**549,99 zł**

- Wydawca: John Wiley & Sons
- Kategoria: Nauka i nowe technologie
- Język: angielski
- Rok wydania: 2017

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.

Ebooka przeczytasz w aplikacjach Legimi na:

Liczba stron: 688

Title Page

Copyright

Dedication

Preface

General Aspects

Fundamentals of EMM Solution Methods

More About the PEEC Method

Teaching Aspects

References

Acknowledgments

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

2.4 Frequency and Time Domain Analyses

2.5 Frequency Domain Analysis Formulation

2.6 Time Domain Analysis Formulations

2.7 General Modified Nodal Analysis (MNA)

2.8 Including Frequency Dependent Models in Time Domain Solution

2.9 Including Frequency Domain Models in Circuit Solution

2.10 Recursive Convolution Solution

2.11 Circuit Models with Delays or Retardation

References

Chapter 3: Maxwell's Equations

3.1 Maxwell's Equations for PEEC Solutions

3.2 Auxiliary Potentials

3.3 Wave Equations and Their Solutions

3.4 Green's Function

3.5 Equivalence Principles

3.6 Numerical Solution of Integral Equations

References

Chapter 4: Capacitance Computations

4.1 Multiconductor Capacitance Concepts

4.2 Capacitance Models

4.3 Solution Techniques for Capacitance Problems

4.4 Meshing Related Accuracy Problems for PEEC Model

4.5 Representation of Capacitive Currents for PEEC Models

References

Chapter 5: Inductance Computations

5.1 Loop Inductance Computations

5.2 Inductance Computation Using a Solution or a Circuit Solver

5.3 Flux Loops for Partial Inductance

5.4 Inductances of Incomplete Structures

5.5 Computation of Partial Inductances

5.6 General Inductance Computations Using Partial Inductances and Open Loop Inductance

5.7 Difference Cell Pair Inductance Models

5.8 Partial Inductances with Frequency Domain Retardation

References

Chapter 6: Building PEEC Models

6.1 Resistive Circuit Elements for Manhattan-Type Geometries

6.2 Inductance–Resistance (Lp,R)PEEC Models

6.3 General (Lp,Pp,R)PEEC Model Development

6.4 Complete PEEC Model with Input and Output Connections

6.5 Time Domain Representation

References

Chapter 7: Nonorthogonal PEEC Models

7.1 Representation of Nonorthogonal Shapes

7.2 Specification of Nonorthogonal Partial Elements

7.3 Evaluation of Partial Elements for Nonorthogonal PEEC Circuits

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

8.3 SPICE Type Geometry Description

8.4 Detailed Properties of Meshing Algorithms

8.5 Automatic Generation of Geometrical Objects

8.6 Meshing of Some Three Dimensional Pre-determined Shapes

8.7 Approximations with Simplified Meshes

8.8 Mesh Generation Codes

References

Chapter 9: Skin Effect Modeling

9.1 Transmission Line Based Models

9.2 One Dimensional Current Flow Techniques

9.3 3D Volume Filament (VFI) Skin-Effect Model

9.4 Comparisons of Different Skin-Effect Models

References

Chapter 10: PEEC Models for Dielectrics

10.1 Electrical Models for Dielectric Materials

10.2 Circuit Oriented Models for Dispersive Dielectrics

10.3 Multi-Pole Debye Model

10.4 Including Dielectric Models in PEEC Solutions

10.5 Example for Impact of Dielectric Properties in the Time Domain

References

Chapter 11: PEEC Models for Magnetic Material

11.1 Inclusion of Problems with Magnetic Materials

11.2 Model for Magnetic Bodies by Using a Magnetic Scalar Potential and Magnetic Charge Formulation

11.3 PEEC Formulation Including Magnetic Bodies

11.4 Surface Models for Magnetic and Dielectric Material Solutions in PEEC

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

12.3 Direct Far-Field Radiation Computation

References

Chapter 13: Stability and Passivity of PEEC Models

13.1 Fundamental Stability and Passivity Concepts

13.2 Analysis of Properties of PEEC Circuits

13.3 Observability and Controllability of PEEC Circuits

13.4 Passivity Assessment of Solution

13.5 Solver Based Stability and Passivity Enhancement Techniques

13.6 Time Domain Solver Issues for Stability and Passivity

13.7 Acknowledgment

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.2 Controlled Source Stamps

References

Appendix C: Computation of Partial Inductances

C.1 Partial Inductance Formulas for Orthogonal Geometries

C.2 Partial Inductance Formulas for Nonorthogonal Geometries

References

Appendix D: Computation of Partial Coefficients of Potential

D.1 Partial Potential Coefficients for Orthogonal Geometries

D.2 Partial Potential Coefficient Formulas for Nonorthogonal Geometries

References

Appendix E: Auxiliary Techniques for Partial Element Computations

E.1 Multi-function Partial Element Integration

References

Index

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Table of Contents

Preface

Begin Reading

Chapter 2: Circuit Analysis for PEEC Methods

Figure 2.1 Resistance circuit example for Kirchhoff's voltage and current laws.

Figure 2.2 Flux patterns and equivalent circuit for two-conductor system.

Figure 2.3 A small example circuit that includes an inductor.

Figure 2.4 A small example circuit that includes an inductor.

Figure 2.5 Directed graph

G

for the circuit in Figure 2.1.

Figure 2.6 Example PEEC model for three fundamental loops, which includes the capacitive part.

Figure 2.7 Two-conductor equivalent circuit example for KCL matrix.

Figure 2.8 Example of frequency domain macromodel with four ports.

Figure 2.9 Equivalent circuit in terms of admittances for a four-terminal synthesis model.

Figure 2.10 Equivalent RL circuit for real pole synthesis.

