reduced-order integral equation methods to solve complex ......utkarsh r. patel philosophical...

Reduced-Order Integral Equation Methods to Solve ComplexElectromagnetic Problems

by

Utkarsh R. Patel

A thesis submitted in conformity with the requirementsfor the degree of Philosophical Doctorate

Graduate Department of The Edward S. Rogers Sr. Department of Electrical &Computer EngineeringUniversity of Toronto

c© Copyright 2019 by Utkarsh R. Patel

Abstract

Reduced-Order Integral Equation Methods to Solve Complex Electromagnetic Problems

Utkarsh R. Patel

Philosophical Doctorate

Graduate Department of The Edward S. Rogers Sr. Department of Electrical & Computer

Engineering

University of Toronto

2019

Despite vast advancements in computational hardware capabilities, full-wave electromagnetic

simulations of many multiscale problems continue to be a daunting task. Multiscale problems

are encountered, for example, when modeling interconnects in an integrated circuit or when

simulating complex electromagnetic structures. In interconnect problems, the main challenge

is to model the multiscale skin effect that develops inside the conductors at high frequency.

Similarly, complex electromagnetic structures are multiscale because these surfaces are tens of

wavelengths large, while each unit cell often contains subwavelength geometrical features.

This thesis presents reduced-order integral equation methods to solve complex multiscale

problems. For interconnect problems, it proposes a single-source surface integral equation

method to model 2-D and 3-D conductors or dielectrics of arbitrary shape. In this approach,

electromagnetic fields inside a conductor or a dielectric object are accurately modeled by a

differential surface admittance operator and an equivalent electric current density on the object’s

surface. Since the proposed method does not use any volumetric unknowns, it is more efficient

than volumetric methods encountered in the literature and commercial solvers, which require a

fine mesh to model the skin effect. Furthermore, since the proposed approach is single-source,

it is more efficient than other surface methods in the literature that require both equivalent

electric and magnetic current densities. Numerical results show that the proposed method

can be over 100× and 20× faster than commercial FEM solvers for 2-D and 3-D problems,

respectively, while consuming significantly lower memory.

The proposed surface method for conductors and dielectrics is further generalized to de-

velop the so-called macromodeling technique to simulate complex scatterers. In this technique,

a heterogeneous scatterer composed of dielectric and PEC objects is accurately modeled by

ii

equivalent electric and magnetic current densities that are introduced on a fictitious surface

enclosing the element. The crux of the technique is to solve for unknowns only on the fictitious

surface, instead of the scatterers, which results in fewer unknowns. Numerical results show that

the proposed macromodeling technique can efficiently simulate electrically large reflectarrays

composed of square patches and Jerusalem crosses, that are difficult to simulate even with

commercial solvers.

iii

Dedicated to my family

for their endless support!

iv

Acknowledgements

I would like to express my deepest gratitude to my advisors Prof. Sean Hum and Prof. Piero

Triverio for their support and guidance. Their co-supervision and complementary expertise

often gave me a different perspective on how to tackle a problem. I am especially thankful for

the freedom they gave me to direct my research and explore various ideas, while being patient

and encouraging when some of the ideas took too long to implement or did not pan out well.

I would also like to thank Prof. Costas Sarris, Prof. George Eleftheriades, and Prof. Mo

Mojahedi for serving on my Ph.D. thesis committee. Their feedback helped improve this thesis.

I also learned a lot from them while taking their courses. I would also like to thank Prof.

Weng Chew for serving on my defense committee as an external examiner. As a graduate

student, I learned many things on computational electromagnetics reading his research papers

and textbooks.

Discussions and interactions with other graduate students in the electromagnetics group

have positively influenced my learning experience, for which I am thankful. As a researcher,

I found it helpful to bounce of ideas off of someone and potentially clear up some doubts or

technical flaws. Outside of my meetings with Prof. Hum and Prof. Triverio, I was mostly able

to do that through long discussions with Dr. Alex Wong, in earlier years of my studies, or

Shashwat Sharma, in later years. Shashwat, who also works on the integral equations method,

also helped a lot with development of research codes, generating results, and proofreading my

papers and thesis. Other students that have helped me in one way or another include: Ciaran

Geaney, Albert He, Parinaz Naseri, Fadime Bekmambetova, Jonathan Anderson, Neeraj Sood,

Hans-Dieter Lang, Trevor Cameron, and Damien Marek.

My Ph.D. degree was funded by various departmental, provincial, and federal scholarships.

I would like to acknowledge financial support from Ontario Student Assistance Program for

awarding me the Ontario Graduate Scholarships (OGS) in the first two years of my degree, the

Natural Sciences and Engineering Research Council of Canada for awarding me the Alexander

Graham Bell Canada Graduate Scholarship (CGS) in years 3 and 4, and the Edward S. Rogers

Department of Electrical and Computer Engineering for awarding me the Doctoral Completition

Award in year 5.

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Contents

1 Introduction 1

1.1 Complex Problems in Electromagnetics . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Wireless Communication and Radar Sensing . . . . . . . . . . . . . . . . 1

1.1.2 Complex Electromagnetic Structures . . . . . . . . . . . . . . . . . . . . . 2

1.1.3 Electronic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Analysis Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 The Finite Difference Time Domain Method . . . . . . . . . . . . . . . . . 7

1.2.2 The Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.3 The Integral Equation Method . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.1 Iterative Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.2 Acceleration Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.3 Macro Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.4 Equivalence Principle Algorithms . . . . . . . . . . . . . . . . . . . . . . . 13

1.4 Thesis Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 The Surface Integral Equation Method 17

2.1 Electromagnetic Fields in a Homogeneous Medium . . . . . . . . . . . . . . . . . 17

2.1.1 Vector Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.2 Superposition Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Electromagnetic Fields in an Inhomogeneous Medium . . . . . . . . . . . . . . . 20

2.2.1 Love’s Equivalence Principle from a Mathematical Perspective . . . . . . 21

2.2.2 Schelkunoff Equivalence Principle . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Scattering from a PEC Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.1 The Method of Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.2 Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Scattering from Penetrable Objects . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4.1 Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4.2 Discretization of Equivalent Current Densities . . . . . . . . . . . . . . . . 33

vi

2.4.3 Discretization of Integral Equations . . . . . . . . . . . . . . . . . . . . . 34

2.4.4 Enforcement of Boundary Conditions . . . . . . . . . . . . . . . . . . . . 36

2.4.5 Elimination of Additional Equations . . . . . . . . . . . . . . . . . . . . . 36

2.4.6 Numerical Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.5 Scattering from Composite Objects . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5.1 Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.5.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.5.4 Numerical Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3 Transmission Line Modeling with a 2-D Surface Method 45

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3 Differential Surface Admittance Operator . . . . . . . . . . . . . . . . . . . . . . 47

3.3.1 Equivalence Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3.2 Contour Integral Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4 Numerical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4.1 Discretization of Fields and Currents . . . . . . . . . . . . . . . . . . . . . 50

3.4.2 Magnetic Field in the Original Configuration . . . . . . . . . . . . . . . . 51

3.4.3 Magnetic Field in the Equivalent Configuration . . . . . . . . . . . . . . . 52

3.4.4 Differential Surface Admittance Operator . . . . . . . . . . . . . . . . . . 53

3.4.5 Choice of C0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.5 Exterior Problem and Impedance Computation . . . . . . . . . . . . . . . . . . . 54

3.6 Calculation of the Admittance Parameters . . . . . . . . . . . . . . . . . . . . . . 56

3.6.1 Equivalence Principle for Electrostatic Problems . . . . . . . . . . . . . . 57

3.6.2 Discretization of Scalar Potential . . . . . . . . . . . . . . . . . . . . . . . 58

3.6.3 Equivalent Charge Density . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.6.4 Capacitance Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.7 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.7.1 Two Conductor Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.7.2 Valley Microstrip Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.7.3 On-chip Transmission Line with Trapezoidal Conductors . . . . . . . . . . 64

3.7.4 A Curved Microstrip Line . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.7.5 A Coated Differential Line . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.8 Chapter Summary and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 68

4 A Magneto-Quasistatic Analysis of 3-D Interconnects with a 2-D Surface

Operator 70

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

vii

4.2 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3 Skin Effect Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3.1 Discretization of Electric and Magnetic Fields . . . . . . . . . . . . . . . . 72

4.3.2 Original Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3.3 Equivalent Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3.4 Equivalent Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.4 Impedance Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75


4.5.1 Loop Conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.5.2 Chip package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.6 Chapter Summary and Contribution . . . . . . . . . . . . . . . . . . . . . . . . . 78

5 Modeling Conductors and Dielectrics with a 3-D Surface Method 80

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2 Differential Surface Admittance Operator . . . . . . . . . . . . . . . . . . . . . . 81

5.2.1 Discretization of Fields and Currents . . . . . . . . . . . . . . . . . . . . . 82

5.2.2 Integral Equation for the Original Problem . . . . . . . . . . . . . . . . . 82

5.2.3 Equivalent Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2.4 Equivalent Electric Current Density . . . . . . . . . . . . . . . . . . . . . 84

5.3 Interconnect Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.3.1 Boundary Conditions on the Terminals . . . . . . . . . . . . . . . . . . . 86

5.3.2 Electric Field Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . 87

5.3.3 Potential Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.3.4 Final System of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . 88

5.4 Numerical Results for Interconnect Modeling . . . . . . . . . . . . . . . . . . . . 88

5.4.1 3-D Inductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.4.2 Multiconductor Bus in a Stratified Medium . . . . . . . . . . . . . . . . . 89

5.5 Scattering from Dielectric Objects . . . . . . . . . . . . . . . . . . . . . . . . . . 91


5.6.1 Dielectric Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.6.2 4 × 4 Array of Dielectric Spheres . . . . . . . . . . . . . . . . . . . . . . . 94


6 A Macromodeling Approach for Complex PEC Scatterers 97

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.2 Macromodel Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.2.1 Fields and Currents Discretization . . . . . . . . . . . . . . . . . . . . . . 100

6.2.2 Equivalence Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.2.3 Stratton-Chu Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.2.4 Stratton-Chu Formulation Applied to the Original Problem . . . . . . . . 103

viii

6.2.5 Stratton-Chu Formulation Applied to the Equivalent Problem . . . . . . . 105

6.2.6 Equivalent Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.2.7 Macromodel For All Array Elements . . . . . . . . . . . . . . . . . . . . . 106

6.3 Exterior Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.4 Acceleration with the AIM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.4.1 Matrix-Vector Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.4.2 The Adaptive Integral Method . . . . . . . . . . . . . . . . . . . . . . . . 109


6.5.1 Array of Spherical Helix Antennas . . . . . . . . . . . . . . . . . . . . . . 113

6.5.2 Two-layer Reflectarray with Jerusalem Cross Elements . . . . . . . . . . . 115

6.5.3 Reflectarray Composed of Elements with Fine Features . . . . . . . . . . 118


7 Simulation of Complex Electromagnetic Structures using a Macromodeling

Technique for Composite Scatterers 123

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.2 Macromodel Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.2.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.2.2 Surface Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

7.2.3 Enforcement of Boundary Conditions . . . . . . . . . . . . . . . . . . . . 128

7.2.4 Macromodel Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.3 Simulation of Electromagnetic Structures . . . . . . . . . . . . . . . . . . . . . . 131

7.3.1 Array of Macromodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.3.2 Inter-Element Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.3.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.3.4 Modeling Connected Equivalent Surfaces . . . . . . . . . . . . . . . . . . 134

7.4 Acceleration for Large Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7.4.1 Iterative solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7.4.2 Evaluation of Matrix-Vector Product with the FFT . . . . . . . . . . . . 136

7.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138


7.5.1 Single-Layer Reflectarray with Square Elements . . . . . . . . . . . . . . . 139

7.5.2 Two-Layer Reflectarray with Jerusalem Cross Elements . . . . . . . . . . 142

7.6 Summary and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

8 Conclusion 146

8.1 Research Achievements: Interconnect Network Modeling . . . . . . . . . . . . . . 146

8.2 Research Achievement: Complex Electromagnetic Structure Modeling . . . . . . 147

8.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

8.4 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

ix

A Integration Routines for the Method of Moments 152

B Evaluation of Singular Integrals in 2-D Problems 158

C Properties of Circulant and Toeplitz Matrices 159

Bibliography 162

x

Table of Acronyms

AIM Adaptive Integral Method

CFIE Combined Field Integral Equation

CG-FFT Conjugate Gradient Fast Fourier Transform

CIM Contour Integral Method

DDM Domain Decomposition Method

EFIE Electric Field Integral Equation

EM Electromagnetic

EPA Equivalence Principle Algorithm

FDTD Finite Difference Time Domain

FEM Finite Element Method

FFT Fast Fourier Transform

FMM Fast Multipole Method

GIBC Generalized Impedance Boundary Condition

GMRES Generalized Minimal Residual

IC Integrated Circuit

IE Integral Equation

LU decomposition Lower-Upper decomposition

MBF Macro Basis Functions

MFIE Magnetic Field Integral Equation

MLFMM Multilevel Fast Multipole Method

MoM Method of Moments

MQS Magneto-quasistatic

PCB Printed Circuit Board

PEC Perfect Electric Conductor

xi

pFFT Precorrected Fast Fourier Transform

PMCHWT Poggio-Miller-Chang-Harrington-Wu-Tsai

PML Perfectly Matched Layers

p.u.l. Per unit length

RWG Rao-Wilton-Glisson

RCS Radar Cross-Section

SIBC Surface Impedance Boundary Condition

T-EFIE Tangential Electric Field Integral Equation

T-MFIE Tangential Magnetic Field Integral Equation

xii

Chapter 1

Introduction

1.1 Complex Problems in Electromagnetics

Understanding the behaviour of the electromagnetic (EM) fields is important to design, ana-

lyze, and optimize many engineering systems in wireless communication, imaging, integrated

circuits, and power systems. Electromagnetic fields are governed by coupled partial differential

equations, which are commonly known as Maxwell’s equations. Analytic solutions of Maxwell’s

equations are limited to very few problems, such as the propagation of waves in waveguides [1]

and stratified media [2]. Therefore, engineers and scientists have relied heavily upon full-wave

numerical methods, such as the finite element method (FEM) [3], the integral equations (IE)

method [4], and the finite difference time domain (FDTD) method [5], to solve complex EM

problems that arise in today’s engineering systems. EM simulation tools are necessary to sustain

rapid advancements in wireless communication, radar sensing, electronic systems, antennas, and

metasurfaces, where EM problems are becoming increasingly complex.

