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Circuit Oriented Electromagnetic Modeling Using the PEEC Techniques

Wiley-IEEE Press
Erschienen am 30. Mai 2017
464 Seiten
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Bridges the gap between electromagnetics and circuits by addressing electrometric modeling (EM) using the Partial Element Equivalent Circuit (PEEC) method
This book provides intuitive solutions to electromagnetic problems by using the Partial Element Equivalent Circuit (PEEC) method. This book begins with an introduction to circuit analysis techniques, laws, and frequency and time domain analyses. The authors also treat Maxwell's equations, capacitance computations, and inductance computations through the lens of the PEEC method. Next, readers learn to build PEEC models in various forms: equivalent circuit models, non-orthogonal PEEC models, skin-effect models, PEEC models for dielectrics, incident and radiate field models, and scattering PEEC models. The book concludes by considering issues like stability and passivity, and includes five appendices some with formulas for partial elements.
* Leads readers to the solution of a multitude of practical problems in the areas of signal and power integrity and electromagnetic interference
* Contains fundamentals, applications, and examples of the PEEC method
* Includes detailed mathematical derivations
Circuit Oriented Electromagnetic Modeling Using the PEEC Techniques is a reference for students, researchers, and developers who work on the physical layer modeling of IC interconnects and Packaging, PCBs, and high speed links.
1. Auflage
John Wiley & Sons
Dateigröße: 21,78 MB
978-1-119-07840-1 (9781119078401)
Schweitzer Klassifikation
Thema Klassifikation
DNB DDC Sachgruppen
Dewey Decimal Classfication (DDC)
BIC 2 Klassifikation
BISAC Klassifikation
Warengruppensystematik 2.0
ALBERT E. RUEHLI is an Adjunct Professor at MST Rolla, Missouri. He received his PhD, EE, at the University of Vermont and an honorary doctorate from Lulea University, Sweden. Ruehli received the Golden Jubilee Medal, the Guillemin-Cauer Prize from the IEEE CAS and the Richard Stoddart Award from the IEEE EMC Society.
GIULI ANTONINI is a Full Professor in the Department of Industrial and Information Engineering and Economics at the Universit?? degli Studi dell'Aquila in L'Aquila, Italy. He received his PhD from the University of Rome "Sapienza." He worked on the development of the PEEC method for more than 15 years.
LIJUN JIANG is an Associate Professor in the Department of EEE at the University of Hong Kong. He received HP STAR Award, Y.T. Lo Outstanding Research Award, IBM Research Technical Achievement Award, and other awards. He serves as the Associate Editor for IEEE Transactions on Antennas and Propagation and for PIER.
  • Cover
  • Title Page
  • Copyright
  • Contents
  • Dedication
  • Preface
  • Acknowledgements
  • Acronyms
  • Chapter 1 Introduction
  • References
  • Chapter 2 Circuit Analysis for PEEC Methods
  • 2.1 Circuit Analysis Techniques
  • 2.2 Overall Electromagnetic and Circuit Solver Structure
  • 2.3 Circuit Laws
  • 2.3.1 Kirchoff's Current Law
  • 2.3.2 Kirchoff's Voltage Law
  • 2.3.3 Branch Impedances
  • 2.3.4 Incomplete Kirchhoff's Current Law
  • 2.4 Frequency and Time Domain Analyses