Figure 2.11 First equivalent circuit for complex pole/residue pair synthesis.

Figure 2.12 Second equivalent circuit for complex pole/residue pair synthesis.

Figure 2.13 Example for impedance circuit.

Figure 2.14 Macromodel for impedance circuit.

Chapter 3: Maxwell's Equations

Figure 3.1 Example for surface and volume for the integrals.

Figure 3.2 Example interface between two different materials.

Figure 3.3 Illustration of surface equivalence principle for closed-surface two-region problem.

Chapter 4: Capacitance Computations

Figure 4.1 Single plate. (a) Capacitance to infinity. (b) Charge distribution.

Figure 4.2 Multiconductor example where some of the conductors are electrically connected.

Figure 4.3 Flux patterns and equivalent circuit for a two-conductor system.

Figure 4.4 Capacitance model for

N - conductor

system.

Figure 4.5 Example for star node N (a) and Delta elimination of node N (b).

Figure 4.6 Mesh cells for charge distribution on the conductor.

Figure 4.7 (a) Parallel plate conductor. (b) Side view and field lines.

Figure 4.8 Mesh for two-dimensional capacitance model.

Figure 4.9 Two-dimensional capacitance model for free space or dielectric region.

Figure 4.10 Two 2D plates embedded in capacitance mesh.

Figure 4.11 Three-dimensional capacitance model for dielectric region. (a) Subdivision of space. (b) Equivalent circuit.

Figure 4.12 Two conductor example for the capacitance sensitivity.

Figure 4.13 Three-conductor problem with overlapping cells.

Figure 4.14 Three-conductor problem with nonoverlapping projection cells.

Figure 4.15 An example of three capacitive cells for a PEEC circuit.

Figure 4.16 Example PEEC capacitance model for small three cell example.

Figure 4.17 Current source capacitance equivalent circuit model for retarded PEEC model.

Figure 4.18 Voltage source capacitance equivalent circuit model for PEEC model.

Chapter 5: Inductance Computations

Figure 5.1 System of coupled conductor loops.

Figure 5.2 Equivalent circuit for the loop geometry.

Figure 5.3 Example loop for flux and vector potential computations.

Figure 5.4 Single loop approximated by four bars.

Figure 5.5 PEEC equivalent circuit for the single loop.

Figure 5.6 Two examples for the flux loops for partial inductances.

Figure 5.7 Example of a device with two pins and an open loop.

Figure 5.8 Example geometry and equivalent circuit for capacitor macromodel.

Figure 5.9 Example geometry for open-loop inductance computation.

Figure 5.10 Open-loop inductance example for geometry in Figure 5.9.

Figure 5.11 Error in for approximate formula.

Figure 5.12 Long loop represented by four bars.

Figure 5.13 Partial self-inductance divided by length for different cross sections.

Figure 5.14 Partial mutual inductance divided by length for different cross sections.

Figure 5.15 Subinductances divided by length for different cross sections.

Figure 5.16 Open-loop inductance for three bars in series.

Figure 5.17 Loop inductance for three bars in series.

Figure 5.18 Finite two-wire line represented in terms of partial inductances.

Figure 5.19 Coupling pairs for two plane conductor cell pair.

Figure 5.20 Transmission line model with line sections and equivalent circuit.

Figure 5.21 Parallel power planes example with IC and decoupling capacitor.

Figure 5.22 Small example geometry for two parallel plane PEEC model.

Figure 5.23 General schematic MNA matrix with additional equations.

Figure 5.24 Mesh at corner interface between coarse and fine mesh.

Figure 5.25 Equivalent circuit for reduction of meshing example for PPP inductance model.

Figure 5.26 Two-bar example for a retarded partial inductance computation.

Figure 5.27 Round wire example for a retarded partial inductance evaluation.

Figure 5.28 Two wires in series for computation of retarded partial inductance.

Figure 5.29 Geometry of two cells with a certain distance.

Figure 5.30 Mutual inductance real part of the coupled cells by the fast multipole algorithm compared with the theoretical result.

Figure 5.31 Mutual inductance imaginary part of the coupled cells by the fast multipole algorithm compared with the theoretical result.

Figure 5.32 Single-loop inductance example.

Figure 5.33 Two-loop example.

Figure 5.34 Single loop over a ground plane and connected to it in a point.

Figure 5.35 Lossy loop over a lossy ground plane.

Figure 5.36 Equivalent circuit for single-loop example.

Figure 5.37 Capacitor model for inductance problem.

Chapter 6: Building PEEC Models

Figure 6.1 Example conductor bar for resistance computations.

Figure 6.2 Subdivision of thin conducting panel into cells and PEEC circuit.

Figure 6.3 Example cell division with both inductive filaments and capacitive cells.

Figure 6.4 Inductive and capacitive cells for thin metal panel.

Figure 6.5 Three zero thickness capacitance cells example.

Figure 6.6 Example PEEC capacitance model for small three-cell example.

Figure 6.7 Current source capacitance equivalent circuit model for retarded PEEC model.

Figure 6.8 Small (Lp,Pp,R,) PEEC equivalent circuit model with current sources.

Figure 6.9 Quasistatic PEEC model for three fundamental loops.

Figure 6.10 Example of sources associated with a PEEC model node.

Figure 6.11 Single loop impedance example.

Chapter 7: Nonorthogonal PEEC Models

Figure 7.1 Example of two long quadrilateral inductive cells.

Figure 7.2 Line in global 3D coordinate system.

Figure 7.3 Basic quadrilateral element with local and global coordinates.

Figure 7.4 Basic hexahedral element or object with local coordinates.

Figure 7.5 Quadrilateral patch with four inductive half cells.