1.1.1 Wireless Communication and Radar Sensing

The quest for higher data transmission rates has stimulated the development of sophisticated

electronic and communication systems and devices. Computational EM solvers are needed,

for example, to design sophisticated antennas and other complex electromagnetic structures

that are used to manipulate EM waves to improve the performance of wireless systems. EM

simulation tools can also give precise information on signal transmission strengths in a wireless

channel [6–8], which can be used to decide locations for wireless hotspots and relays inside

a building or a crowded street. Wireless channel modeling is a multiscale problem in which

the transmitting antenna, for example in a cellphone, is typically subwavelength in dimension,

while geometrical features of the surrounding environment may be a few hundred wavelengths.

Simulating such problems with full-wave EM solvers requires prohibitively large memory and

computation times.

Antenna placement on electrically large platforms, such as an airplane or a ship, is another

1

Chapter 1. Introduction 2

(a) Antenna on an airplane (b) Antenna on a ship

Figure 1.1: Field induced on an airplane and a ship due to antennas on the platform. Simulation resultswere obtained with FEKO [9].

class of problems that needs efficient EM simulation tools. In such problems, the platform

can significantly influence the radiation pattern of the antenna by, for example, shifting null-

locations or increasing the side-lobe levels. Optimal placement of an antenna on a platform

is also important to mitigate electromagnetic interference between the antenna and other elec-

tronic components on the platform. As shown in Fig. 1.1, the field distribution due to an

antenna on a large platform is usually nontrivial and can only be computed with full-wave EM

solvers. Solving for EM field distributions for antenna placement application is a multiscale

problem. In this problem, an antenna is typically a few wavelengths large with potentially even

smaller geometrical features, whereas the platform is a few hundred wavelengths large.

Computational EM tools are also used to predict the radar cross-section of airplanes or

cars. In autonomous vehicles, radar systems are used to identify the position and velocity of

neighbouring vehicles and other obstacles. Radar technology is also heavily relied upon to guide

air-traffic.

1.1.2 Complex Electromagnetic Structures

Electromagnetic surfaces such as reflectarrays [11] and metasurfaces [12,13] are increasingly used

to manipulate EM waves for various applications. For example, reflectarrays are a low cost, low

profile solution to produce focused antenna beams with applications in satellite communication,

terrestrial links, and remote sensing. Reflectarrays are inhomogeneous surfaces that are designed

to reflect the incident wave with a different phase shift at each unit cell location so that the

radiated fields from all elements add up constructively in the desired beam direction(s). They

can also be reconfigured with varactor diodes and other electrical components to steer the main

beam in different directions [14]. A sample reflectarray composed of PEC square patches on a

dielectric substrate backed by a PEC ground plane is shown in Fig. 1.2.

Like reflectarrays, metasurfaces are commonly used to manipulate EM waves for communi-

cation applications. Metasurfaces, like reflectarrays, are periodic with each unit cell designed

to manipulate EM waves locally. In the past, surfaces have been designed as spectral filters


(a) Top view (b) Radiation pattern

Figure 1.2: (a): Top view of a sample reflectarray. (b): Side view of the 3-D radiation pattern of thereflectarray when it is excited by a horn antenna.

Figure 1.3: Top view of a three-layer metasurface designed to refract a plane wave at f = 20 GHz [10].

that block the transmission of EM waves in certain frequency bands, while allowing propaga-

tion of waves at others frequencies. Such surfaces are called frequency-selective surfaces and

are commonly used in microwave ovens and radomes. Metasurfaces have also been designed to

manipulate the EM waves spatially. Metasurfaces are also commonly used to refract EM waves

to design, for example, thin lenses. A sample metasurface is shown in Fig. 1.3.

Presently, reflectarrays and metasurfaces are typically designed with simulation tools that

solve Maxwell’s equations with periodic boundary conditions. However, most of these surfaces

are not periodic because adjacent unit cells may have different shapes. By assuming perfect

periodicity, the mutual coupling between the unit cells is not captured accurately. Addition-

ally, periodic boundary conditions neglect the edge effects due to finite-size of these surfaces1.

1Absorbing boundary conditions may be used to account for edge effect. However, individually simulatingevery unit cell configuration with absorbing boundary conditions can be computationally expensive.


Figure 1.4: An illustration of the architecture of the AMD Radeon R9 GPU. This 3-D integratedarchitecture shows how the memory is stacked in the GPU in the vertical direction so that distance fromthe processor to the memory is minimized, and the latency to retrieve data from the memory is reduced.[credit: amd.com]

Furthermore, they do not account for spillover losses from these surfaces. Therefore, it is impor-

tant to perform full-wave simulation, before fabricating such structures, to ensure that all the

design specifications are met. Full-wave simulation tools can allow the designer to predict the

scattered fields, calculate the directivity, sidelobe levels, spillover losses, and other important

design parameters. Efficient full wave simulation tools can also allow optimizing the surfaces.

For example, they can allow one to design aperiodic surfaces with non-periodic spacing between

the elements.

Full-wave modeling of complex electromagnetic structures such as reflectarrays and meta-

surfaces is a challenging multiscale problem. These surfaces are typically tens-to-hundreds of

wavelengths long. At the same time, each unit cell of the array might have subwavelength geo-

metrical features that are challenging to model. Hence, a full-wave simulation of such surfaces is

very expensive to perform, and in many cases impossible. For example, the 30× 30 reflectarray

in Fig. 1.2 is 13.5λ × 13.5λ large. Running a full-wave simulation of this simple reflectarray

required over a day of computational time and over 256 GB of memory with a commercial

solver. Many reflectarrays are much more complicated than the one shown in Fig. 1.2, and

often contain multiple layers of dielectrics and complex unit cell geometries. Running full-wave

EM simulations of such structures is impossible, even with commercial solvers, without using

large computing clusters.

1.1.3 Electronic Systems

Increasing clock speeds and miniaturization over the years have reduced electrical dimensions

of most electronic systems. Due to miniaturization, circuit components on a printed circuit

board (PCB) are tightly-packed, leading to signal integrity issues such as cross-talk and signal

distortion [15]. Since PCB fabrication and testing can be expensive, most signal integrity issues

need to be resolved at the design stage using EM simulation tools. However, a full-wave EM


(a) A coated differential pair (b) Valley interconnect

Figure 1.5: Sample transmission lines used in electronic systems. Dielectrics are shown with differentshades of yellow, while dark grey shows copper.

Figure 1.6: Current distribution on an interconnect network.

analysis of an entire PCB is usually not possible because a typical PCB contains thousands of

circuit components, signal traces, and input/output (IO) ports. Hence, PCB designers create

compact EM circuit models for individual components on a PCB based on EM simulation

results. For a typical electronic board, this requires creating models for integrated circuits (IC)

driving the signal, IC packaging, PCB traces routing the signal, and additional interconnects,

like cables, that connect the board to external components.

Electromagnetic solvers are also becoming necessary to design ICs. The International Tech-

nology Roadmap for Semiconductors (ITRS) predicted in 2015 that the size of transistors could

stop shrinking by 2021 [16]. Reducing the current size of 7 nm transistors may be cost ineffec-

tive. Therefore, chip manufacturers, like Intel, AMD, and IBM, have been looking for new ways

to increase the level of integration and performance. One of the ways that is currently being

explored by companies is 3-D integration. In 3-D integration, silicon wafers are stacked on top

of one another. Signal transmission between two layers is made possible by through silicon vias

(TSVs). 3-D integration offers two main advantages. First, it allows manufacturers to continue

to increase the number of transistors that can be integrated in an IC. Second, it reduces latency

in signal transmission by having circuit blocks closer to each other, as illustrated in Fig. 1.4.

However, miniaturization and 3-D integration also give rise to higher electromagnetic coupling

between components and traces in the IC, posing modeling challenges for electrical engineers.

Interconnects at system, board, and chip levels are one of the major sources of signal integrity

issues. Typically, interconnects are made up of lossy conductors, such as copper. Electromag-

netic solvers are needed to properly characterize inductive, capacitive, and resistive effects of


(a) 700 Hz (b) 7 kHz (c) 70 kHz

Figure 1.7: Normalized current distribution inside a round copper wire of radius 1 mm showing thedevelopment of skin effect.

interconnects with either a 2-D approach or a 3-D approach. In the 2-D approach, transmission

line theory is invoked to model signal propagation along interconnects. EM solvers are used to

calculate the multiconductor transmission line per-unit length (p.u.l.) parameters. Figure 1.5

shows sample 2-D transmission lines found in electronic systems. While computationally faster

than the 3-D approach, the 2-D approach has some limitations. For example, the interconnect

is assumed to be invariant along the propagation direction and its cross-section is assumed to

be electrically small. The 3-D approach, on the other hand, solves 3-D Maxwell’s equations

without any geometrical assumptions. The 3-D approach can allow, for example, bends and

non-uniform cross-section along the length of the interconnect. A sample 3-D interconnect

network is shown in Fig. 1.6.

The main challenge with both the 2-D and 3-D approaches is the accurate modeling of skin

effect inside conductors, which is a multiscale phenomenon. As shown in Fig. 1.7, due to skin

effect, current crowds at the surface of the conductors at high frequency, but is distributed

uniformly at low frequency. Therefore, commercial finite-element method solvers need very fine

discretization to accurately model conductors at high frequency, requiring a large amount of

memory and long solution times. For example, the interconnect network in Fig. 1.6 could not

be simulated with a commercial EM solver on a computer with 128 GB of memory.

1.2 Analysis Methods

Numerical methods to solve Maxwell’s equations can be categorized into time-domain meth-

ods and frequency-domain methods. In time-domain methods, EM fields are computed at

discrete time steps. The most popular time-domain method is the finite difference time do-

main (FDTD) method. Other commonly used time domain methods are the finite integration

time (FIT) domain method [17] and the time-domain integral equation (TD-IE) method [18].

Time-domain methods are typically preferred for problems that require a broadband solution

of Maxwell’s equations. They are commonly used to solve highly inhomogeneous problems,

such as some biomedical problems that have a variety of tissues with different electrical prop-

erties. They are also used for problems that couple non-linear or multiphysics effects, such as

circuit-electromagnetic or thermal-electromagnetic co-simulations. In multiphysics problems,


the rate of change of EM fields may be significantly different from the circuit or thermal physics,

which impacts the time-step of the simulation. On the other hand, frequency-domain methods

solve Maxwell’s equations in time-harmonic form. Broadband results can be obtained with

frequency-domain methods by running the solver multiple times. Commonly used frequency-

domain methods include the finite-element method (FEM) and the integral equation method

(IE) method.

Numerical methods can be further classified into differential or integral methods. Both

formulations have strengths and weaknesses that must be properly considered before apply-

ing these techniques. As the name suggests, differential methods solve the differential form

of Maxwell’s equations. The FDTD method and FEM are examples of differential methods.