  • 2.5 Frequency Domain Analysis Formulation
  • 2.6 Time Domain Analysis Formulations
  • 2.6.1 Numerical Integration of Time Domain Differential Equations
  • 2.6.2 List of Integration Methods for PEEC Solver
  • 2.6.3 Initial Conditions for Time Solver with Delays
  • 2.7 General Modified Nodal Analysis (MNA)
  • 2.7.1 Matrix Kirchhoff's Current Law and Stamps
  • 2.7.2 Matrix Kirchhoff's Voltage Law
  • 2.7.3 Matrix KCL Solution of MNA Equations for PEEC
  • 2.7.4 Matrix KCL for Conductor Example
  • 2.8 Including Frequency Dependent Models in Time Domain Solution
  • 2.9 Including Frequency Domain Models in Circuit Solution
  • 2.9.1 Equivalent Circuit for Rational Approximation of Transfer Functions
  • 2.9.2 Inclusion of Frequency Domain Models in a Time Domain Circuit Solver
  • 2.9.3 General Inclusion of Frequency Domain Admittance Models
  • 2.9.4 State-Space and Descriptor Representations
  • 2.10 Recursive Convolution Solution
  • 2.10.1 Conventional Convolution
  • 2.10.2 Recursive Convolution
  • 2.11 Circuit Models with Delays or Retardation
  • 2.11.1 Inclusion of Delays in the Circuit Domain
  • Problems
  • References
  • Chapter 3 Maxwell's Equations
  • 3.1 Maxwell's Equations for PEEC Solutions
  • 3.1.1 Maxwell's Equations in the Differential Form
  • 3.1.2 Maxwell's Equations in the Integral Form
  • 3.1.3 Maxwell's Equations and Kirchhoff's Circuit Laws
  • 3.1.4 Boundary Conditions
  • 3.2 Auxiliary Potentials
  • 3.2.1 Magnetic Vector Potential A and Electric Scalar Potential e
  • 3.2.2 Electric Vector Potential F and Magnetic Scalar Potential m
  • 3.2.3 Important Fundamental Relationships
  • 3.3 Wave Equations and Their Solutions
  • 3.3.1 Wave Equations for E and H
  • 3.3.2 Wave Equations for A, F, and e
  • 3.3.3 Solution of the Helmholtz Equation
  • 3.3.4 Electric Field Integral Equation
  • 3.4 Green's Function
  • 3.4.1 Notation Used for Wave Number and Fourier Transform
  • 3.4.2 Full Wave Free Space Green's Function
  • 3.5 Equivalence Principles
  • 3.5.1 Volume Equivalence Principle
  • 3.5.2 Huygens' Equivalence Principle
  • 3.6 Numerical Solution of Integral Equations
  • Problems
  • References
  • Chapter 4 Capacitance Computations
  • 4.1 Multiconductor Capacitance Concepts
  • 4.2 Capacitance Models
  • 4.2.1 Capacitance Models for Multiconductor Geometries
  • 4.2.2 Short Circuit Capacitances
  • 4.2.3 Coefficient of Potential Matrix Pp
  • 4.2.4 Capacitance of Conductor Systems
  • 4.2.5 Elimination of a Floating Conductor Node
  • 4.2.6 Floating or Reference Free Capacitances
  • 4.3 Solution Techniques for Capacitance Problems
  • 4.3.1 Differential Equation (DE) Methods for Capacitance Computations
  • 4.4 Meshing Related Accuracy Problems for PEEC Model
  • 4.4.1 Impact of Meshing on Capacitances and Stability and Passivity
  • 4.5 Representation of Capacitive Currents for PEEC Models
  • 4.5.1 Quasistatic Capacitance-based Model
  • 4.5.2 Current Source-Based Model for the Capacitances
  • 4.5.3 Potential-Based Model for the Capacitances
  • Problems
  • References
  • Chapter 5 Inductance Computations
  • 5.1 Loop Inductance Computations
  • 5.1.1 Loop Inductance Computation in Terms of Partial Inductances
  • 5.1.2 Circuit Model for Partial Inductance Loop