Figure 7.6 Geometry with rectangular and quadrilateral elements.

Figure 7.7 PEEC model for quad with four nodes in Figure 7.5.

Figure 7.8 Model for volume capacitance from Section 10.4.6.

Figure 7.9 Corner of cell for continuity equation.

Figure 7.10 Example for numerical solution of direction layered representation.

Figure 7.11 Example for subdividing quadrilateral sheet with triangles or rectangles for evaluation of integral for quadrilateral external shapes.

Figure 7.12 Orthogonal quadrilateral for partial self-potential term.

Figure 7.13 Orthogonal coplanar quadrilateral for partial mutual term.

Figure 7.14 Relevant geometry for the double-line integrals.

Figure 7.15 Orthogonal quadrilateral for partial self-potential term.

Chapter 8: Geometrical Description and Meshing

Figure 8.1 A wire section that is meshed with long, shallow angle cells.

Figure 8.19 Triangle to quadrilateral subdivision.

Figure 8.2 Inductive and capacitive mesh for sheet or block.

Figure 8.3 Inductance volume cells for rectangular conductor.

Figure 8.4 3D meshing of a corner for finite thickness plane.

Figure 8.5 Submeshing for a flat nonorthogonal quadrilateral block patch.

Figure 8.6 An example for the introduction of additional boundary nodes.

Figure 8.10 Example of shorting between two hexahedral bodies.

Figure 8.7 Contact and macromodel for contact.

Figure 8.17 Via connection without meshed hole in plane.

Figure 8.8 Basic quadrilateral object with local coordinates.

Figure 8.9 Basic hexahedral element or object with local coordinates.

Figure 8.11 Example of conductive sheet of thickness 2D with skin-effect layers.

Figure 8.12 Example mesh for increasing layer thickness.

Figure 8.13 Example of -projection for layers under narrow conductor.

Figure 8.14 Reduction of four nodes (top) to three nodes (bottom) surfaces.

Figure 8.15 Example for node joining with tolerance circle test.

Figure 8.16 Node relaxation with rectangular cells.

Figure 8.18 Zero thickness approximation of an L-shaped 3D object.

Figure 8.20 Example of small thickness meshed structure EMC problem.

Figure 8.21 Examples of possible

split line

.

Figure 8.22 Example node point arrangements created by generatrix equation.

Figure 8.23 Spiral of node points generated with generatrix equation.

Figure 8.24 Small section of mesh formed with two generatrices.

Figure 8.25 Outline of steps in a PEEC-oriented mesher for zero thickness structures.

Chapter 9: Skin Effect Modeling

Figure 9.1 Conductor interface for 1D skin-effect model.

Figure 9.2 Section of ground plane with layers for current flow.

Figure 9.3 PEEC loop with additional skin-effect impedance source.

Figure 9.4 Two-dimensional skin-effect circuit for the layered model in Figure 9.2.

Figure 9.5 Comparison between PM-GSI and 1D-EXP model internal inductance.

Figure 9.6 Comparison between PM-GSI and 1D-EXP model internal resistance.

Figure 9.7 Resistance and inductance for for 2 layer.

Figure 9.8 Section of round wire for internal skin-effect models.

Figure 9.9 Cross section for round-wire skin-effect model.

Figure 9.10 Skin-effect diffusion equivalent circuit for cylinder.

Figure 9.11 Internal skin-effect resistance in kiloohms for cylinder.

Figure 9.12 Resultant inner differential and external partial inductance in microhenry.

Figure 9.13 Volume filament subdivisions for a single conductor bar with 1D current flow.

Figure 9.14 Equivalent circuit for 1D current VFI section with shorts.

Figure 9.15 Two volume filament (VFI) model bars separated by distance .

Figure 9.16 Shorts connections between faces for the two-conductor example.

Figure 9.17 VFI circuit model for the two-conductor problem in Figure 9.16.

Figure 9.18 Resistance of bar versus frequency.

Figure 9.19 Inductance of bar versus frequency.

Figure 9.20 Mutual inductance of bar versus frequency.

Figure 9.21 Horizontal partitioning of one layer of a metal sheet.

Figure 9.22 Example of vertical cell to connect the skin-effect layers.

Figure 9.23 VFI conductor internal 3D Skin-effect PEEC circuit.

Figure 9.30 Resistance for HShoe, .

Figure 9.31 Inductance for Hshoe, .

Figure 9.32 Resistance for HShoe with conductor with GIBC solver.

Figure 9.33 Inductance for HShoe with conductor with GIBC solver.

Figure 9.24 Geometry of U-shaped test problem called

HShoe

.

Figure 9.25 Shorted L-shaped conductor over ground-plane model.

Figure 9.26 Inductance for HShoe with 1 conductor with thin GSI solver.

Figure 9.27 Resistance for HShoe with 1 conductor with thin GSI solver.

Figure 9.28 Inductance for thin LShape conductor over ground plane.

Figure 9.29 Resistance for thin LShape conductor over ground plane.

Figure 9.34 Two rectangular conductors.

Figure 9.35 A rectangular bar with thickness.

Chapter 10: PEEC Models for Dielectrics

Figure 10.1 Fourth-order Debye model for DriClad. (a) Magnitude of permittivity. (b) Loss tangent.

Figure 10.2 Rectangular block of dielectric material.

Figure 10.3 Equivalent circuit for a one-pole Debye medium.

Figure 10.4 Complex pole equivalent circuit for a Lorentz medium.

Figure 10.5 N-pole Debye dielectric equivalent circuit.

Figure 10.6 FR-4 loss tangent representation for an increasing number of poles (example in Section 10.5.2).

Figure 10.7 FR-4 permittivity representation for an increasing number of poles.