Integral methods, instead, exploit the linearity of Maxwell’s equations and solve for EM fields

through the use of Green’s functions. Since the optimal choice of the numerical method is based

on the type of problem, let us discuss briefly the advantages and disadvantages of the FDTD

method, FEM, and the IE method.

1.2.1 The Finite Difference Time Domain Method

In the FDTD method, the problem domain is first discretized with a structured grid, i.e.

rectangular for 2-D and cubical for 3-D problems. The electric and magnetic fields are defined

on staggered grid cells called Yee cells. Both field quantities are computed at each time step

by numerically solving Maxwell’s equations. The spatial and time derivatives in Maxwell’s

equations are approximated with finite differences, and so the accuracy of the method is easily

controllable with the size of the grid and order of the finite difference scheme.

In comparison to FEM and IE method, FDTD method has the following useful proper-

ties [19]:

3 Inversion: FDTD is generally an explicit method (although implicit formulations exist)

that does not require inversion or LU factorization, which is an expensive step in the IE

method and FEM.

3 Broadband response: Broadband frequency response can be obtained with a single

FDTD simulation via Fourier transform.

3 Simplicity: Compared to FEM and the IE method, FDTD method is intuitive to un-

derstand and simpler to implement.

3 Nonlinear problems: The FDTD method can simulate nonlinear problems very well.

Some of the disadvantages of the FDTD method are the following:

7 Non-rectangular structures: Since FDTD method requires a structured grid, it does

not lend itself very nicely to modeling non-rectangular objects.


7 High permittivity dielectrics and lossy conductors: The grid size and time step

in the FDTD method are dictated by the material properties of the medium. Hence,

simulation of high permittivity dielectric objects or lossy conductors is a challenging

problem that requires fine spatial discretization.

7 Open boundary problems and PML: For open boundary problems, FDTD method

requires discretizing a very large area to mimic open boundaries. Perfectly matched layers

(PMLs), fortunately, alleviate this issue to a certain extent. However, they are somewhat

complicated to implement.

Due to the above properties, it is difficult to apply FDTD method to model interconnects or

complex electromagnetic surfaces because these applications require modeling lossy conductors

and multiscale geometrical features.

1.2.2 The Finite Element Method

In the FEM, the domain is discretized with non-structured mesh elements, such as triangles

for 2-D and tetrahedra for 3-D problems. Electromagnetic fields are expanded in terms of

basis functions that are defined on the mesh. The differential form of Maxwell’s equations in

frequency-domain is solved using the variational principle to obtain an approximate solution of

fields inside the domain. This step involves setting up and solving a linear system.

Compared to the FDTD and the IE method, FEM has the following advantages [20]:

3 Sparse linear system: FEM requires solving a linear system of the form Ax = b. Since

A is sparse, the system solution can be obtained very efficiently. Furthermore, the cost

to store A grows only linearly with the number of mesh elements.

3 Unstructured mesh: Unlike the FDTD method, FEM can handle objects of arbitrary

shapes since it allows the use of a non-structured mesh.

3 Inhomogeneous materials: It is simpler to model inhomogeneous materials in FEM

compared to the IE method.

Disadvantages of FEM are the following:

7 High permittivity dielectrics and lossy conductors: Like in the FDTD method,

modeling high permittivity dielectrics or lossy conductors requires fine meshing in the

FEM in order to capture rapid field variations. Hence, interconnect modeling with FEM

is expensive.

7 Volumetric meshing: For large problems, volumetric meshing algorithms can be a

bottleneck. Meshing algorithms are also difficult to parallelize.

7 Open boundary problems and PML: Similar to the FDTD method, FEM requires

meshing the entire domain with volumetric mesh elements. It also requires termination of


open boundaries with the PML. Due to these issues, FEM is not suitable to model large

complex electromagnetic surfaces.

The above properties make FEM unsuitable to model interconnects or complex electromagnetic

structures that requires modeling lossy conductors and multiscale geometrical features.

1.2.3 The Integral Equation Method

In the frequency-domain IE method, Maxwell’s equations are solved in the integral form using

the superposition principle and the Green’s function of the Helmholtz equation. In this method,

we discretize the surface electric current density on all perfect electric conductor (PEC) surfaces.

Then, a linear system of equations is formulated to solve for unknown electric current densities

by discretizing frequency-domain integral equations using the method of moments (MoM).

The IE method has the following advantages over FEM and FDTD [21]:

D Stratified media: Stratified media that exist in many problems can be modeled with

the multilayer Green’s function. This approach is more efficient than the FEM and FDTD

method because unknowns do not have to be introduced on the interface of each layer,

or within the volume of each layer. This property makes IE method very appealing for

modeling interconnects in a multilayer PCB. It can also be used to compute scattering

from an extremely large electromagnetic surface, such as a reflectarray or a metasurface.

D Background medium: Unlike FEM and FDTD, the IE method does not require mesh-

ing the background medium, which is modeled through the Green’s function. This prop-

erty results in computational savings when simulating radiation and scattering problems.

Some of the drawbacks of the IE method are:

7 Dense linear system: The main disadvantage of the IE method is that it requires

solving a linear system of the form Ax = b, where A is a full matrix. Hence, generating

A and solving the linear system are CPU- and memory-intensive operations.

7 Inhomogeneous media: Unlike FDTD and FEM, it is challenging to model inhomoge-

neous media with the IE method. Piecewise homogeneous media can, however, still be

modeled efficiently with the surface IE method.

7 Broadband robustness: Broadband formulation with surface IE method is a challeng-

ing problem due to numerical issues such as low-frequency breakdown [22] and interior

resonances [23].

The IE method can also be used to model dielectric objects and good conductors in the

simulation domain. Dielectric objects are typically modeled with the Poggio-Miller-Chang-

Harrington-Wu-Tsai (PMCHWT) formulation [21]. In PMCHWT, the object is replaced by


the background medium, and its effect is modeled with unknown equivalent electric and mag-

netic current densities introduced on its surface. However, PMCHWT is inaccurate when

applied to conductive media because of high contrast in wavenumbers of conductors and those

of the surrounding medium. Therefore, alternative surface formulations have been proposed to

model good conductors. One such formulation is the surface impedance boundary condition

(SIBC) [24], which models conductors with a surface current density and an analytic surface

impedance operator that is accurate under the assumption of smooth conductor geometry and

pronounced skin effect. Other surface methods that are accurate over a broad frequency range

include the generalized impedance boundary condition (GIBC) [25], FastImp [26], and two-

region surface integral equation methods [27, 28]. These methods require solving at least two

sets of unknowns on the surface of conductors: equivalent electric current density and equiva-

lent magnetic current density. The differential surface admittance operator approach is another

interesting formulation [29]. This formulation is single-source, and it models conductors with

only equivalent electric current density and an admittance operator. Since the approach re-

lies on the analytic solution of Maxwell’s equations, this approach was previously limited to

2-D round, rectangular, and triangular conductors [29–32, 32, 33]. Recently, the approach was

also extended to 3-D rectangular [34] and cylindrical conductors [35]. However, the differential

surface admittance approach has not been applied to model skin effect inside conductors of

arbitrary shapes. A similar single-source formulation for 2-D dielectric objects was derived pre-

viously [36, 37]. Compared to the differential surface admittance formulation, this approach is

slightly more complicated as it requires computation of more integral operators. Furthermore,

the differential surface admittance operator approach is more intuitive, as it is derived directly

from the equivalence principle.

Good conductors in the IE method can also be modeled by volumetric methods. In a volu-

metric method, the EM fields inside conductors are discretized with volumetric filaments. Com-

monly used volumetric approaches to model conductors include: conductor partitioning [38–40],

PEEC [41, 42], volumetric integral equation [43–45], FastHenry [46], and FastMaxwell [47].

These formulations tend to be robust over a wide frequency range. However, at high frequen-

cies, when the skin depth inside a conductor is much smaller than its cross-section, the number

of mesh elements required to accurately model the field distribution becomes prohibitively large.

For conductors with a rectangular cross-section, non-uniform meshing [48] has been proposed

to reduce the problem size at high frequencies, but generalizing such techniques to arbitrary

geometries is quite difficult. It is worth noting that commercial tools such as EMX [49] use a

volumetric approach to model conductors.

1.3 Literature Review

This thesis focuses on interconnect and complex electromagnetic surface modeling with the IE

method. Solving these two classes of problems poses two main challenges: solving electrically


large problems and modeling multiscale features. As discussed in Sec. 1.2.3, the IE method

requires solving a linear system of form Ax = b, where A is a dense matrix. If the simulation

domain is electrically large, then computational time and memory consumption to store A and

solve the linear system Ax = b grows very quickly. Therefore, realistically, structures larger

than a couple of wavelengths cannot be simulated efficiently with the basic IE formulation.

Furthermore, if the simulation domain contains multiscale features, then mesh needs to be finer

in some places to accurately resolve geometry and field variations. Fine meshes also hurt the

convergence of iterative solvers. Techniques to tackle large-scale and multiscale problems can

be classified into the following categories: iterative methods, acceleration methods, macro basis

functions, and equivalence principle algorithms.

1.3.1 Iterative Methods

As discussed in Sec. 1.2.3, the IE method requires solving a system of the form Ax = b.

Generally, A is formed by computing reaction integrals between source and test basis functions.

The (n, n′)-th entry of A is, therefore, the field on the n-th basis function due to the n′-th source

basis function. The matrix A is a dense matrix because an infinitesimally small current basis

function produces non-zero fields at all points. The complexity of solving a dense linear system

with direct methods is O(n3). Therefore, the linear system Ax = b is solved iteratively with

Krylov methods, like GMRES or conjugate gradient descent, for electrically large problems.

The total computational cost of an iterative method is given by MtMV , where M is the number

of iterations needed to meet the convergence criteria and tMV is the cost of the matrix-vector

product in each iteration.

Preconditioners can significantly reduce the number of iterations required for solution to

converge. Preconditioners are applied by solving a modified linear system of the form

AP−1︸︷︷︸A

Px︸︷︷︸x

= b , (1.1)

where P is the preconditioning matrix, A is the modified system matrix and x is the modified

vector of unknowns. For good preconditioners, A has eigenvalues that are clustered so that the

condition number of the system is low.

The preconditioner matrix P is typically chosen such that P−1 can be obtained easily.

Commonly used preconditioners include the block-preconditioner [50] and incomplete LU pre-

conditioner [51,52]. Advanced preconditioners such as Calderon preconditioner [53] and sparse-

approximate inverse preconditioner [54] can also significantly reduce the number of iterations.

1.3.2 Acceleration Methods

While efficient preconditioners are used to reduce the number of iterations, acceleration methods

are used to reduce the cost of each iteration. Generating and storing A for an electrically large


problem with many unknowns is infeasible in terms of computation time and memory. In

acceleration methods, A is decomposed into two parts: near-field and far-field. For near-

field entries of A, in which source and observation basis functions are nearby, the entries are

computed accurately and stored in a sparse format. However, far-field interactions between

source and observation basis functions are computed in a fast yet approximate way. There are

two main ways to accelerate the matrix-vector product: the multilevel fast multipole method

(MLFMM) and the fast Fourier transform (FFT).

In the fast multipole method (FMM) [55], source basis functions are grouped into several

clusters. The interaction between two clusters is computed by first aggregating fields radiated

from all basis functions in the source cluster through spherical wave functions, then computing

the radiated field on the observation cluster due to the source cluster, and finally by interpolating

fields on each basis function inside the observation cluster. The procedure is demonstrated

pictorially in Fig. 1.8. In the MLFMM [50] each cluster is subdivided into many levels to

obtain computational complexity of O(n log n) [50,56].

The MLFMM works well for scatterers in homogeneous media [57] for which the radiated

field from sources can be expanded with spherical wave harmonics. There are some works that

apply FMM for thin stratified media [58] by expressing the multilayer Green’s function as a

series of homogeneous Green’s function using the discrete complex image method (DCIM) [59].

However, it is difficult to generalize the FMM to stratified media [60].

Alternative acceleration algorithms based on the FFT are suitable for solving problems in

both stratified and homogeneous media. Conjugate fast Fourier transform (CG-FFT), pre-

corrected fast Fourier transform (pFFT), and the adaptive integral equation method (AIM)

are examples of FFT-based acceleration algorithms. The CG-FFT has also been applied to

simulate microstrip structures [61, 62]. However, the CG-FFT requires modeling geometries

on a uniform rectangular grid, which means the grid size is limited by the smallest feature on

the scatterer. For irregular geometries, the AIM [63] has been proposed. To compute far-field

interactions with AIM, source basis functions are first approximated with point sources on a

rectangular grid. Then, radiated fields on all grid points due to all point sources are computed

with FFT. Finally, fields on observation basis functions are computed through interpolation.