  • 5.2 Inductance Computation Using a Solution or a Circuit Solver
  • 5.3 Flux Loops for Partial Inductance
  • 5.4 Inductances of Incomplete Structures
  • 5.4.1 Open-Loop Inductances
  • 5.4.2 Open-Loop Macromodels
  • 5.4.3 Examples for Open-Loop Inductances
  • 5.5 Computation of Partial Inductances
  • 5.5.1 Approximate Formulas for Partial Inductances
  • 5.5.2 Inductance Computations for Large Aspect Ratio Conductors
  • 5.6 General Inductance Computations Using Partial Inductances and Open Loop Inductance
  • 5.6.1 Closing the Loop for Open-Loop Problems
  • 5.7 Difference Cell Pair Inductance Models
  • 5.7.1 Inductances for Transmission Line-Type Geometries
  • 5.7.2 Approximate Inductive Coupling Calculation Between Difference Cell Pairs
  • 5.7.3 Inductance of Finite and Semi-Infinite Length TL
  • 5.7.4 Plane Pair PEEC Models Based on Difference Currents
  • 5.7.5 Parallel Plane PEEC Modeling
  • 5.7.6 PEEC Inductance Plane Model with Orthogonal Meshing
  • 5.7.7 Mesh Reduction Without Couplings of Nonparallel Inductances
  • 5.8 Partial Inductances with Frequency Domain Retardation
  • 5.8.1 Thin Wire Example for Retarded Partial Inductances
  • 5.8.2 General Case for Separated Conductor Partial Inductances with Retardation
  • Problems
  • References
  • Chapter 6 Building PEEC Models
  • 6.1 Resistive Circuit Elements for Manhattan-Type Geometries
  • 6.2 Inductance-Resistance (Lp,R)PEEC Models
  • 6.2.1 Inductance-Resistance (L,R)PEEC Model for Bar Conductor
  • 6.3 General (Lp,Pp,R)PEEC Model Development
  • 6.3.1 Continuity Equation and KCL
  • 6.3.2 Relaxation Time for Charge to Surface
  • 6.3.3 Physical Aspect of the Capacitance Model
  • 6.3.4 Equivalent Circuits for PEEC Capacitances
  • 6.3.5 (Pp,R)PEEC Resistive Capacitive Inductor-Less Models
  • 6.3.6 Delayed (Lp,Pp,R, )PEEC Models
  • 6.3.7 Simple Full-Wave (Lp,Pp,R, )PEEC Models Implementation
  • 6.4 Complete PEEC Model with Input and Output Connections
  • 6.4.1 Full-Wave Models
  • 6.4.2 Quasistatic PEEC Models
  • 6.4.3 Input and Output Selectors
  • 6.4.4 Power/Energy Type Circuit Model
  • 6.4.5 Resistances, Inductance, and Capacitive Terms
  • 6.5 Time Domain Representation
  • Problems
  • References
  • Chapter 7 Nonorthogonal PEEC Models
  • 7.1 Representation of Nonorthogonal Shapes
  • 7.1.1 Hexahedral Bodies
  • 7.1.2 Derivatives of the Local Coordinates
  • 7.2 Specification of Nonorthogonal Partial Elements
  • 7.2.1 Discretization of Conductor and Dielectric Geometries
  • 7.2.2 Continuity Equation and KCL for Nonorthogonal Geometries
  • 7.3 Evaluation of Partial Elements for Nonorthogonal PEEC Circuits
  • 7.3.1 Analytic Solution for Quadrilateral Cells in a Plane
  • 7.3.2 General Case for Evaluation of Integral Ip
  • 7.3.3 Evaluation of Integral Ip When Two Sides l Coincide
  • Problems
  • References
  • Chapter 8 Geometrical Description and Meshing
  • 8.1 General Aspects of PEEC Model Meshing Requirements
  • 8.2 Outline of Some Meshing Techniques Available Today
  • 8.2.1 Meshing Example for Rectangular Block
  • 8.2.2 Multiblock Meshing Methods
  • 8.2.3 Meshing of Nonorthogonal Subproblems
  • 8.2.4 Adjustment of Block Boundary Nodes
  • 8.2.5 Contacts Between the EM and Circuit Parts
  • 8.2.6 Nonorthogonal Coordinate System for Geometries