Figure 10.8 Equivalent circuit for a general dispersive dielectric.

Figure 10.9 Two infinite dielectric layers geometry for images.

Figure 10.10 Solution for region above the dielectric interface.

Figure 10.11 Solution for region below the dielectric interface.

Figure 10.12 Reflection and image for the four Green's functions.

Figure 10.13 Example geometry for three dielectric layers.

Figure 10.14 Images for where observation and source points are in layer 1.

Figure 10.15 Images for where the source point is in region 1 and the observation point is in region 2.

Figure 10.16 Images for where the source point is in region 1 and the observation point is in region 3.

Figure 10.17 Images for where the source point is in region 2 and the observation point is in region 1.

Figure 10.18 Images for where the observation and source points are in region 2.

Figure 10.19 Dielectric block between two zero thickness metal plates.

Figure 10.20 One section example of equivalent circuit for dielectric.

Figure 10.21 Equivalent circuit for a Debye medium with finite electrical conductivity.

Figure 10.22 Transmission line example for lossy dielectrics.

Figure 10.23 Time domain output port voltage with Debye model with and different values of .

Figure 10.24 FR-4 loss tangent for increasing number of poles (example in Section 10.5.2).

Figure 10.25 FR-4 permittivity for increasing number of poles (example in Section 10.5.2).

Figure 10.26 Output port voltage (example in Section 10.5.2).

Figure 10.27 Magnitude spectrum of the output port voltage (example in Section 10.5.2).

Figure 10.28 Coplanar microstrip line (example in Section 10.5.3).

Figure 10.29 Near-end (a) and far-end (b) voltages for the coplanar lines. The solid line refers to the results obtained using the proposed methodology in the time domain (PEEC–TD–MNA). The dash–dot line refers to the results obtained using the finite integration technique (FIT) technique in the frequency domain via-inverse fast Fourier transform (IFFT) (FIT–FD–IFFT).

Chapter 11: PEEC Models for Magnetic Material

Figure 11.1 Simple example bar of magnetic material.

Figure 11.2 Magnetic transformer core example.

Figure 11.3 Example bar of magnetic material for reluctance computation.

Figure 11.4 Example problem of a three-bar magnetic circuit.

Figure 11.5 Magnetic equivalent circuit for three-bar problem.

Figure 11.6 Equivalent circuit for transformer.

Figure 11.7 Interface between the two regions with different permeability.

Figure 11.8 Example geometry for conductor and magnetic material problem.

Figure 11.9 Meshing of magnetic body into conventional PEEC cells.

Figure 11.10 Example of two cell volumes for inductive coupling for .

Figure 11.11 Basic PEEC loop that includes the coupling source for the magnetic coupling.

Figure 11.12 Side view of two regions.

Figure 11.13 PEEC equivalent circuit for electrical surface equation.

Figure 11.14 PEEC equivalent circuit for magnetic surface equation.

Figure 11.15 Short transmission line model with finite dielectric block.

Figure 11.16 Imaginary part of input current.

Figure 11.17 Real part of input current.

Chapter 12: Incident and Radiated Field Models

Figure 12.1 Example cell with component of incident field.

Figure 12.2 PEEC model with voltage sources for an incident electric field.

Figure 12.3 PEEC model with current induced by an incident electric field.

Figure 12.4 Electric field sensors for all three directions.

Figure 12.5 Magnetic field sensors loops for all three directions.

Figure 12.6 Example -loop for inductive coupled voltages.

Figure 12.7 Example of approximation for far-field computation.

Chapter 13: Stability and Passivity of PEEC Models

Figure 13.1 Example of a stable and an unstable response.

Figure 13.2 Current source connected to quasistatic (QS)PEEC model.

Figure 13.3 Two port non-passive circuit example.

Figure 13.6 Very-high-frequency resonance using the PEEC model for patch antenna.

Figure 13.4 Port measurement for passivity for EM solver.

Figure 13.5 Small two-patch antenna where source is connected between the two patches.

Figure 13.7 Example loop problem geometry for passivity study.

Figure 13.8 Real part of input impedance for loop problem.

Figure 13.9 Scattering response for loop problem.

Figure 13.10 Result from energy integral for the two-patch antenna for fast input signal.

Figure 13.11 Response and energy integral results for the two-patch antenna for slower input signal.

Figure 13.12 Example of EM solver with stability/passivity enhancement.

Figure 13.13 Example of surface cells with phase subdivisions.

Figure 13.14 Real part of subsurface partitioning of path antenna for three subdivision cases.

Figure 13.15 Results for reflection coefficient test for patch antenna. (a) Results without subdivisions. (b) Results with subdivisions.

Figure 13.16 PEEC model example with damping resistors.

Figure 13.17 Real and imaginary parts of response below for patch antenna with and without damping resistors.

Figure 13.18 Real part response above with resistive damping.

Figure 13.19 Imaginary part above with and without resistive damping.

Figure 13.20 Magnitude example for retarded partial inductance .

Figure 13.21 Phase example for retarded partial inductance .

Figure 13.22 Pole–residue approximation of : near-field case. (a) Magnitude. (b) Phase.

Figure 13.23 Induced voltage in the near-field case.

Figure 13.24 Response for VFI model with center-to-center delays compared to quasistatic model.

Figure 13.25 Skin-effect diffusion equivalent circuit for cylinder.

Figure 13.26 Loss resistance for small cylinder conductor example in kiloohms.

Figure 13.27 Modification of equivalent circuit to include the loss resistance for self-term.

Figure 13.28 Total real part in kiloohms for high frequencies.

Figure 13.29 Inductance for equivalent circuit which includes .

Figure 13.30 Two tube or wire segments in series for coupling calculations.