This algorithm has a computational complexity of O(n log n). The AIM has been used to simu-

late microstrip antennas [64,65]. A more efficient formulation called precorrected FFT (pFFT)

was proposed to simulate electrostatic and electrodynamic problems [26,66]. In pFFT, the rect-

angular grid can be much coarser than the AIM, which results in a more efficient formulation.

For scattering problems, pFFT has been applied to free space problems [67] and microstrip

antenna structures [60].

1.3.3 Macro Basis Functions

One way to reduce the number of unknowns in the IE method is to use macro basis functions

(MBFs) to expand unknown current densities. MBFs are higher-order basis functions that may


(a) Original Problem

(b) Aggregation

R |~r − ~r ′|

(c) Translation

R |~r − ~r ′|

(d) Disaggregation

Figure 1.8: Main stages in the fast multipole method [21]. (a): A sample problem with several basisfunctions (dots) that are grouped into four clusters (shown with boxes). Red dots represent source basisfunctions and blue dots represent observation basis functions. (b): First, fields are computed on the unitsphere enclosing source cluster due to all source basis functions in the source cluster. (c): The radiationfrom the source cluster is then propagated to calculate fields on the unit sphere enclosing the observationcluster. (d): Fields on each basis function inside the observation cluster is calculated in the last stagevia interpolation.

be interpreted as a linear combination of lower-order basis functions, like Rao-Wilton-Glisson

(RWG) basis functions [68]. Therefore, MBFs span a much larger domain than a single RWG

basis function. The MBFs [69], characteristic basis functions [70], synthetic basis functions [71],

and eigencurrent basis functions [72] are all examples of works that create MBFs from a linear

combination of RWG basis functions using either the eigenvalue decomposition or the singular

value decomposition. By using the eigenvalue decomposition or singular value decomposition,

one can ensure that strong radiating modes are preserved by MBFs, while weak radiating modes

are discarded. This process results in a reduction in the number of basis functions needed to

accurately capture the current distribution.

To the best of our knowledge, MBFs have only been applied to stratified medium antenna

array problems, where MBFs are created for each unit cell in isolation of other cells, i.e. inter-

element coupling is neglected when creating the macro basis functions. However, the strategy

has not been applied to finite-substrate reflectarrays and metasurfaces where it is not obvious

how to truncate the domain to create macro basis functions.

1.3.4 Equivalence Principle Algorithms

The equivalence principle algorithm (EPA) is yet another class of reduced-order modeling tech-

niques in the literature [73–82]. The EPA is based on the Love’s equivalence principle [83].


In the EPA, each scatterer of an array is enclosed by a fictitious (equivalent) closed surface.

Electromagnetic interactions between two or more scatterers are characterized by the scattering

and translation matrices. The scattering matrix relates the incident and scattered tangential

electric and magnetic fields on each equivalent surface. The coupling between two scatterers is

captured by a translation matrix, which relates the tangential electric and magnetic fields on

each pair of equivalent surfaces. This method computes scattered field from an object by solving

for unknowns only on the equivalent surfaces. Hence, the method can efficiently model multi-

scale scatterers. Several works have focused on improving the accuracy of the EPA [74,78] and

reducing the computational cost of the EPA with higher-order basis functions and acceleration

algorithms [75–77].

There are many open problems in regards to the EPA method. From a theoretical viewpoint,

there is a scope for improvement with regards to the choice of basis functions. Typically, RWG

basis functions are used with the EPA. However, due to poor convergence behavior of the basis

functions, many problems need a large number of RWG basis functions to properly model the

(near) fields on the equivalent surfaces. Furthermore, most works in the literature only consider

the case where scatterer is inside the fictitious equivalent surface. It has not been demonstrated

how the method works when scatterers are touching the equivalent surface. Finally, modeling

dielectrics inside the equivalent surface is not well-discussed, especially when dielectrics touch

the equivalent surface. When applying the EPA to simulate electromagnetic surfaces, equivalent

surface always cuts through the dielectric substrate. From an application viewpoint, EPA has

been applied to compute scattering from an array of spheres or cubes [73, 78]. However, EPA

has not been applied to solve for scattering from complex electromagnetic surfaces composed

of dielectric substrates and PECs.

Domain decomposition methods (DDMs) are techniques similar to the EPA [84, 85]. In

the DDM, a large domain is decomposed into several subdomains. Electromagnetic fields on

these subdomains are solved iteratively, enforcing continuity on the intersection of two or more

subdomains. In the past, domain decomposition methods have also been applied to solve array

problems [86–88].

1.4 Thesis Goals

This thesis addresses challenges in simulating interconnects and complex electromagnetic sur-

faces by proposing reduced-order integral equation methods. Simulation techniques developed

for interconnect modeling and complex electromagnetic surfaces are closely related as they are

based on the surface integral equation method and the equivalence principle. We have classified

the proposed techniques as reduced-order techniques because, in comparison to standard MoM

techniques, the proposed techniques reduce the number of unknowns that need to be solved.

First, a surface method based on the differential surface admittance operator [29] is proposed

to model homogeneous lossy conductors of arbitrary shapes for 2-D and 3-D problems. For 2-D


problems, the thesis presents the surface method to compute the per-unit length impedance

and admittance parameters that are inclusive of skin and proximity effects. For 3-D problems,

the thesis presents a technique to compute the multiport scattering matrix of an interconnect

network. From a theoretical viewpoint, the proposed method generalizes the differential sur-

face admittance approach [29] that was previously limited to model 2-D and 3-D conductors

of canonical shapes. The proposed technique can be considered as a reduced-order method

because it turns a volumetric problem into a surface problem by modeling skin effect inside the

conductors with an equivalent surface electric current density and an admittance operator. Due

to the absence of a volumetric discretization, the proposed method is significantly more efficient

than volumetric methods [38–47]. Furthermore, the proposed technique is more efficient than

other surface methods [25–28] because it only requires a single equivalent current density to

model conductors, which results in fewer unknowns than other surface methods.

In the second part, the thesis also presents a macromodeling approach to simulate complex

electromagnetic surfaces. The macromodeling approach can be considered as a reduced-order

method to model heterogeneous objects, which are composed of dielectrics and PECs, with

equivalent surface currents. In this approach, each unit cell of a complex electromagnetic

surface is modeled by unknown current densities that are introduced on a fictitious surface

enclosing each unit cell. This results in a reduced-order method in which a complex electro-

magnetic surface is turned into an equivalent problem with current densities on an array of

equivalent surfaces. This approach can be considered as a form of EPA with application to

the modeling of complex electromagnetic surfaces. The proposed macromodeling approach is

first developed for PEC scatterers. This single-source macromodeling technique is more effi-

cient than the EPA because it requires only an electric equivalent current density to model the

scatterer [74]. Then, a dual-source macromodeling technique is developed for scatterers com-

posed of PEC and dielectric objects. The proposed developments provide a solution to the open

problem of modeling continuous dielectric substrates with the EPA. This extension allows effi-

cient simulation of complex electromagnetic surfaces such as reflectarrays, transmitarrays, and

metasurfaces. Furthermore, to simulate electrically large array problems with many elements,

the macromodeling technique is accelerated with FFT-based algorithms.

1.5 Thesis Organization

This thesis is organized as follows:

• Chapter 2 presents background material on the integral equation method by reviewing

the superposition principle, Green’s functions, the equivalence principle, the method of

moments, and the surface integral equation formulation for PEC and dielectric objects.

• Chapter 3 presents the differential surface admittance approach to model 2-D conductors

and dielectrics. As an application, the chapter shows how the surface method can be used


to compute the per-unit length impedance and admittance parameters of multiconductor

transmission lines that are composed of lossy conductors and dielectrics.

• Chapter 4 uses the 2-D surface operator to model 3-D conductors with straight segments.

The technique can be used to compute, for example, the parasitic impedance of a chip

package.

• Chapter 5 generalizes the differential surface admittance operator to 3-D conductors and

dielectric objects. The presented technique can accurately model skin effect inside con-

ductors of arbitrary shapes without any volumetric discretization.

• Chapter 6 presents a single-source macromodeling technique for antenna arrays composed

of PECs. This is a further generalization of the proposed surface method in which reduced-

order models are formed for complicated PEC scatterers. With the proposed method, each

antenna is modeled compactly with an equivalent current density that is introduced on a

fictitious surface enclosing the element and an admittance operator.

• Chapter 7 presents the macromodeling technique for antenna arrays composed of PECs

and dielectrics. In this technique, a multilayered unit cell of a complex electromagnetic

surface is modeled by equivalent electric and magnetic current densities introduced on

a fictitious surface enclosing the unit cell. We accelerated the technique using the fast

Fourier transform to simulate electrically large surfaces such as reflectarray.

• Chapter 8 summarizes the contributions of this thesis and discusses possible ways to

improve the presented work.

Chapter 2

The Surface Integral Equation

Method

2.1 Electromagnetic Fields in a Homogeneous Medium

2.1.1 Vector Wave Equation

Time-harmonic electromagnetic fields inside a simple medium (linear, isotropic, and homoge-

neous) are governed by the following Maxwell equations:

∇× ~E(~r) = −jωµ ~H(~r)− ~M(~r) (2.1a)

∇× ~H(~r) = jωε ~E(~r) + σ ~E(~r) + ~J(~r) (2.1b)

∇ · ~E(~r) =qeε

(2.1c)

∇ · ~H(~r) =qmµ, (2.1d)

where ~E(~r) is the electric field intensity with units of V/m, ~H(~r) is the magnetic field intensity

with units of A/m, qe is the electric charge density with units of C/m3, qm is the magnetic

charge density with units of Wb/m3, ~J(~r) is the impressed or source electric current density

with units of A/m2, and ~M(~r) is the impressed or source magnetic current density with units of

V/m2. Throughout this thesis, unless explicitly stated, all vectorial quantities are in the phasor

domain. Magnetic permeability, electric permittivity, and electric conductivity are denoted by

µ, ε, and σ, respectively. Even though the magnetic current density and magnetic charge density

are non-physical, they are included here in the Maxwell equations to support the duality theory,

which eases our understanding for certain types of problems [89].

By substituting the expression for ~H(~r) from (2.1a) into (2.1b) and simplifying the resulting

equation, we obtain the vector wave equation [83]

∇×∇× ~E(~r)− k2 ~E(~r) = −∇× ~M(~r)− jωµ ~J(~r) , (2.2)

17

Chapter 2. The Surface Integral Equation Method 18

where k =√ωµ (ωε− jσ) is the wavenumber inside the medium. Similarly, the vector wave

equation for the magnetic field is obtained by substituting (2.1b) into (2.1a)

∇×∇× ~H(~r)− k2 ~H(~r) = ∇× ~J(~r)− (jωε+ σ) ~M(~r) . (2.3)

2.1.2 Superposition Integrals

We can simplify (2.2) by using the vector identity ∇×∇× ~A = ∇(∇ · ~A)−∇2 ~A to obtain

∇(∇ · ~E(~r)

)−∇2 ~E(~r)− k2 ~E(~r) = −∇× ~M(~r)− jωµ ~J(~r) . (2.4)

Furthermore, by taking the divergence of (2.1b)1 and substituting the result in (2.4), we obtain

∇2 ~E(~r) + k2 ~E(~r) = − 1

(jωε+ σ)∇(∇ · ~J(~r)

)+∇× ~M(~r) + jωµ ~J(~r) ,

= jωµ

[1 +∇∇·k2

]~J(~r) +∇× ~M(~r) . (2.5)

From (2.5), the total electric field can be interpreted as the solution of the Helmholtz equation

due to all the source terms on the right-hand side of (2.5). Due to the linearity of Maxwell’s

equations, the electric field can be written using superposition integrals as

~E(~r) = −jωµˆVG(~r, ~r ′)

[1 +∇′∇′k2·]~J(~r ′)dV ′ −

ˆVG(~r, ~r ′)∇′ × ~M(~r ′)dV ′ , (2.6)

where dV ′ is the differential volume element, the integration is performed over the region of

space where ~J(~r) or ~M(~r) is nonzero, and G(~r, ~r ′) is the scalar Green’s function of the Helmholtz

equation. The scalar Green’s function is the solution of the scalar Helmholtz equation for a

point source

∇2G(~r, ~r ′) + k2G(~r, ~r ′) = −δ(~r, ~r ′) , (2.7)

where δ(~r, ~r ′) is used to denote the Dirac delta function which is nonzero only if ~r = ~r ′ and

satisfies the property´R3 δ(~r, ~r

′)dV ′ = 1. For a 3-D homogeneous medium, the Green’s function

is [21]