  • 8.3 SPICE Type Geometry Description
  • 8.3.1 Shorting of Adjoining Bodies
  • 8.4 Detailed Properties of Meshing Algorithms
  • 8.4.1 Nonuniform Meshing Algorithm for Efficient PEEC Models
  • 8.4.2 Cell Projection Algorithm
  • 8.4.3 Smoothing and Tolerancing
  • 8.4.4 Node Relaxation
  • 8.5 Automatic Generation of Geometrical Objects
  • 8.5.1 Automatic Meshing Techniques for Thin and Other Objects
  • 8.5.2 Looping Algorithm Example
  • 8.6 Meshing of Some Three Dimensional Pre-determined Shapes
  • 8.6.1 Generation Techniques and Meshing of Special Shapes Like Circles
  • 8.6.2 Bodies Generated by Using Generatrices
  • 8.7 Approximations with Simplified Meshes
  • 8.8 Mesh Generation Codes
  • Problems
  • References
  • Chapter 9 Skin Effect Modeling
  • 9.1 Transmission Line Based Models
  • 9.1.1 Anomalous Skin-Effect Loss and Surface Roughness
  • 9.1.2 Current Flow Direction and Coordinate Dependence
  • 9.2 One Dimensional Current Flow Techniques
  • 9.2.1 Analytical 1D Current Flow Models
  • 9.2.2 Narrow Band High-Frequency Skin-Effect Models
  • 9.2.3 Approximate GSI Thin Conductor Skin-Effect Model
  • 9.2.4 Physics-Based Macromodel
  • 9.2.5 Frequency Domain Solver for Physics-Based Macromodel
  • 9.2.6 Approximate Thin Wire Skin-Effect Loss Model
  • 9.3 3D Volume Filament (VFI) Skin-Effect Model
  • 9.3.1 Approximate 3D VFI Model with 1D Current Flow
  • 9.3.2 Shorts at the Intersections
  • 9.3.3 Proximity Effect
  • 9.3.4 Circuit Equations for Proximity Effect Study
  • 9.3.5 Full 3D Current Flow Skin-Effect Models
  • 9.3.6 Equivalent Circuit for 3D VFI Model
  • 9.3.7 Surface Equivalence Theorem-Based Skin-Effect Model
  • 9.4 Comparisons of Different Skin-Effect Models
  • 9.4.1 Thin Conductor Results
  • 9.4.2 Thick Conductor Results
  • 9.4.3 Comparison of Example Results
  • Problems
  • References
  • Chapter 10 PEEC Models for Dielectrics
  • 10.1 Electrical Models for Dielectric Materials
  • 10.1.1 Frequency and Time Domain Models for Dielectric Materials
  • 10.1.2 Models for Lossy Dielectric Materials
  • 10.1.3 Permittivity Properties of Dielectrics
  • 10.1.4 Electrical Permittivity Model for Time Domain
  • 10.1.5 Causal Models for Dispersive and Lossy Dielectrics
  • 10.2 Circuit Oriented Models for Dispersive Dielectrics
  • 10.2.1 Simple Debye Medium Circuit Model for Dielectric Block
  • 10.2.2 Simple Capacitance Model for Lorentz Media
  • 10.3 Multi-Pole Debye Model
  • 10.3.1 Combined Debye and Lorentz Dielectric Model
  • 10.4 Including Dielectric Models in PEEC Solutions
  • 10.4.1 Models for Uniform, Lossless Dielectrics
  • 10.4.2 Green's Functions for Dielectric Layers Based on the Image Theory
  • 10.4.3 Green's Function for One Dielectric Interface
  • 10.4.4 Three Dielectric Layers Green's Functions
  • 10.4.5 Dielectric Model Based on the Volume Equivalence Theorem
  • 10.4.6 Discretization of Dielectrics
  • 10.4.7 Dispersive Dielectrics Included in the Volume Equivalence Theorem Model
  • 10.4.8 Dispersive Dielectrics with Finite Electrical Conductivity
  • 10.4.9 Convolution Formulation for General Dispersive Media
  • 10.5 Example for Impact of Dielectric Properties in the Time Domain
  • 10.5.1 On-Chip Type Interconnect
  • 10.5.2 Microstrip Line with Dispersive, Lossy dielectric