Figure 13.31 Absolute stability regions for (a) FE and (b) BE methods.

Figure 13.32 Low-pass circuit used for digital filter.

Figure 13.33 Example of two-cell zero thickness sheet.

Figure 13.34 Test circuit for numerical damping of integration method.

Chapter 2: Circuit Analysis for PEEC Methods

Table 2.1 Matrix element stamp for resistor .

Table 2.2 Small list of circuit elements for which we need MNA matrix stamps.

Table 2.3 Matrix element stamp for capacitor in frequency domain.

Table 2.4 Matrix element stamp for capacitor in time domain.

Table 2.5 Time integration methods of importance.

Table 2.6 Matrix element stamp for capacitor in time domain.

Table 2.7 Matrix element stamp for admittance .

Table 2.8 Matrix element stamp for single pole.

Table 2.9 Matrix element stamp for a complex pole pair.

Table 2.10 Matrix stamp for two ports of circuit in Fig. 2.8.

Table 2.11 Poles and residues of the open-ended input impedance of a transmission line.

Chapter 3: Maxwell's Equations

Table 3.1 Notation used for propagation parameters.

Chapter 4: Capacitance Computations

Table 4.1 Matrix element stamp for two terminal capacitances into short circuit capacitance matrix.

Table 4.2 Sensitivity factor for capacitance to meshing.

Chapter 5: Inductance Computations

Table 5.1 Comparison of section-to-section coupling inductance decay.

Table 5.2 Comparison of TL inductance v.s. the number of sections.

Table 5.3 Some results for the PPP method for cm planes.

Chapter 7: Nonorthogonal PEEC Models

Table 7.1 Local corner specification for quadrilateral surface.

Table 7.2 Local corner specification for hexahedral body.

Chapter 8: Geometrical Description and Meshing

Table 8.1 Global corner specification for quadrilateral block surface example.

Chapter 9: Skin Effect Modeling

Table 9.1 Dimensions in micrometers for the example problems.

Chapter 10: PEEC Models for Dielectrics

Table 10.1 FR-4 Debye model parameters.

Table 10.2 Parameters for an FR-4 Debye model.

Chapter 11: PEEC Models for Magnetic Material

Table 11.1 Equivalent variables for the electric and magnetic integral equation.

Table 11.2 Run time for volume compared to surface PEEC formulations.

Chapter 13: Stability and Passivity of PEEC Models

Table 13.1 Values of partial elements for example problem.

Appendix A: Table Of Units

Table A.1 Important variable and constants in SI units.

Table A.2 Important variable and constants in normalized microelectronics units.

Albert E. Ruehli

Missouri University of Science and Technology, Rolla, MO

Giulio Antonini

Università degli Studi dell'Aquila, Italy

Lijun Jiang

Anatomy Instructor, Auburn University, Auburn, AL, USA

Copyright © 2017 by The Institute of Electrical and Electronics Engineers, Inc. All rights reserved.

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Library of Congress Cataloging-in-Publication Data:

Names: Ruehli, A. E. (Albert E.), 1937- author. | Antonini, Giulio, 1969-

author. | Jiang, Lijun 1970- author.

Title: The partial element equivalent circuit method for electro-magnetic and

circuit problems : a paradigm for EM modeling / Albert E. Ruehli, Giulio

Antonini, Lijun Jiang.

Description: Hoboken, New Jersey : John Wiley & Sons, 2016. | Includes

bibliographical references and index.

Identifiers: LCCN 2016026830 (print) | LCCN 2016049198 (ebook) | ISBN

9781118436646 (cloth) | ISBN 9781119078395 (pdf) | ISBN 9781119078401

(epub)

Subjects: LCSH: Electric circuits–Mathematical models. |

Electromagnetism–Mathematical models.

Classification: LCC TK3001 .R68 2016 (print) | LCC TK3001 (ebook) | DDC

621.301/51–dc23

LC record available at https://lccn.loc.gov/2016026830

Cover Design: Wiley

Cover Image: © Vectorig/Gettyimages

The book is dedicated to our families without whose patience and support it would have been impossible to write.

Albert: to my wife, Kristina, and all members of my extended family. This includes not only the children but also the grandchildren who are our future.

Giulio: to my wife Francesca, “the one, the beloved one, the most beautiful” (Ancient Egyptian Poem) and to our loved son Andrea.

Lijun: to my dear wife, Tao, and to my blessed daughters.

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.

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.

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.

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,

USA

Giulio Antonini,

L'Aquila,

Italy

Lijun Jiang,

Hong Kong

January, 2017

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2. C. R. Paul.

Introduction to Electromagnetic Compatibility

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3. A. E. Ruehli. Inductance calculations in a complex integrated circuit environment.

IBM Journal of Research and Development

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(5):470–481, September 1972.

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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21

(2):76–82, February 1973.

5. A. E. Ruehli. Equivalent circuit models for three dimensional multiconductor systems.

IEEE Transactions on Microwave Theory and Techniques

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MTT-22

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6. P. B. Johns and R. L. Beurle. Numerical solution of 2-dimensional scattering problems using a transmission-line matrix.

Proceedings of the IEEE

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7. G. Kron. Equivalent circuit for the field equations of Maxwell.

Proceedings of the IRE

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8. A. Ramachandran, A. Ramachandran, and A. C. Cangellaris. SPICE-compatible stamps for semi-discrete approximations of Maxwell's equations. In

International Journal of Numerical Modelling: Electronic Networks, Devices and Fields

, Volume 21, pp. 265–277, October 2008.

9. K. S. Yee. Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media.

IEEE Transactions on Antennas and Propagation

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10. T. Weiland. Eine Methode zur Loesung der Maxwellschen Gleichungen fuer sechskompoentige Felder auf diskreter Basis.