G(~r, ~r ′) =1

4π

e−jk|~r−~r′|

|~r − ~r ′| . (2.8)

By duality, the magnetic field due to ~J(~r) and ~M(~r) is given by

~H(~r) = − (jωε+ σ)

ˆVG(~r, ~r ′)

[1 +∇′∇′k2·]~M(~r ′)dV ′ +

ˆVG(~r, ~r ′)∇′ × ~J(~r ′)dV ′ . (2.9)

In our expressions, the derivatives in ∇ and ∇′ act on observation coordinate ~r and source

1By taking the divergence of (2.1b) and using (2.1c), we obtain the continuity equation.


coordinate ~r ′, respectively. The operator ∇′∇′ in (2.6) and (2.9) is undesirable because it

requires that the source term must be twice differentiable. In numerical simulations, this poses

a limitation that the basis functions chosen to expand the source terms have a non-zero second

derivative. However, we can transfer one or both del operators ∇′ from source coordinates to

the observation coordinates and the Green’s function using vector calculus identities [21]

∇ ·(ψ ~A)

= ψ∇ · ~A+ ~A · ∇ψ (2.10a)

∇ (ψφ) = φ∇ψ + ψ∇φ (2.10b)

∇×(ψ ~A)

= ψ(∇× ~A

)+∇ψ × ~A , (2.10c)

and Gauss’ theorem [89] ˆV∇ · ~A(~r)dV =

˛Sn · ~A(~r)dS , (2.11)

where dS is the differential surface element, S is the surface bounding the volume V and n is

the outward-pointing unit vector on the surface. We also need to use the following property of

the homogeneous Green’s function

∇G(~r, ~r ′) = −∇′G(~r, ~r ′) . (2.12)

After turning all ∇′ operators to ∇ in (2.6) and (2.9), we obtain the following superposition

integrals for the electric and magnetic fields

~E(~r) = −jωµ[~L ~J(~r ′)

](~r)−

[~K ~M(~r ′)

](~r) (2.13)

~H(~r) = − (jωε+ σ)[~L ~M(~r ′)

](~r) +

[~K ~J(~r ′)

](~r) (2.14)

where we introduced the integro-differential operators[~L ~X(~r ′)

](~r) =

[1 +

1

k2∇∇·

]ˆVG(~r, ~r ′) ~X(~r ′)dV ′ (2.15a)[

~K ~X(~r ′)]

(~r) = ∇×ˆVG(~r, ~r ′) ~X(~r ′)dV ′ . (2.15b)

The ~K operator is not well-behaved because of the singularity that exists for ~r → ~r ′. This

singularity may be extracted analytically using [21]

lim~r→~r ′

(n(~r)×

[~K ~X(~r ′)

](~r))

=~X(~r)

2. (2.16)

Therefore, (2.15b) can be evaluated numerically as

n×[~K ~X(~r ′)

](~r) = n×∇×

ˆVG(~r, ~r ′) ~X(~r ′)dV ′


= n× p.v.

[∇×

ˆVG(~r, ~r ′) ~X(~r ′)dV ′

]+

~X(~r)

2, (2.17)

where p.v. [·] stands for the principal value. Singularity in the ~L operator is handled via analytic

integration, as discussed in Appendix A.

Many works in the literature choose to express the superposition integrals in (2.13) and (2.14)

in terms of the dyadic Green’s function as

~E(~r) = −ˆV∇×G(~r, ~r ′) · ~M(~r ′)dV ′ − jωµ

ˆVG(~r, ~r ′) · ~J(~r ′)dV ′ , (2.18)

~H(~r) =

ˆV∇×G(~r, ~r ′) · ~J(~r ′)dV ′ − (jωε+ σ)

ˆVG(~r, ~r ′) · ~M(~r ′)dV ′ , (2.19)

where G(~r, ~r ′) satisfies the following differential equation [90]

∇×∇×G(~r, ~r ′)− k2G(~r, ~r ′) = Iδ(~r, ~r ′) . (2.20)

In (2.20), I is the identity tensor that is defined as

I = xx+ yy + zz (2.21)

where x, y, and z are unit vectors in the Cartesian coordinate system. For a homogeneous

medium, the dyadic Green’s function is related to the scalar Green’s function through

G(~r, ~r ′) =

[I +∇∇k2

]G(~r, ~r ′) (2.22a)

∇×G(~r, ~r ′) = ∇×(IG(~r, ~r ′)

)= ∇G(~r, ~r ′)× I . (2.22b)

In this thesis, we will use the dyadic Green’s function to derive the equivalence principle from

first principles in Sec. 2.2. Throughout the rest of the thesis, we will be using the scalar

Green’s function G(~r, ~r ′). For other complex media, such as a stratified medium, the dyadic

Green’s function is more complicated and cannot be expressed in terms of a single scalar Green’s

function [90].

2.2 Electromagnetic Fields in an Inhomogeneous Medium

In Sec. 2.1, we discussed how to evaluate the electromagnetic fields inside a homogeneous

medium due to a given set of electric and magnetic current distributions. Most practical elec-

tromagnetic problems, however, have sources inside an inhomogeneous medium that contain a

mixture of perfect electric conductors, lossy electric conductors, dielectrics, magnetic materials,

etc. In the surface integral equation method, modeling of such materials is facilitated by the

use of the equivalence principle.


S~Mi, ~Ji

Vi

Vo

~Mo, ~Jo ni no

S∞

Figure 2.1: Sample problem to demonstrate the equivalence principle

2.2.1 Love’s Equivalence Principle from a Mathematical Perspective

Most electromagnetic textbooks present the final result of the equivalence principle without a

detailed derivation. Since our developments will heavily rely on this theorem, we go through

its derivation to obtain greater insight. Therefore, let us derive the so-called Love’s equivalence

principle from the wave equation and the Green’s theorem. We adopt the procedure following

one discussed previously [91].

For this demonstration, let us consider the setup shown in Fig. 2.1, which contains two sets

of sources:(~Ji(~r), ~Mi(~r)

)and

(~Jo(~r), ~Mo(~r)

). Set

(~Ji(~r), ~Mi(~r)

)is inside volume Vi that is

bounded by S; and set(~Jo(~r), ~Mo(~r)

)is inside volume Vo that is bounded inside by S and

bounded outside by S∞, which is a sphere whose radius approaches infinity. We will assume

that both Vi and Vo are homogeneous media, but with different material properties.

Equations (2.13)–(2.14) cannot be used to evaluate the electromagnetic fields due to two

sets of sources because we do not know the Green’s function of the inhomogeneous medium in

the setup. Instead, we need to apply the equivalence principle. So, let us first find fields due to

both sources when observation point ~r ∈ Vi. Since Vi is a homogeneous medium, we do know

from (2.2) that inside Vi the vector wave equation for electric field reads

∇×∇× ~E(~r)− k2i~E(~r) = −∇× ~Mi(~r)− jωµ ~Ji(~r) . (2.23)

Also, the dyadic Green’s function for the electric field in Vi is Gi(~r, ~r′) and it satisfies the

following dyadic differential equation

∇×∇×Gi(~r, ~r ′)− k2iGi(~r, ~r

′) = Iδ(~r, ~r ′) , (2.24)

where ki is the wavenumber inside Vi. Recall that the dyadic Green’s function Gi(~r, ~r′) is

related to the scalar Green’s function by (2.22a).

Our goal is to compute ~E(~r) in terms of the Green’s function. As such, we right-multiply (2.23)

by Gi(~r, ~r′) and subtract from it (2.24) left-multiplied by ~E(~r), and then integrate the result


over Vi to obtain

ˆVi∇×∇× ~E(~r)·Gi(~r, ~r ′)− ~E(~r, ~r ′) · ∇ ×∇×Gi(~r, ~r ′)dV = −

ˆVi∇× ~Mi(~r) ·Gi(~r, ~r ′)dV

− jωµˆVi

~Ji(~r) ·Gi(~r, ~r ′)dV −ˆVi

~E(~r) · Iδ(~r, ~r ′)dV︸︷︷︸~E(~r ′) if ~r ′ ∈ Vi0 if ~r ′ ∈ Vo

. (2.25)

Note that we have used the sifting property of the Dirac delta function to simplify the last term

on the right-hand side in (2.25). Using the vector identity [91]

∇×∇× ~E(~r) ·G(~r, ~r ′)− ~E(~r) · ∇ ×∇×G(~r, ~r ′)

= ∇ ·[~E(~r)×∇×G(~r, ~r ′) +∇× ~E(~r)×G(~r, ~r ′)

](2.26)

and by applying the Gauss’s theorem in (2.11), (2.25) can be rewritten as

−˛Sni·[~E(~r)×∇×G(~r, ~r ′) +∇× ~E(~r)×G(~r, ~r ′)

]dV = −

ˆVi∇× ~Mi(~r) ·Gi(~r, ~r ′)dV

− jωµˆVi

~Ji(~r) ·Gi(~r, ~r ′)dV −

~E(~r ′) ~r ′ ∈ Vi0 ~r ′ ∈ Vo

. (2.27)

Note that we have replaced the δ function in (2.11) by a case statement in (2.27). Next, we

substitute Faraday’s Law (2.1a) into (2.27), swap ~r and ~r ′, and take the transpose of the

resulting equation2 to obtain

−˛S∇×G(~r, ~r ′) ·

[−n′i × ~E(~r ′)

]dS′ − jωµ

˛SG(~r, ~r ′) ·

[n′i × ~H(~r ′)

]dS′

−ˆVi∇×Gi(~r, ~r ′) · ~Mi(~r

′)dV ′ − jωµˆViGi(~r, ~r

′) · ~Ji(~r ′)dV ′ =

~E(~r) ~r ∈ Vi0 ~r ∈ Vo

, (2.28)

Equation (2.28) is the mathematical form of the Love’s equivalence theorem. It says that the

electric field inside Vi is due to the radiation from sources that are within Vi, i.e. ~Ji(~r) and

~Mi(~r), and equivalent sources that are introduced on S. The equivalent electric and magnetic

current densities on S have values of

~Jeq,i(~r) = ni × ~H(~r) (2.29a)

~Meq,i(~r) = −ni × ~E(~r) . (2.29b)

It is straightforward to derive a similar expression for evaluating the magnetic fields with the

2Transpose is taken strictly to obtain a conventional form where a dyad is dot-multiplied by a vector.


~Mi, ~Ji

Vi

Vo

~Jeq,i(~r)

~Meq,i(~r)

Null Fields

ni no

(a) Inner equivalence

Vi

Vo

~Jeq,o(~r)

~Meq,o(~r)

Null Fields

~Mo, ~Jo ni no

(b) Outer equivalence

Figure 2.2: Inner and outer equivalent problems in the Love’s equivalence principle.

same procedure. Since (2.28) does not include ~Jo(~r) and ~Mo(~r), it must be that the radiation

from ~Jo(~r) and ~Mo(~r) is captured by the equivalent sources on S. Also note that in (2.28)

if the observation point falls outside Vi, then the right-hand side equals to zero. In other

words, for such cases, the radiation from the equivalent currents cancels out the radiation from

~Ji(~r) and ~Mi(~r), producing null fields in Vo. This is the extinction theorem [91]. Therefore,

the formulation described by (2.28) is equivalent to the actual problem only for ~r in Vi, but

produces fictitious fields for ~r ∈ Vo.To compute fields inside Vo, we can again start from the vector wave equation and dyadic

Green’s function for Vo and carry out an analogous procedure. The final result is

−˛S∇×G(~r, ~r ′) ·

[−n′o × ~E(~r ′)

]dS′ − jωµ

˛SG(~r, ~r ′) ·

[n′o × ~H(~r ′)

]dS′

−ˆVo∇×Go(~r, ~r ′) · ~Mo(~r

′)dV ′ − jωµˆVoGo(~r, ~r

′) · ~Jo(~r ′)dV ′ =

~E(~r) ~r ∈ Vo0 ~r ∈ Vi

(2.30)

with equivalent current densities given by

~Jeq,o(~r) = no × ~Ho(~r) (2.31a)

~Meq,o(~r) = −no × ~Eo(~r) . (2.31b)

Radiation due to ~Jeq,o(~r) and ~Meq,o(~r) can be computed using the homogeneous Green’s func-

tion of Vo.The interior equivalent problem modeled by (2.28) and the exterior equivalent problem

modeled by (2.30) are shown in Fig. 2.2. While the expressions in (2.28) and (2.30) allow us to

evaluate the electric field anywhere in space, the irony is that these expressions depend on the

actual values of the electric and magnetic fields on S, which are not readily available. Therefore,

we first need to solve for these fields using, for example, the method of moments, before we can


~Mi, ~Ji

Vi

Vo

~Jeq,i(~r)

~Meq,i(~r)

~E′, ~H ′

~M ′o,~J ′o

ni no

(a) Inner Equivalence

~M ′i ,~J ′i

Vi

Vo

~Jeq,o(~r)

~Meq,o(~r)

~E′, ~H ′

~Mo, ~Jo ni no

(b) Outer Equivalence

Figure 2.3: Schelkunoff equivalence principle

evaluate fields anywhere in space.