  • 10.5.3 Coplanar Microstrip Line Example
  • Problems
  • References
  • Chapter 11 PEEC Models for Magnetic Material
  • 11.1 Inclusion of Problems with Magnetic Materials
  • 11.1.1 Magnetic Circuits for Closed Flux Type Class of Problems
  • 11.1.2 Example for Inductance Computation
  • 11.1.3 Magnetic Reluctance Resistance Computation
  • 11.1.4 Inductance Computation for Multiple Magnetic Paths
  • 11.1.5 Equivalent Circuit for Transformer-Type Element
  • 11.2 Model for Magnetic Bodies by Using a Magnetic Scalar Potential and Magnetic Charge Formulation
  • 11.2.1 Magnetic Scalar Potential
  • 11.2.2 Artificial Magnetic Charge
  • 11.2.3 Magnetic Charge Integral Equation for Surface Pole Density
  • 11.2.4 Magnetic Vector Potential
  • 11.3 PEEC Formulation Including Magnetic Bodies
  • 11.3.1 Model for Magnetic Body
  • 11.3.2 Computation of Inductive Magnetic Coupling
  • 11.3.3 Relation Between Magnetic Field, Current, and Magnetization
  • 11.4 Surface Models for Magnetic and Dielectric Material Solutions in PEEC
  • 11.4.1 PEEC Version of Magnetic Field Integral Equation (MFIE)
  • 11.4.2 Combined Integral Equation for Magnetic and Dielectric Bodies
  • Problems
  • References
  • Chapter 12 Incident and Radiated Field Models
  • 12.1 External Incident Field Applied to PEEC Model
  • 12.2 Far-Field Radiation Models by Using Sensors
  • 12.2.1 Radiated Electric Field Calculations Using Sensors
  • 12.2.2 Evaluation of z- Direction Inductive Coupling Term for the E-Field Sensor
  • 12.2.3 Potential Coefficient Coupling Contribution
  • 12.2.4 Summary of E-Field Calculation with eSensor
  • 12.2.5 Magnetic Field Calculation Using Sensors
  • 12.2.6 Time Domain Solution for H-Field Sensor
  • 12.2.7 Frequency Domain Solution for H-Field Sensor
  • 12.3 Direct Far-Field Radiation Computation
  • 12.3.1 General Radiated Field
  • 12.3.2 Radiated Field Computation Based on the PEEC Computation Results
  • 12.3.3 Approximate Computation of Far Fields
  • Problems
  • References
  • Chapter 13 Stability and Passivity of PEEC Models
  • 13.1 Fundamental Stability and Passivity Concepts
  • 13.1.1 Time Domain Stability
  • 13.1.2 Time Domain Passivity
  • 13.1.3 Causality
  • 13.1.4 Positive Real Function and Passivity
  • 13.1.5 Example Circuit for Non- or Limited Passivity
  • 13.2 Analysis of Properties of PEEC Circuits
  • 13.2.1 Ports and Nodal Potentials (Voltages)
  • 13.2.2 Passivity for Quasistatic PEEC Port Impedance
  • 13.3 Observability and Controllability of PEEC Circuits
  • 13.3.1 General Properties
  • 13.3.2 Passivity at Ports for PEEC Circuit in the Frequency Domain
  • 13.3.3 Time Domain Stability and Passivity Issues
  • 13.4 Passivity Assessment of Solution
  • 13.4.1 Port-Based Passivity Assessment in Frequency Domain
  • 13.4.2 Port-Based Passivity Assessment in Time Domain
  • 13.5 Solver Based Stability and Passivity Enhancement Techniques
  • 13.5.1 Solver Enhancement Techniques for Time and Frequency Domains
  • 13.5.2 Passivity Enhancement by Subdivision of Partial Elements
  • 13.5.3 Passivity Enhancement Using Resistive Damping
  • 13.5.4 Partial Elements Delay Macromodels for Passivity Enhancement
  • 13.5.5 Passivity Enhancement for Model with VFI Skin-Effect Models
  • 13.5.6 Physics-Based Skin-Effect Macromodel for Partial Elements