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The authors of this book are aware that progress in any area, including science and technology, is driven by the requirement for innovation. As part of this contribution, we also learn that it is not pursued in isolation. Our own thinking is greatly impacted in this process in our various areas of research.

We acknowledge contributions by many people who helped to debug the book to various degrees. To remove the errors was a daunting challenge. In particular, Giulio Antonini wants to acknowledge Professor Antonio Orlandi for having initially introducing him to the PEEC method. Giulio Antonini is then deeply grateful to Albert E. Ruehli for his guidance in the PEEC modeling, his long lasting friendship and for the many evenings spent together, making integrals and solving equations, having a lot of fun.

Kristina Ruehli read and corrected each of the chapters multiple times and modified the text written by Swiss-German, Italian, and Chinese thinkers to conform with English grammatical forms of speech, syntax, and punctuation.

We decided to find an alternate approach for acknowledgment of contributions by other researchers, since it is clearly impossible to add all the appropriate references to the book chapters. We added a list of M.S. and Ph.D. works that have contributed to the partial element equivalent circuit (PEEC) method and thereby acknowledge the contributions by their advisors.

Albert E. Ruehli

University of Vermont

PhD 1972

Chien-Nan Kuo

National Taiwan University (NTU)

MS 1990

Jean-Luc Schanen

Université Grenoble Alpes – CNRS

PhD 1994

Carla Giachino

Politecnico di Torino

MSc 1996

Andrea Giuliano

Politecnico di Torino

MSc 1996

Edith Clavel

Université Grenoble Alpes – CNRS

PhD 1996

Jan E. Garrett

The University of Kentucky

PhD 1997

William P. Pinello

The University of Arizona

PhD 1997

Hao Shi

Missouri University of Science and Technology

PhD 1997

Joel Reuben Phillips

Massachusetts Institute of Technology

PhD 1997

Giulio Antonini

Università degli Studi dell'Aquila

PhD 1998

Youssef Moez

Università Grenoble Alpes – CNRS

PhD 1998

Mattan Kamon

Massachusetts Institute of Technology

PhD 1998

Nicola Caporale

Università degli Studi dell'Aquila

MS 1999

Nadége Piette

Université Grenoble Alpes – CNRS

PhD 1999

Gabriele Di Fazio

Università degli Studi dell'Aquila

MS 1999

Junfeng Wang

Massachusetts Institute of Technology

PhD 1999

Yehia Mahmoud Massoud

Massachusetts Institute of Technology

PhD 1999

Jun Fan

Missouri University Science and Technology

PhD 2000

Karen M. Coperich

University of Illinois at Urbana – Champaign

PhD 2001

Jean-Michel Guichon

Université Grenoble Alpes – CNRS

PhD 2001

Maxime Besacier

Université Grenoble Alpes – CNRS

PhD 2001

Andre Görisch

Otto von Guericke University Magdeburg

PhD 2002

Ting-Yi Huang

National Taiwan University (NTU)

MS 2002

Valerio Di Fulvio

Università degli Studi dell'Aquila

MS 2003

Chun-Te Wu

National Taiwan University (NTU)

PhD 2002

Jonas Ekman

Lulea University of Technology

PhD 2003

Luca Daniel

University of California, Berkeley

PhD 2003

Gernot Steinmair

Johannes Kepler University, Linz (A)

PhD 2003

Enrico Vialardi

Politecnico di Torino

PhD 2003

Zhenhai Zhu

Massachusetts Institute of Technology

PhD 2004

Carlo Pariset

Università degli Studi dell'Aquila

MS 2004

Andrea Viaggi

Università degli Studi dell'Aquila

MS 2004

Luca Goffi

Università degli Studi dell'Aquila

MS 2004

Corrado Pistilli

Università degli Studi dell'Aquila

MS 2004

Dipanjan Gope

University of Washington

PhD 2005

Lina Colangelo

Università degli Studi dell'Aquila

MS 2005

Jean-Paul Gonnet

Université Grenoble Alpes – CNRS

PhD 2005

Christian Martin

Université Grenoble Alpes – CNRS

PhD 2005

Thomas Jonas Klemas

Massachusetts Institute of Technology

PhD 2005

Frederik Schmid

Lulea University of Technology

MSc 2005

Shaofeng Yuan

Missouri University of Science and Technology

PhD 2005

Mattia Di Prinzio

Università degli Studi dell'Aquila

MS 2005

Chuanyi Yang

University of Washington

PhD 2005

Amilcare Tiberti

Università degli Studi dell'Aquila

MS 2005

Swagato Chakraborty

University of Washington

PhD 2005

Marco Sabatini

Università degli Studi dell'Aquila

MS 2005

Chuanyi Yang

University of Washington

PhD 2005

Xin Hu

Massachusetts Institute of Technology

PhD 2006

Magnus Olovsson

Lulea University of Technology

MSc 2006

Liang Li

Missouri University Science and Technology

MS 2006

Francesco Ferranti

Università degli Studi dell'Aquila

PhD 2007

Michael Cracraft

Missouri University of Science and Technology

PhD 2007

Martin Ludwig Zitzmann

Friedrich-Alexander-Universität Erlangen-Nürnberg

PhD 2007

Alfredo Centinaro

Università degli Studi dell'Aquila

MS 2008

Xu Gao

Missouri University of Science and Technology

MS 2008

Giuseppe Miscione

Università degli Studi dell'Aquila

MS 2008

Carla Monterisi

Università degli Studi dell'Aquila

MS 2008

Sergey Kochetov

Otto von Guericke University Magdeburg

Habil. 2008

Luca De Camillis

Università degli Studi dell'Aquila

MS 2008

Tore Lindgren

Lulea University of Technology

PhD 2009

Jeremy Aimé

Université Grenoble Alpes – CNRS

PhD 2009

Volker Vahrenholt

Technische Universität Hamburg-Harburg

PhD 2009

Miao Zhang

Tsinghua University

MS 2009

Fan Zhou

Missouri University of Science and Technology

MS 2010

Vincent Ardon

Université Grenoble Alpes – CNRS

PhD 2010

Peter Scholz

Technische Universität Darmstadt

PhD 2010

Eleonora Palma

Università degli Studi dell'Aquila

MS 2011

Tarek A. El Moselhy

Massachusetts Institut of Technology

PhD 2010

Ruey-Bo Sun

National Taiwan University (NTU)