2.2.2 Schelkunoff Equivalence Principle

Figure 2.2a shows an equivalent setup that produces correct field distribution inside Vi. There

are in fact a countless number of such equivalent setups. Figure 2.3a shows the Schelkunoff

equivalence principle for Vi. In this setup, the electric and magnetic fields inside Vi remain

unchanged and are given by ~E(~r) and ~H(~r), respectively. However, fields in Vo take fictitious

values ~E′(~r) and ~H ′(~r). It is important to realize that the fictitious values of the electric and

magnetic fields inside Vo is a consequence of either a change in material properties or a change

of sources inside Vo. For the aforementioned fields to exist inside Vi and Vo, equivalent electric

and magnetic current densities

~Jeq,i(~r) = ni ×[~H(~r)− ~H ′(~r)

], (2.32)

~Meq,i(~r) = −ni ×[~E(~r)− ~E′(~r)

](2.33)

are needed on S to satisfy the boundary conditions. Therefore, the total radiated field inside

Vi is due to ~Jeq,i(~r), ~Meq,i(~r), ~Ji(~r), and ~Mi(~r).

While the Schelkunoff equivalence principle is interesting from a theoretical viewpoint, it

does have a drawback. By replacing the material and fields outside Vo, we have made the

equivalent problem inhomogeneous, and therefore we cannot use the homogeneous Green’s

function to compute the radiated fields from ~Jeq,i(~r) and ~Meq,i(~r). For this reason, most of the

integral equation formulations are based on the Love’s equivalence principle which only requires

the homogeneous Green’s function of inner and outer media. Love’s equivalence principle can

be seen as a special case of the Schelkunoff equivalence principle, where we introduced null

fields outside Vo. Since null fields are a solution to Maxwell’s equations for any set of material

parameters, we can replace the material of Vo with the material of Vi, thus homogenizing the


S

PEC

n

~Escat~Einc

~J(~r)

Figure 2.4: Scattering from a PEC scatterer

medium. This allows us to use the Green’s function of a homogeneous medium with material

properties of Vi to evaluate fields due to ~Jeq,i and ~Meq,i. The Schelkunoff equivalence principle

for the exterior problem can be obtained similarly, as shown in Fig. 2.3b.

2.3 Scattering from a PEC Object

In Secs. 2.1–2.2, we discussed how to evaluate the electromagnetic fields due to electric and

magnetic current distributions. However, in most problems, the electric and magnetic current

distributions are not known and need to be solved for with, for example, the method of moments.

In this section, we review how the method of moments can be applied to compute scattering

from a PEC object. In Sec. 2.4-2.5, we will discuss how the same procedure may be generalized

to dielectric scatterers and composite scatterers.

2.3.1 The Method of Moments

For illustration purposes, let us consider the setup shown in Fig. 2.4. This setup contains a

PEC scatterer excited by an incident electromagnetic fields ~Einc(~r) and ~H inc(~r). We assume

here that the background medium is free space.

When ~Einc(~r) impinges on the PEC surface, it induces an electric current density ~J(~r) on

the surface of the PEC, which in turn radiates back in free space. According to the boundary

conditions, the tangential electric field on the PEC surface is zero, that is

n× ~E(~r) = n×[~Einc(~r) + ~Escat(~r)

]= 0 for ~r ∈ S , (2.34)

where ~Escat(~r) is the scattered field from the current density induced on the PEC. The scattered

field can be written using the ~L operator as in (2.13) to obtain the integral equation

−jωµ0n×[~L ~J(~r ′)

]= −n× ~Einc(~r) , (2.35)

which is valid when ~r ∈ S. Equation (2.35) may also be viewed as an application of the


Love’s equivalence theorem in which we remove the PEC scatterer and model its effect through

equivalent electric and magnetic current density on S. However, since the tangential electric

field on S is zero, the magnetic current density is equal to zero and the total radiated scattered

field is due to only the (equivalent) electric current density.

In (2.35), the current density is not known3. In order to solve it numerically, we must first

discretize it using a set of basis functions as

~J(~r) =

N∑n=1

jn ~fn(~r) , (2.36)

where ~f1(~r), . . . , ~fN (~r) are the basis functions and j1, . . . , jN are the weights for each basis

function. Since the predicted current distribution spans the set of basis functions, the choice

of basis function is crucial for accuracy and robustness. Basis functions can be classified as

either local basis functions or global (or entire-domain) basis functions. As the name implies,

local basis functions are non-zero over a small region of the computational domain, while global

(or entire-domain) basis functions are non-zero over the entire domain. Global basis functions

are typically used for problems where the behavior of the current is known approximately. For

example, sinusoidal basis functions may be used to model currents on a thin dipole. For a

majority of the problems, however, the currents are expanded with local basis functions, which

are simpler to generate and use. We will discuss all the local basis functions used in this thesis

in Sec. 2.3.2.

The weights jn of the current distribution are now the unknowns of the problem. Since

there are N basis functions (and unknowns), we need to introduce N equations to solve the

problem. We can substitute the expansion (2.36) back into (2.35) to obtain

−jωµ0

N∑n=1

jnn×[~L~fn(~r ′)

]= n× ~Einc(~r) (2.37)

for ~r ∈ S. In order to discretize (2.37), we test the equation with N different testing basis

functions, which are denoted by ~g1(~r), . . . , ~gN (~r). Testing procedure involves taking a dot

product of gm(~r) and (2.37) for m = 1, . . . , N and integrating over the support of the testing

function. We can then write N linear equations, one from each testing function, that can be

written in matrix form as

−jωµ0

⟨g1(~r), n×

[~L~f1(~r ′)

]⟩. . .

⟨g1(~r), n×

[~L~fN (~r ′)

]⟩...

......⟨

gN (~r), n×[~L~f1(~r ′)

]⟩. . .

⟨gN (~r), n×

[~L~fN (~r ′)

]⟩j1...

jN

3Since the unknowns are inside the integral operator, the equation is termed as the “integral equation.”


=

⟨~g1(~r), n× ~Einc(~r)

⟩...⟨

~gN (~r), n× ~Einc(~r)⟩ , (2.38)

where the reaction inner product of two functions is defined to be [91]⟨~gm(~r), ~fn(~r)

⟩=

ˆ~gm(~r) · ~fn(~r)dV , (2.39)

where the integration is performed over the region of support where both ~gm(~r) and ~fn(~r) are

non-zero. Equation (2.38) is a linear system that may be solved with direct methods such as

LU decomposition or with iterative methods such as conjugate gradient descent or GMRES.

The choice of testing functions is also important to obtain a well-conditioned linear system.

Commonly, the testing functions are chosen to be the same as the basis functions used to

expand the unknown current distribution. This testing procedure is termed as the Galerkin

testing method. After solving for vector[j1 . . . jN

]Tin (2.38), we can compute the scattered

field everywhere outside S using the ~L operator.

2.3.2 Basis Functions

There are many possible basis functions that can be used in the method of moments procedure.

In this thesis, we will encounter, (2-D) pulse basis functions, panel basis functions, RWG basis

functions, and dual RWG basis functions. Properties of these basis functions are summarized

below.

Pulse Basis Functions

A common choice of local basis function for 2-D problems is the pulse basis function Π(~r).

Pulse basis functions are defined by partitioning a contour into several smaller segments. The

n-th pulse basis function is defined to be

Πn(~r) =

1 ~r on n-th segment

0 Otherwise. (2.40)

Panel Basis Functions

Panel basis functions are very similar to the pulse basis functions, except they are for 3-D

problems and are defined over a surface. The surface is first discretized with polygons and the

n-th panel basis function is defined to be

Πn(~r) =

1 ~r on n-th polygon

0 Otherwise. (2.41)


Both the pulse and the panel basis functions have low cost when assembling the method of

moments system. However, one typically requires a lot of panel basis functions to model smooth

field distributions.

RWG Basis Functions

Rao-Wilton-Glisson (RWG) basis functions are vectorial basis functions that are commonly

used in the 3-D surface integral equation method [68]. RWG basis functions are defined over a

triangular mesh of a surface. The n-th RWG basis function is defined over a pair of connected

triangles, T+n and T−n as

~Λn(~r) =

ln

2A+n

(~r − ~Q+

n

)if ~r is on T+

n

− ln2A−n

(~r − ~Q−n

)if ~r is on T−n

0 otherwise

, (2.42)

where ln is the length of the edge joining the two triangles, A+n and A−n are, respectively, the

areas of T+n and T−n , and ~Q+

n and ~Q−n are free vertices on T+n and T−n , respectively. Figure 2.5

shows the vectorial distribution of the RWG basis function over its supporting triangles.

RWG basis functions possess useful properties which make them ideal basis functions to

model current distributions in the method of moments. First, the divergence of the RWG basis

function is constant on each of the two triangles and is equal to [68]

∇ · ~Λn(~r) =

lnA+

nif ~r is on T+

n

− lnA−n

if ~r is on T−n

0 otherwise

. (2.43)

RWG basis functions are typically used to expand the surface current density. Since we know

that charge density is proportional to the divergence of current density, we can readily compute

the charge distribution on the surface using (2.43). Furthermore, if we integrate ∇ · ~Λn(~r)

over the region of support of the basis function, then the result is zero. Hence, the basis

function has a net charge of zero. This property is important because it automatically enforces

conservation of charge. The class of basis functions that possess this property are known

as divergence-conforming basis functions. As seen from Fig. 2.5, RWG basis function does

not enforce continuity of current density. The normal component of the current density is

continuous on the shared edge, but the tangential components are not continuous. Lack of

continuity, however, does not affect most simulations as the issue can be circumvented by mesh

refinement.


T+n T−n~Q+

n~Q−n

Figure 2.5: An RWG basis function is defined on a pair of connected triangles.

Dual RWG Basis Functions

Expanding both the electric and magnetic current densities with RWG basis functions in a

MoM-based integral equation formulation leads to some integral operators to be poorly-tested,

requiring special treatment [92] as discussed later in this thesis. Since the tangential electric

and magnetic fields are typically perpendicular, a robust MoM formulation requires that the

tangential magnetic field is expanded with RWG basis functions, but the tangential electric

field is expanded with different basis functions that are orthogonal to RWG basis functions, or

vice versa. One possible choice for orthogonal basis functions is n× ~Λn(~r). However, this basis

function is not divergence-conforming and requires special treatment to enforce charge neutral-

ity [68]. The so-called dual RWG basis function is another basis function that is approximately

orthogonal to an RWG basis function, and it is divergence-conforming [93]. A vectorial plot for

the dual RWG basis function is shown in Fig. 2.6c. The region of support of its corresponding

RWG basis function is highlighted in Fig. 2.6a. The RWG basis function over the triangle pair

(5, 6) in Fig. 2.6a is directed in the vertical direction. In contrast, the dual RWG basis function

is approximately perpendicular and is directed horizontally as shown in Fig. 2.6c.

The dual RWG basis functions are defined on the same triangular mesh as the RWG basis

functions using the following procedure:

1. Each triangle (T ) in the original mesh is divided into 6 subtriangles (T ′). The three

vertices of each T ′ are: the centroid of T , the midpoint of one of the edges of T , and one

of the vertices of T . Fig. 2.6a shows a sample triangular grid composed of 10 triangles.

Each of the 10 triangles is subdivided into 6 subtriangles as shown in Fig. 2.6b. Dual

RWG functions are defined on this dual grid.

2. The n-th dual RWG basis function spans over two polygons, P+ and P−. P+ is composed

of all T ′ surrounding the first vertex of the n-th RWG edge. Similarly, P− is composed of

all T ′ surrounding the second vertex of the n-th RWG edge. Fig. 2.6a shows T+ (in blue)

and T− (in red) for one of the RWG basis function. As shown in Fig. 2.6b, its dual RWG

basis function is created by two polygons: P+ (in blue) and P− (in red). These polygons


(a) Regular grid (b) Subgrid

(c) Dual RWG basis function

Figure 2.6: (a): Sample triangular grid for RWG basis functions with the region of support of one ofthe RWG basis functions highlighted. Label at the centroid of each triangle indicates a unique triangleidentification number.(b): Dual triangular grid with the region of support of one of the dual RWG basisfunctions highlighted. (c): Vectorial distribution of the dual RWG basis function.

are composed of all subtriangles surrounding the two common vertices of the RWG basis

function.