  • 13.5.7 Mutual Coupling Inductance Terms with Retardation
  • 13.6 Time Domain Solver Issues for Stability and Passivity
  • 13.6.1 Impact of Time Integration on Stability
  • 13.6.2 Impact of Numerical Damping of Integration Method
  • 13.6.3 Digital Waveform Filtering
  • Acknowledgment
  • Problems
  • References
  • Appendix A Table of Units
  • A.1 Collection of Variables and Constants for Different Applications
  • Appendix B Modified Nodal Analysis Stamps
  • B.1 Modified Nodal Analysis Matrix Stamps
  • B.1.1 Resistor
  • B.1.2 Capacitor
  • B.1.3 Independent Voltage Source
  • B.1.4 Independent Voltage Source with Series Elements
  • B.1.5 Independent Current Source
  • B.1.6 Short Circuit Connection
  • B.1.7 Coupled Inductances
  • B.1.8 Ideal Transformer Model
  • B.2 Controlled Source Stamps
  • B.2.1 Current Controlled Voltage Source (CCVS)
  • B.2.2 Voltage Controlled Voltage Source (VCVS)
  • B.2.3 Current Controlled Current Source (CCCS)
  • B.2.4 Voltage Controlled Current Source (VCCS)
  • References
  • Appendix C Computation of Partial Inductances
  • C.1 Partial Inductance Formulas for Orthogonal Geometries
  • C.1.1 Lp12 for Two Parallel Filaments
  • C.1.2 Lp11 for Round Wire
  • C.1.3 Lp12 for Filament and Current Sheet
  • C.1.4 Lp11 for Rectangular Zero Thickness Current Sheet
  • C.1.5 Lp12 for Two Parallel Zero Thickness Current Sheets
  • C.1.6 Lp12 for Two Orthogonal Rectangular Current Sheets
  • C.1.7 Lp11 for Rectangular Finite Thickness Bar
  • C.1.8 Lp12 for Two Rectangular Parallel Bars
  • C.1.9 1/R3 Kernel Integral for Parallel Rectangular Sheets
  • C.1.10 1/R3 Kernel Integral for Orthogonal Rectangular Sheets
  • C.2 Partial Inductance Formulas for Nonorthogonal Geometries
  • C.2.1 Rotation for Different Nonorthogonal Conductor Orientations
  • C.2.2 Lp for Arbitrary Oriented Wires in the Same Plane z = 0
  • C.2.3 Lp for Wire Filaments with an Arbitrary Direction
  • C.2.4 Lp for Two Cells or Bars with Same Current Direction
  • C.2.5 Lp for Arbitrary Hexahedral Partial Self-Inductance
  • C.2.6 Lp for Arbitrary Hexahedral Partial Mutual Inductance
  • References
  • Appendix D Computation of Partial Coefficients of Potential
  • D.1 Partial Potential Coefficients for Orthogonal Geometries
  • D.1.1 Pp12 for Two Parallel Wires
  • D.1.2 1/R3 Integral Ip12 for Two Parallel Wires
  • D.1.3 1/R3 Integral Ip12 for Two Orthogonal Filaments
  • D.1.4 Pp11 for Round Tube Cell Shape
  • D.1.5 Pp12 for a Sheet and a Filament
  • D.1.6 Pp11 for Rectangular Sheet Cell
  • D.1.7 Pp12 for Two Parallel Rectangular Sheet Cells
  • D.1.8 Pp12 for Two Orthogonal Rectangular Sheet Cells
  • D.2 Partial Potential Coefficient Formulas for Nonorthogonal Geometries
  • D.2.1 Pp12 for Wire Filaments with an Arbitrary Direction
  • D.2.2 Pp12 for a Pair of General Quadrilaterals on Same Plane
  • References
  • Appendix E Auxiliary Techniques for Partial Element Computations
  • E.1 Multi-function Partial Element Integration
  • E.1.1 Appropriate Numerical Integration Methods
  • E.1.2 Numerical Solution for Singular Self-Coefficients Lp11 or Pp11
  • E.1.3 Analytical and Numerical Integral Solutions with Variable Subdivisions for Nonself-Partial Elements
  • References
  • Index
  • EULA