PhD 2011

Hsiang-Yuan Cheng

National Taiwan University (NTU)

PhD 2011

Tung Le Duc

Université Grenoble Alpes – CNRS

PhD 2011

Pasquale Caravaggio

Università degli Studi dell'Aquila

MS 2012

Danesh Daroui

Lulea University of Technology

PhD 2012

Ali Jazzar

Université Grenoble Alpes – CNRS

PhD 2012

Andrea De Luca

Università degli Studi dell'Aquila

MS 2012

Hanfeng Wang

Missouri University of Science and Technology

PhD 2012

Michela Miliacca

Università degli Studi dell'Aquila

MS 2012

Trung Son Nguyen

Université Grenoble Alpes – CNRS

PhD 2012

Maria De Lauretis

Università degli Studi dell'Aquila

MS 2012

Anca Goleanu

Université Grenoble Alpes – CNRS

PhD 2012

Andreas M. Muessing

ETH Zürich

PhD 2012

Thomas de Oliveira

Université Grenoble Alpes – CNRS

PhD 2012

Ivana F. Kovaćević

ETH Zürich

PhD 2012

Nenghong Xia

Hong Kong Polytechnic University

MS 2013

Tao Wang

Missouri University of Science and Technology

MS 2013

Xu Gao

Missouri University of Science and Technology

PhD 2014

Thanh Trung Nguyen

Université Grenoble Alpes – CNRS

PhD 2014

Sohrab Safavi

Lulea University of Technology

PhD 2014

Leihao Wei

Rose-Hulman Institute of Technology

MS 2015

Natalia Bondarenko

Missouri University of Science and Technology

PhD 2015

Tamar Makharashvili

Missouri University of Science and Technology

MS 2015

Daniele Romano

Università degli Studi dell'Aquila

PhD 2017

Yves Hackl

Technische Universität Darmstadt

PhD 2016

Luigi Lombardi

Università degli Studi dell'Aquila

PhD 2018

Carmine Gianfagna

Università degli Studi dell'Aquila

PhD 2019

Albert, Giulio, Lijun

ABC

absorbing boundary condition

BE

backward Euler method, BD1

BD2

backward differentiation method, Gear 2

CAD

computer-aided design

Ckt

circuit

EFIE

electric field integral equation

EM

electromagnetic

EMM

electromagneticmodeling

FDTD

finite difference time domain

FE

forward Euler method

FEM

finiteelement method

FFT

fast Fourier transform

FIR

digital filter nonrecursive

FIT

finiteintegration technique

IIR

digital filter with feedback

KCL

Kirchhoff's current law

KVL

Kirchhoff's voltage law

MFM

multifunction method

MFIE

magnetic field integralequation

MNA

modified nodal analysis

MOR

model order reduction

NI

noise integrity

PCB

printed circuit board

PEEC

partial element equivalent circuit

PI

power integrity

PDE

partial differential equation

PEC

perfect electric conductor

PMC

perfect magnetic conductor

PML

perfect matched layer

PPP

parallel plane PEEC model

PWTD

plane wave time domain

RCS

radar cross section

ROM

reduced order model

SI

signal integrity

SPICE

Simulation Program withIntegrated Circuit Emphasis

TEM

transverse electromagnetic

Theta,Θ

theta integration method

TL

transmission line

TLM

transmission line modeling method

TR

trapezoidal method

VFI

volume filament

WRM

weighted residual method

The history of the fundamental general techniques that are applied in electromagnetic (EM)-solvers today is interesting. Many techniques were devised in the decades between 1960 and 1980. The first work in the EM field for the finite element (FE) technique was presented in 1965 [1]. A paper on the foundation of the numerical differential equation (DE)-based finite difference time domain (FDTD) method [2] was published in 1966.

Interestingly, a second set of techniques used in commercial solvers today had been devised in the decade from 1970 to 1980. The circuit-oriented DE-based transmission line matrix (TLM) method originated in 1971 [3]. The circuit-oriented partial element equivalent circuit (PEEC) method, which is based on an integral equation (IE) formulation, originated in 1972 [4]. Finally, the DE-based finite integration (FIT) approach was devised in 1977 [5]. Since then, many different submethods have been developed based on all the cited techniques.

We note that most of the numerical solutions are solved using the weighted residuals method (WRM) [6]. The notation is not always the same since sometimes the notation MWR is used. Importantly, the WRM applies to the majority of fundamental solution techniques that include FE, DE, or IE based, for example, Refs [6, 7].

In 1968, a key paper on the numerical implementation of integral equation-based approach was presented in Ref. [8] and it was called the method of moments (MoM). The name is used in an inconsistent way since the MoM is a subset of the WRM methods that applies to a subclass of all formulations [9]. The MoM name originated in 1932 [10] as one of the WRM methods where the two approximation functions are not the same unlike for the Galerkin method. Further, there is no relation between MoM and the moment matching method used for macromodeling [11, 12]. To avoid confusions, we do not use MoM as a name for impedance-type IE solution as is done by some researchers.