3. Next, we introduce an RWG basis function for each interior edge of P+ and P−, except

those interior edges that are co-aligned with the original RWG basis functions. We also

introduce two interior edges at the intersection of P+ and P−. In the sample basis function

shown in Fig 2.6b, there are 24 interior RWG edges. An RWG edge is not introduced

between triangles pairs (25, 31) and (26, 32) because these edges co-align with the original

RWG basis function. The n-th dual RWG basis function is thus given by the superposition

of these newly-introduced RWG basis functions as

~Λ′n(~r) =M∑m=1

cn,m~Λn,m(~r) , (2.44)


where ~Λn,m(~r) is the m-th RWG basis function and M is the total number of interior

RWG basis functions.

4. The coefficients cn,m in (2.44) are chosen such that the divergence of the dual RWG basis

function is

∇ · ~Λ′n(~r) =

lPA+ if ~r ∈ P+

− lPA− if ~r ∈ P−

, (2.45)

where lP is the length of intersection of P+ and P−. It is evident from (2.45) that the

dual RWG basis functions are divergence-conforming because integrating the divergence

of ~Λ′n(~r) over its supporting polygons gives zero. A different set of values for cn,m can lead

to Buffa-Christiansen basis functions, which are used for Calderon preconditioners [53].

Despite the advantages of dual RWG basis functions, the usage of the dual RWG basis func-

tions in MoM formulations is limited due to their computational complexity, which is approx-

imately six times higher than the RWG basis functions when assembling MoM linear system.

Furthermore, it is difficult to model junctions with these basis functions. More information on

the dual RWG basis functions can be obtained in the literature [93].

2.4 Scattering from Penetrable Objects

2.4.1 Integral Equations

In Sec. 2.3, we reviewed the MoM applied to compute scattering from a PEC object. Let

us now generalize the procedure to compute scattering from a penetrable object, such as a

dielectric. For illustration purposes, let us consider the setup shown in Fig. 2.7. It consists of a

dielectric object (denoted by Vi) with permittivity ε embedded in free space (denoted by Vo).The dielectric object is bounded by surface S. There also exists a current source in Vo, which

produces excitation fields ~Einc(~r) and ~H inc(~r). We will solve this problem with the PMCHWT

formulation [94]. In the PMCHWT formulation, we invoke the Love’s equivalence principle

and decompose the problem into the exterior equivalent problem and the interior equivalent

problem, as shown in Fig. 2.7.

For the exterior equivalent problem shown in Fig. 2.7b, recall that from the equivalence

theorem (2.28), the total electric field ~Eo(~r) in Vo is due to the excitation field ~Einc(~r) from the

sources within Vo and the scattered field ~Escat(~r) due to equivalent currents on S. Mathemati-

cally, for both the electric and magnetic fields, this is given by

~Eo(~r) = ~Einc(~r) + ~Escat(~r) (2.46a)

~Ho(~r) = ~H inc(~r) + ~Hscat(~r) (2.46b)


S

ε, µ0

ε0, µ0

noni

~Escat~Einc

ViVo

(a) Dielectric scatterer

(0, 0)ε0, µ0

~Mo(~r)

~Escat~Einc

~Jo(~r)

(b) Exterior equivalence

ε, µ0

(0, 0)

~Mi(~r)

~Ji(~r)

(c) Interior equivalence

Figure 2.7: Scattering from a dielectric scatterer

for ~r ∈ Vo. The scattered field is, as given in (2.28),

~Escat(~r) = −jωµ0

[~Lo ~Jo(~r ′)

](~r)−

[~Ko ~Mo(~r

′)]

(~r) (2.47a)

~Hscat(~r) = −jωε0

[~Lo ~Mo(~r

′)]

(~r) +[~Ko ~Jo(~r ′)

](~r) , (2.47b)

where ~Lo and ~Ko operators are as defined in (2.15a)-(2.15b), but with the homogeneous Green’s

function of the outer medium. The equivalent electric and magnetic currents introduced on Sto model the effects of Vi are given by

~Jo(~r) = no × ~Ho(~r) (2.48a)

~Mo(~r) = −no × ~Eo(~r) . (2.48b)

By substituting the expressions for the scattered fields in (2.47a)–(2.47b) into expressions for

the total fields in (2.46a)–(2.46b), we obtain

~Eo(~r) = ~Einc(~r)− jωµ0

[~Lo ~Jo(~r ′)

](~r)−

[~Ko ~Mo(~r

′)]

(~r) (2.49a)

~Ho(~r) = ~H inc(~r)− jωε0

[~Lo ~Mo(~r

′)]

(~r) +[~Ko ~Jo(~r ′)

](~r) , (2.49b)

which are commonly referred to as the electric field integral equation (EFIE) and the magnetic

field integral equation (MFIE), respectively. In (2.49a)–(2.49b), field quantities ~Jo(~r) and ~Mo(~r)


are unknowns that need to be solved with the MoM. Equations (2.49a)–(2.49b) contain normal

and tangential components of electric and magnetic fields. However, from the equivalence

theorem, we know that tangential components of fields are sufficient to fully solve the problem.

Therefore, we filter out tangential components by left-multiplying (2.49a)–(2.49b) by no× no×to obtain

−no × ~Mo(~r) = no × no × ~Einc(~r)− jωµ0no × no ×[~Lo ~Jo(~r ′)

](~r)

− no × no ×[~Ko ~Mo(~r

′)]

(~r) (2.50a)

no × ~Jo(~r) = no × no × ~H inc(~r)− jωε0no × no ×[~Lo ~Mo(~r

′)]

(~r)

+ no × no ×[~Ko ~Jo(~r ′)

](~r) , (2.50b)

where we have written the tangential fields in terms of the currents using (2.48a)–(2.48b).

Equations (2.50a)–(2.50b) are refered to as the tangential EFIE (T-EFIE) and the tangential

MFIE (T-MFIE), respectively, in the literature.

For the interior equivalent problem shown in Fig. 2.7c, the electric field ~Ei(~r) inside Vi is

due to equivalent sources. The T-EFIE and T-MFIE for the interior equivalent problem are

−ni × ~Mi(~r) = −jωµ0ni × ni ×[~Li ~Ji(~r ′)

](~r)− ni × ni ×

[~Ki ~Mi(~r

′)]

(~r) , (2.51a)

ni × ~Ji(~r) = −jωεni × ni ×[~Li ~Mi(~r

′)]

(~r) + ni × ni ×[~Ki ~Ji(~r ′)

](~r) , (2.51b)

where ~Li and ~Ki operators are as defined in (2.15a)-(2.15b), but with the homogeneous Green’s

function of the inner medium. Furthermore, the equivalent electric and magnetic currents

radiating inside Vi are

~Ji(~r) = ni × ~Hi(~r) (2.52a)

~Mi(~r) = −ni × ~Ei(~r) . (2.52b)

2.4.2 Discretization of Equivalent Current Densities

In order to evaluate electromagnetic fields everywhere, we need to solve for the four unknowns:

~Ji(~r), ~Jo(~r), ~Mi(~r), and ~Mo(~r). We first discretize S with triangular mesh elements. Edge

lengths of the triangles are set to be between λ/8 – λ/12 in order to accurately resolve current

distributions. Next, let us expand all the equivalent current densities with RWG basis functions

as

~Ji(~r) =

N∑n=1

ji,n~Λn(~r) (2.53a)

~Jo(~r) =N∑n=1

jo,n~Λn(~r) (2.53b)


~Mi(~r) =

N∑n=1

mi,n~Λn(~r) (2.53c)

~Mo(~r) =

N∑n=1

mo,n~Λn(~r) , (2.53d)

and collect the expansion coefficients into vectors Ji =[ji,1, . . . , ji,N

]T, Jo =

[jo,1, . . . , jo,N

]T,

Mi =[mi,1, . . . ,mi,N

]Tand Mo =

[mo,1, . . . ,mo,N

]T.

2.4.3 Discretization of Integral Equations

Now, let us discretize integral equations for the exterior problem (2.50a)–(2.50b) and the interior

problem (2.51a)–(2.51b). For the exterior problem, we substitute (2.53b) and (2.53d) into the

T-EFIE (2.50a), and test the resulting equation with RWG basis functions ~Λm(~r) to obtain

−⟨~Λm(~r), no × no × ~Einc(~r)

⟩=

N∑n=1

jo,n

⟨~Λm(~r),−jωµono × no ×

[~Lo~Λn(~r ′)

](~r)⟩

+N∑n=1

mo,n

⟨~Λm(~r),−no × no ×

[~Ko~Λn(~r ′)

](~r)⟩

+N∑n=1

mo,n

⟨~Λm(~r), no × ~Λn(~r)

⟩, (2.54)

for m = 1, . . . , N . Similarly, by testing T-MFIE (2.50b) for the outer problem with ~Λm(~r) we

obtain

−⟨~Λm(~r), no × no × ~H inc(~r)

⟩=

N∑n=1

mo,n

⟨~Λm(~r),−jωεono × no ×

[~Lo~Λn(~r ′)

](~r)⟩

+N∑n=1

jo,n

⟨~Λm(~r), no × no ×

[~Ko~Λn(~r ′)

](~r)⟩

−N∑n=1

jo,n

⟨~Λm(~r), no × ~Λn(~r)

⟩(2.55)

for m = 1, . . . , N . Integrals to compute the inner products in (2.54)–(2.55) are evaluated with

a mixture of numerical integration using the Gaussian quadratures and analytic integration to

extract singularities. Details on integration can be found in Appendix A. Equations (2.54)–

(2.55) may now be assembled into matrix form as[LEo KE

o

KHo LHo

][Jo

Mo

]=

[VE

VH

], (2.56)


where superscripts E and H are used to indicate that the matrices stem from the T-EFIE and

T-MFIE, respectively. The (m,n)-th entry of LEo , KEo , KH

o , and LHo is

[LEo](m,n)

=⟨~Λm(~r),−jωµono × no ×

[~Lo~Λn(~r ′)

](~r)⟩

(2.57a)[KEo

](m,n)

=⟨~Λm(~r),−no × no ×

[~Ko~Λn(~r ′)

](~r)⟩

+⟨~Λm(~r), no × ~Λn(~r)

⟩, (2.57b)[

LHo](m,n)

=⟨~Λm(~r),−jωεono × no ×

[~Lo~Λn(~r ′)

](~r)⟩

(2.57c)[KHo

](m,n)

=⟨~Λm(~r), no × no ×

[~Ko~Λn(~r ′)

](~r)⟩−⟨~Λm(~r), no × ~Λn(~r)

⟩, (2.57d)

and the m-th entry of VE and VH is

[VE]m

= −⟨~Λm(~r), no × no × ~Einc(~r)

⟩(2.58a)[

VH]m

= −⟨~Λm(~r), no × no × ~H inc(~r)

⟩. (2.58b)

The matrices in (2.56) have several important properties:

• Matrices LEo and LHo are well-tested. When applied to ~fn(~r ′), the ~L operator produces

fields that are dominant in the direction of ~fn(~r ′). Since both LEo and LHo are gener-

ated with identical source and testing functions, both matrices end up being diagonally

dominant4. Matrices LEo and LHo , however, have two potential issues. First, they suffer

from the so-called low-frequency breakdown due to which its condition number increases

as the frequency approaches zero. Second, they suffer from dense-mesh breakdown due to

which its condition number increases with smaller mesh size. Low-frequency breakdown

problems can be alleviated using loop-star decomposition [95] or by introducing charge

density into the integral equation formulation [96].

• Matrices KEo and KH

o are poorly tested. The (m,n)-th term of KEo matrix is⟨

~Λm(~r), no × ~Λn(~r)⟩

+⟨~Λm(~r),−no × no ×

[~K~Λn(~r ′)

](~r)⟩

(2.59)

= −⟨~Λm(~r), no × no × p.v.

[∇×

ˆVG(~r, ~r ′)~Λm(~r ′)dV ′

]⟩+

1

2

⟨~Λm(~r), no × ~Λn(~r)

⟩where we have substituted in (2.17). The first term on the right-hand side is evaluated in

the principal value sense. Therefore, the first term is zero when the source and observation

triangles are the same. Hence, the matrix associated with this term has zeros along its

main diagonal. The matrix may still be full-rank and well-conditioned, depending on the

wavenumber. The second term on the right-hand side tests the RWG basis functions with

no× RWG basis functions. This operation results in a singular matrix whose rank is equal

to the number of triangles on the surface, instead of the number of edges on the surface.