General Aspects

Electromagnetic (EM) modeling has been of interest to the authors of this book for a large portion of their careers. Giulio Antonini has been involved with partial element equivalent circuit (PEEC) for over 15 years at the Università degli Studi dell'Aquila, Italy, where he is now a professor. Both Albert Ruehli and Lijun Jiang worked as Research Staff members at the IBM Research Laboratory in Yorktown Heights, New York, on electrical interconnect and package modeling and electromagnetic compatibility (EMC) issues. Lijun Jiang is now a professor at the University of Hong Kong, Hong Kong, and Albert Ruehli is now an adjunct professor at the University of Science and Technology, Rolla, Missouri. We all continue to work today on different aspects of the PEEC method.

We welcome the opportunity to share the product of our experience with our readers. Fortunately, electromagnetic modeling (EMM) is a field of increasing importance. Electronic systems have been and will continue to increase in complexity over the years leading to an ever increasing set of new problems in the EM and circuit modeling areas. The number of electronic systems and applications expands every day. This leads to an ever-increasing need for electrical modeling of such systems.

EMM has been a key area of interest to the authors for quite a while. About 40 years ago, the general field of EMM was very specialized and more theoretical. The number of tools in this area and consequent applications were much more limited. Research is driven by the desire to discover new ways and potential applications as well as the need for solutions of real life problems.

Waveguides that mostly were interesting mechanically complex structures were physically large due to the lower frequencies involved. Some of the main topics of interest were antennas and waveguides as well as transmission lines. EM textbooks usually demanded an already high level of education in the theory and they were sometimes removed from realistic problems.

Transmission lines were the most accessible devices from both a theoretical and a practical point of view. Very few tools were available for practical computations especially before computers were widely available. Computers were mostly used for specialized applications. Problems were solved with a combination of theoreticalanalysis and measurements as well as insight that was a result of years of experience.

In contrast, today electromagnetic solver tools are available for the solution of a multitude of problems. Hence, the theoretical and intuitively ascertained solutions have been replaced with numerical method-based results. However, this does not eliminate the need for a thorough understanding of the EM fundamentals and the methods used in EM tools. The advanced capabilities available in the tools require a deeper understanding of the formulations on which the tools are based. We are well aware that the interaction of tools and theory leads to advances.

Textbooks such as Ramo and Whinnery [1] have evolved over many years. Meanwhile, many new excellent introductory textbooks have been written that treat different special subjects such as EMC [2]. Our book is oriented toward a diverse group of students at the senior to graduate level as well as professionals working in this general area. In our text, we clearly want to emphasize the utility of the concepts for real-life applications, and we tried to include as many relevant references as possible.

Fundamentals of EMM Solution Methods

We have to distinguish between two fundamentally different types of circuit models for electromagnetics. Some of them are based on a differential equation (DE) formulation of Maxwell's equations, while others are based on integral equation (IE) form.

The DE forms are commensurate with the system of equations that results from the formulation of a problem in terms of DEs. This results in circuit models that have neighbor-to-neighbor coupling only. The most well-known form is the finite difference time domain (FDTD) method, which is a direct numerical solution of Maxwell's equations. The advantages of DE methods is that very sparse systems of equation result. At the same time, these systems are larger than the ones obtained from IE-based methods.

On the other hand, the IE-based methods will result in systems that have element-to-element couplings. Hence, this results in smaller, denser systems of equations. The finite element (FE) method is a somewhat hybrid technique since it involves local integrations while the overall coupling is local as in the DE methods. This also results in a large and sparse system of equations. Among the formulations used today, there are two circuit-oriented ones: (a) the DE-based transmission line modeling (TLM) method; and (b) the PEEC method. In this text, we mainly consider the IE-based PEEC method.

