adaptation o f “Spice3” to Simulation of Lossy Multiple-Coupled Transmission Lines, IEEE Trans on Components, Packaging and Manufacturing Technology-Part B: Advanced Packaging, Vol. 17, No.2, May 1994, pp 126-133. [52] Gaofeng Wang and Guang-Wen Pan, Full Wave Analysis o f Microstrip Floating Line Structures by Wavelet Expansion Method, IEEE Trans MTT-43, N o.l, Jan 1995, pp 131-142. [53] B.M. Azizur Rahman, F.Anibal Fernandez, and J.Brian Davies, Review of Finite Element Methods for Microwave and Optical Waveguides, Proceedings o f the IEEE, Vol.79, No. 10, Oct 1991, pp 1442-1448. [54] Peter B. Johns, A symmetrical Condensed Node for the TLM method, IEEE trans MTT-35, No.4, April 1987, pp 370-377. 147 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [55] Zhizhang Chen, Michel M.Ney and Wolfgang J.R. Hoefer, Absorbing and Connecting Boundary Connections for the TLM Method, IEEE Tms MTT-41, N o.ll, November 1993, pp 2016-2024. [56] Dominique-Lynda Paul, Nick M. Pothecary and Chris J.Railton, Calculation of the Dispersive Characteristics o f Open Dielectric Structures by the Finite-Difference TimeDomain Method, IEEE Trans MTT-42, No.7, July 1994, pp 1207-1211. [57] Dok Hee Choi and Wolfgang J.R. Hoefer, The Finite-Difference Time-Domain Method and its Application to Eigenvalu Problems, IEEE Trans MTT-34, No. 12, Dec 1986, pp 1464-1469. [58] John Litva, Chen Wu, Ke-Li Wu and Ji Chen, Some Considerations for Using Finite Difference Time Domain Technique to Analyze Microwave Integrated Circuits, IEEE Microwave and Guided Wave Letters, Vol.3, No.2, Dec 1993, pp 438-440. [59] Wojciech K.. Gwarek, Comments on “ On the Relationship Between TLM and Finite-Difference Methods for Maxwell’s Equations”, IEEE Trans MTT-35, No.9, Sep 1987, pp 872-873. [60] Tsugumichi Shibata, Toshio Hayashi and Tadakatsu Kimura, Analysis of Microstrip Circuits Using Three-Dimensional Full-Wave Electromagnetic Field Analysis in the Time Domain, IEEE Trans MTT-36, No.6 , June 1988, pp 1064-1070. [61] Kai Chang, Microwave Solid-State Circuits and Applications, John Wiley and Sons, Inc, NewYork: 1994. 148 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [62] Pierce A.Brennan, Norman Raver and A.E.Ruehli, Three-Dimensional Inductance Computations with Partial Element Equivalent Circuits, IBM Jour. Res and Develop, Vol-23, No.6 , Nove 1979, pp 661- 668 . [63] Chung-Wen Ho and Albert E. Ruehli, The Modified Nodal Approach to Network Analysis, IEEE Trans on Circuits and Systems, Vol. CAS-22, No. 6 , June 1975, pp 504509. [64] T. Vu Dinh, B.Cabon and J.Chilo, SPICE Simulation o f Lossy and Coupled Interconnection Lines, IEEE Trans on Components, Packaging and Manufacturing Technology-Part B: Advanced Packaging, Vol. 17, No.2, May 1994, pp 134-146. [65] Keith Nabors, Songmin Kim and Jacob White, Fast Capacitance Extraction of General Three-Dimensional Structures, IEEE Trans MTT-40, No. 7, July 1992, pp 14961506. [66 ] T. Sakurai and K. Tamaru, Simple Formulas for Two- and Three-Dimensional Capacitances, IEEE Trans on Electron Devices, Vol.ED-30, No.2, Feb 1983, pp 183-185. [67] Jose E. Schutt-Aine and Raj Mittra, Nonlinear Transient Analysis of Coupled Transmission Lines, IEEE Trans on Circuits and Systems, Vol-36, No.7, July 1989, pp 959-967. [68 ] K. Nabors and J.White, Fastcap: A multipole accelerated 3D capacitance extraction, IEEE Trans. Computer-Aided Design, Vol. 11, Oct. 1991, pp 1447-1459. [69] Terry Edwards, Foundations for Microstrip Circuit Design, John Wiley & sons 1982. 149 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. _ 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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