In this book, we consistently use the WRM notation. We also use it for the PEEC method since the finite, circuit-based solution of the PEEC method can be viewed as being solved using a WRM technique.

Several book chapters have been written on PEEC techniques. An introduction to the capacitance circuit elements and partial inductances is given in Ref. [13] and a book dedicated to partial inductances is [14], while sections of otherbooks are dedicated to PEEC methods in Refs [15–17], and [18].

Making the teaching of PEEC for electromagnetic solution techniques easier and more concrete is one of the aims of the authors. Many entry-level students are discouraged with the complexity of the EM subject. Quite a number of excellent textbooks on the fundamentals of electromagnetic exist today, such as Balanis [19], Chew [20], Collin [21], Kong [22], Paul [23], Plonsey and Collin [24] and Ramo et al. [25]. Hence, the underlying physics and the derivation of Maxwell's equations are well covered in these books. Most of them do an excellent job in presenting the subject in a comprehensive way.

The aim of this book is different. The fact that the PEEC approach is circuits based should help many students and readers understand the electromagnetic concepts. Most electrical engineering students will learn basic circuit analysis before taking a course in electromagnetics. Our aim is to present the concepts in an application-oriented way. Hence, the understanding of the concepts presented in this book should be of practical as well as of theoretical interest.

The PEEC method has evolved from the initial work, for example, Ref. [26] to a multitude of works by many researchers. The technique has expanded from its original focus in the interconnect modeling area to a wider range of applications. The modeling of combined electromagnetic and circuit (EM/Ckt) problems is one of the strengths of this technique. For this reason, we also included an introduction to the necessary circuit concepts in this book.

Many new systems involve circuit aspects as well as EM parts. Therefore, it is very desirable to be able to solve combined EM/Ckt problems. Circuit-based EM approaches are usually differential equation based. This includes the transmission line method (TLM) method. Other equivalent circuit methods are derived from DEs. The IE-based PEEC method is treated in this book.

In this text, we are interested in all types of electronic systems. Unlike some other approaches, PEEC also provides a stable solution, which is important for many realistic EM/Ckt problems. Today, many applications for low-frequency problems are in the power engineering area. Quasistatic PEEC solutions lead to conventional SPICE-type circuits [27]. For higher frequencies above the quasistatic frequency range, the approach results in full-wave solutions. Unfortunately, conventional SPICE solvers cannot be employed for full-wave solutions since the resultant PEEC circuits include delays. This leads to circuit solvers with delays.

Full-wave solutions are becoming more relevant in many cases. Miniaturization and other aspects are important for the growth in electrical systems. Due to the improvements and miniaturization of the semiconductor devices, the maximum frequencies are reaching into the 1000 GHz range. At these higher frequencies and component densities, coupling among the components has become a key issue. As an extreme case, power engineering systems may include integrated circuits withcurrents in the microampere range in the vicinity of bars conducting hundreds of amperes. The sizes of the coupled subsystems range from micrometers to meters.

Full-wave solutions may also be of importance due to the increasing spectrum in the noise signal frequencies and due to physical largeness of the systems. Electromagnetic compatibility (EMC) is another area of growing interest where the ever-increasing frequencies represent new challenges. The use of semiconductor devices in power electronic systems leads to higher frequency noise. From an EM-modeling point of view, these challenges represent many new and interesting problems to be solved.

The fundamental technical idea of the PEEC approach is to convert an IE-based solution of Maxwell's equations into appropriate equivalent circuits, which can then be used in conjunction with other different linear or nonlinear circuits in a circuit solver mode. We should not assume that this will compromise the solution from an electromagnetic point of view. In many situations, the opposite is true. Solutions can very often be found in the circuit domain, which are much more difficult to obtain without circuits. Besides, we can borrow from the large number of techniques that are available from circuit theory as well as from the implementation of today's SPICE-type circuit solvers.

The recent rapid increase in performance of today's PC computers provides many new possibilities for the EM-modeling area. In addition, the processor and increased memory size have made EM-modeling affordable for everyone. The availability of a large memory is key for the solution of larger problems since the complexity of the solution increases rapidly with the problem size. The speedup of PEEC solutions for large models is one aspect that we do not cover in this text. Parallel processing is a way to enhance the compute power. Fortunately, processors also have become widely available at a low cost. All EM-modeling techniques need to be tailored to computer systems to take full advantage of these changes. However, these issues for PEEC are not included in this book.

In this book, we aim to introduce electromagnetic PEEC models in a practical useful way. The book is written such that the concepts are ready for real-life applications. We hope that this helps the understanding of the fundamental concepts. Also, it should make the book useful to industry and will help to emphasize the importance of the subject to new students in the EM field. We are also attempting to make the mathematical formulations as transparent and readable as possible. This should extend the overall readability of the book for self-study for everyone.

Besides the necessary introduction of the basic concepts in the circuit and electromagnetic theories, the techniques are presented first starting with the circuit concepts. All aspects of building PEEC models are presented in a logical way. We find it useful to implement PEEC in a SPICE circuit solver-like implementation such that both the time and frequency domain solutions can be provided without the need for the Fourier transform of the frequency domain solution. Linear equivalent circuits can lead to models thatcan be used equally well in both the time and frequency domains. Clearly, the SPICE solver input language represents an excellent implementation of this fact since sources can be specified such that they apply both in the time and the frequency domains. This also helps the flexibility of the overall solution.

In Chapter 2, we give an introduction to circuit analysis necessary for a PEEC solution. Our solution approach and most SPICE circuit solvers today are based on the modified nodal analysis (MNA) method. Importantly, this approach also leads to the