The PEEC method has evolved over the years from its start in the early 1970s [3-5]. Interestingly, this is about the same time when the other circuit-oriented EM approach - the TLM method - was first published [6]. Some early circuit-oriented work for DE solutions of Maxwell's equations was done by Kron in the 1940s [7]. However, the solution of the large resultant systems was impossible to solve without a computer. Hence, the work was of little practical importance. Recently, matrix stamps for FDTD models have been presented [8].

Around the same time, numerical DE methods made important progress. The FDTD method was conceived in 1966 [9]. Also, the finite integration technique (FIT) technique was published in 1977 [10]. All these methods have made substantial progress since the early work was published.

More About the PEEC Method

The PEEC method evolved in a time span of more than 40 years. From the start, the approach has been tailored for EMM of electronic packages or Electronic Interconnect and Packaging called signal integrity (SI). Power integrity (PI) and noise integrity (NI) as well as EMC problems. In the beginning, only high-performance computer system modeling needed accurate models for the electrical performance of the interconnects and power distribution in the package and chips. In main frame computes the speed of the circuits was much faster than that of conventional computer circuits such as the early personal computers.

Quasistatic solutions were adequate then even for the highest performance systems. Problems such as the transient voltage drops due to large switching currents were discovered very early. This prompted and extended the work on partial inductance calculations for problems of an ever-increasing size. In the 1990s, the modeling of higher performance chips and packages became an issue with the race for higher clock rates in computer chips. This led to the need for full-wave solutions. As a consequence, stability and passivity issues became important. Today, aspects such as skin-effect loss and dielectric loss models are required for realistic models.

Numerous problems can be solved besides package and interconnect and microwave problems. Approximate physics-based PEEC equivalent circuit models can be constructed, which are very helpful for a multitude of purposes. Further, PEEC is one of the methods used in some of the EMM tools. Fortunately, PEEC models can easily be augmented with a multitude of additional circuit models. This leads to other real advantages. Further, techniques have been found to improve the efficiency of these methods. As we show, PEEC is ideally suited for small simple models. Also, the wealth of circuit solution techniques that are available today can be employed. One example of this is the use of the modified nodal analysis (MNA) approach, which helps PEEC for low-frequency and a dc solution that other techniques may not provide.

Teaching Aspects

We hope that this text can be used as an effective tool to introduce EM to new students. We think that a key advantage of the PEEC method is its suitability for an introductory course in EM.

The teaching of the PEEC method can be approached from several different points of view. It may be used as a way to introduce EMM, since most engineering students are more familiar with circuit theory rather than EM theory. This is also the case since circuit courses are taught at a lower level than EM courses. Alternatively, one may want to start with the introduction of the quasistatic PEEC models in a first EM course.

We prefer to use concepts that can be understood in lieu of the introduction of more advanced topics and mathematical notation. As a second course, general PEEC methods could be covered. This could be done, perhaps, in conjunction with introduction of concepts such as interconnect modeling and other chip and package design concepts.

Albert E. Ruehli,

Windham, New Hampshire,


Giulio Antonini,



Lijun Jiang,

Hong Kong

January, 2017


  1. 1. S. Ramo, J. R. Whinnery, and T. Van Duzer. Fields and Waves in Communication Electronics. John Wiley and Sons, Inc., New York, 1994.
  2. 2. C. R. Paul. Introduction to Electromagnetic Compatibility. John Wiley and Sons, Inc., New York, 1992.
  3. 3. A. E. Ruehli. Inductance calculations in a complex integrated circuit environment. IBM Journal of Research and Development, 16(5):470-481, September 1972.
  4. 4. A. E. Ruehli and P. A. Brennan. Efficient capacitance calculations for three-dimensional multiconductor systems. IEEE Transactions on Microwave Theory and Techniques,...

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