Since in most cases the second term is dominant over the first term, the overall condition

4The contribution of ~L is the strongest when the test and source basis functions are closest to each other.


number of the KHo and KE

o is very high and the ~K operator is not well-tested [92]. One way

to improve the condition number of KEo and KH

o is using the dual RWG basis functions

~Λ′n(~r) to expand the magnetic current densities. This will result in the second term on

the right-hand side to be full-rank and well-conditioned, resulting in KEo and KH

o to be

well-tested and well-conditioned.

The linear system of equations in (2.56) can be solved with LU factorization. However,

it does not give accurate solution because T-EFIE and T-MFIE are linearly dependent5. To

properly model the physics, we need to capture the effects of material inside S. For this, we

discretize the T-EFIE and T-MFIE for the interior equivalent problem in a similar manner.

By casting all four equations (two for the exterior equivalent problem and two for the interior

equivalent problem) as a matrix, we obtainLEo KE

o 0 0

KHo LHo 0 0

0 0 LEi KEi

0 0 KHi LHi

Jo

Mo

Ji

Mi

=

VE

VH

0

0

, (2.60)

where LEi , LHi , KHi , and KE

i are similar to LEo , LHo , KHo , and KE

o , except they are generated

with the homogeneous Green’s function of interior medium.

2.4.4 Enforcement of Boundary Conditions

We have not yet enforced the boundary conditions for the electromagnetic fields on the interface

of two regions. Since there are no additional sources on S, the tangential boundary conditions

on the interface are

no × ~Eo(~r) = −ni × ~Ei(~r) (2.61a)

no × ~Ho(~r) = −ni × ~Hi(~r) , (2.61b)

which translate to Ji = −Jo and Mi = −Mo.

2.4.5 Elimination of Additional Equations

After we enforce the boundary conditions in (2.61), we can eliminate two unknowns. So, we

now need to solve for two sets of unknowns, let us say Jo and Mo, and we have four equations

in (2.60). In the PMCHWT formulation, the additional equations are eliminated by adding the

T-EFIE and T-MFIE for the outer problem to the T-EFIE and T-MFIE for the inner problem

5Due to the choice of basis functions, testing procedure, and numerical errors, the discretized T-EFIE andT-MFIE are linearly independent.


to get the final system of equations [21][LEo + LEi KE

o + KEi

KHo + KH

i LHo + LHi

][Jo

Mo

]=

[VE

VH

]. (2.62)

Equation (2.62) can be solved with LU factorization or with iterative solvers.

If the penetrable object is a lossy conductor with high conductivity, then (2.62) does not

provide good results because of strong contrast in the wavenumber of the background medium

and the conductive medium. For such cases, a different elimination procedure is typically

employed for accurate results. For example, the generalized impedance boundary condition

(GIBC) method [25] discards T-MFIE for the exterior medium and T-EFIE for the interior

medium to obtain [LEo KE

o

KHi LHi

][Jo

Mo

]=

[VE

0

]. (2.63)

2.4.6 Numerical Validation

The MoM formulation discussed in this section was implemented in C++. Here, we validate

the C++ implementation of the PMCHWT formulation against other simulation tools. Later

in the thesis, the in-house solver will be used as a reference to evaluate the algorithms that will

be proposed.

A Dielectric Sphere

For the first validation example, we consider a dielectric sphere of radius R = 0.5 m with

εr = 2.2. Figure 2.8a shows the monostatic radar cross-section of this sphere over the frequency

range of 70 MHz–500 MHz. The RCS is computed with three solvers: our in-house PMCHWT

solver, FEKO [9], and Scuff-EM [97]. The plot also shows analytic solution for the dielectric

sphere [90,98,99]. As evident from the plots, the in-house solver produces accurate results over

a wide frequency range.

A Dielectric Box

Next, let us consider scattering from a dielectric box of dimension 1 m × 1 m × 0.4 m with

εr = 2.2. The dielectric box is excited by a plane wave traveling in the −z direction. The radar

cross-section obtained with the in-house PMCHWT solver, FEKO, and Scuff-EM is shown in

Fig. 2.8b. The results from the in-house PMCHWT solver match results obtained from FEKO,

Scuff-EM, and analytic formulas.


100 150 200 250 300 350 400 450 5000

0.5

1

1.5

2

2.5

Frequency [MHz]

RC

S [

m2]

Analytic

FEKO

Scuff−EM

MoM

(a) RCS of a sphere

100 150 200 250 300 350 400 450 5000

1

2

3

4

5

6

Frequency [MHz]

RC

S [

m2]

FEKO

Scuff−EM

MoM

(b) RCS of a box

Figure 2.8: Monostatic radar cross-section (RCS) of the dielectric sphere and cube considered inSec. 2.4.6.

V1

V2

V3

V0

S12 S2

2 S32

S11 S2

1 S31

S13 S2

3 S33

S12 S2

2 S32

S41

S43

S42 S5

2S20

S10

S30

S40

Figure 2.9: Sample geometry of a composite scatterer composed of 3 dielectric regions V1, V2, and V3.In the figure, S23 = S22 and S21 = S22 are PEC subsurfaces.

2.5 Scattering from Composite Objects

2.5.1 Integral Equations

So far we have discussed the MoM to compute scattering from a PEC or a dielectric object.

Let us now generalize the procedure to simulate structures composed of both dielectrics and

PECs [94]. This generalization will allow us to simulate structures like patch antennas. A

sample composite scatterer is shown in Fig. 2.9.

Suppose that the scatterer under consideration is composed of M regions, where the m-th

region Vm has material properties µ0 and εm. For generality, let us suppose that the background

free space medium is the 0-th region. Let us denote the surface bounding the m-th region by

Sm. Surface Sm may be a union of P smaller subsurfaces which are denoted with notation Spmfor p = 1, . . . , P . Each subsurface may be PEC or non-PEC. It should be apparent from the

notation that the interface between two regions m1 and m2 is discretized with two subsurfaces

Sp1m1 and Sp2

m2 for some values for p1 and p2. We will assume that triangular meshes of Sp1m1 and

Sp2m2 are identical. The scatterer is excited by electric field ~Einc(~r) and magnetic field ~H inc(~r)


due to some sources in V0.

In order to solve this problem, we invoke the Love’s equivalence principle and model fields

in each region through equivalent current densities that are introduced on the surface bounding

the region. In the m-th region, the electromagnetic fields can be written by T-EFIE (2.50a)

and T-MFIE (2.50b) as

−nm × ~Mm(~r) =nm × nm × ~Einc(~r)− jωµ0nm × nm ×[~Lm ~Jm(~r ′)

](~r)

− nm × nm ×[~Km ~Mm(~r ′)

](~r) (2.64a)

nm × ~Jm(~r) =nm × nm × ~H inc(~r)− jωεmnm × nm ×[~Lm ~Mm(~r ′)

](~r)

+ nm × nm ×[~Km ~Jm(~r ′)

](~r) , (2.64b)

where nm is the unit normal vector pointing into the m-th region, and ~Lm and ~Km are evaluated

with the homogeneous Green’s function of the m-th region. The incident fields in (2.64a)–

(2.64b) are non-zero only for m = 0. The equivalent electric and magnetic current densities

are

~Jm(~r) = nm × ~Hm(~r) (2.65)

~Mm(~r) = −nm × ~Em(~r) . (2.66)

If some of the subsurfaces of Sm are PECs, then the magnetic current on those surfaces will be

zero. Furthermore, the T-MFIE is not applicable on those surfaces. These boundary conditions

can be easily enforced through an incidence matrix as discussed in Sec. 2.5.3.

2.5.2 Discretization

In order to apply the MoM, we discretize the equivalent electric and magnetic current densities

on Sm with RWG basis functions as

~Jm(~r) =

Nm∑n=1

jm,n~Λn(~r) (2.67a)

~Mm(~r) =

Nm∑n=1

mm,n~Λn(~r) , (2.67b)

for the m = 0, . . . ,M . The expansion coefficients of the electric and magnetic current densities

for m-th region are collected into vectors Jm and Mm, respectively. Next, we discretize the

T-EFIE (2.64a) and T-MFIE (2.64b) by substituting (2.67a)–(2.67b) into (2.64b)–(2.64a) and

testing the result with ~Λn(~r). The discretized equations can be compactly written for all M


mo = 0

mi = 0

(a) PEC

mo = −mi, jo = −ji

ji,mi

(b) Dielectric interface

jo

ji

(c) PEC interface

jo = −ji

ji jk = ji

(d) Dielectric junction

jo = −ji

ji

(e) PEC-dielectric junc-tion

Figure 2.10: Boundary conditions discussed in Sec. 2.5.3. In the diagrams, jo, ji, and jk denote electriccurrent density coefficients, while mo and mi denote magnetic current density coefficients. Arrows areused to denote the direction of the RWG basis functions.

regions as

[LE0 KE

0

KH0 LH0

]0 . . . 0

0

[LE1 KE

1

KH1 LH1

]. . . 0

0 0. . . 0

0 0 . . .

[LEM KE

M

KHM LHM

]

︸︷︷︸

A

[J0

M0

][

J1

M1

]...[

JM

MM

]

︸︷︷︸

x

=

[VE

0

VH0

][

0

0

]...[0

0

]

︸︷︷︸

b

(2.68)

where the notation for matrices is similar to that in (2.60). For simplicity, we will just use

A to denote the system matrix, x to denote the list of current coefficients, and b to denote

the right-hand side in (2.68). From an implementation viewpoint, it is important to store A

in (2.68) in a sparse format such as compressed sparse column (CSC) to avoid storing all the

zeros [21]. Alternatively, since the system matrix is block diagonal, we can store M + 1 blocks

in a dense format. The dimension of A is N ×N , where N =∑M

m=0 2Nm. Vectors x and b are

of size N × 1.

2.5.3 Boundary Conditions

Now we apply the boundary conditions and eliminate unnecessary unknowns. To efficiently

and easily implement the boundary conditions we can make use of incidence matrices. Suppose

that, following the removal of redundant unknowns, the final set of unknowns is collected into

vector x =[j1 . . . j

Nm1 . . . m

N

]T, where N and N are, respectively, the number of


unique electric and magnetic current coefficients. We can then write the electric and magnetic

current coefficients on all regions in terms of the product of incidence matrix UR and vector x,

i.e.

x = URx. (2.69)

The dimension of UR is N ×(N + N

). The matrix UR is a highly sparse matrix with at most

one non-zero entry per row. Therefore, it is efficient to store UR in a sparse format.

When modeling composite structures, the following boundary conditions must be enforced [94]:

1. PEC surface: If triangle T+i or T−i of the i-th RWG basis function is PEC, then the

magnetic current coefficient associated with this edge is zero, i.e. mi = 0 as shown in

Fig. 2.10a. This boundary condition translates to having a qmi -th row of the UR to be

zero, where qmi is the index of mi in x.

2. Interface of two regions: On an interface between two regions, the tangential electric

and magnetic fields are continuous. In terms of the equivalent currents this means that

the electric and magnetic current basis functions on either side of the interface have the

same magnitude, but opposite directions. This is shown in Fig. 2.10b. Therefore, we only

need one set of electric and magnetic current coefficients as unknowns, and the other set

of coefficients can be eliminated. The electric field continuity translates to having 1 in the

entry (qji , qji ) and −1 in the entry (qjo, q

ji ), where qji and qjo are, respectively, the index of ji

and jo in x and qji is the index of ji in x. Similarly, continuity of the tangential magnetic

field can be enforced by setting 1 in the entry (qmi , qmi ) and −1 in the entry (qmo , q

mi ).

3. PEC surface on the interface of two regions: The electric current density on either

side of a PEC interface may have different values, as shown in Fig. 2.10c. Therefore, two

electric current coefficient unknowns are needed, one for either side of the interface, to

model the PEC. In UR this boundary condition translates to having 1 in entries (qji , qji )

and (qjo, qjo).

4. Dielectric junction: A junction is defined to be an intersection of three or more regions

as shown in Fig 2.10d. Continuity of the tangential electric and magnetic fields translates

to having only one set of unknowns on a dielectric junction. In UR, continuity of the

tangential magnetic field is enforced by setting ±1 in the entry (qj∗, qji ) where ∗ indicates

all edges that form the junction. The sign is chosen to satisfy the boundary conditions,

as shown in Fig. 2.10d. Continuity of the tangential electric fields can be enforced in the

same way.

5. PEC–dielectric junction: For a junction between a PEC surface and a dielectric

surface, half of an RWG basis function lies on the PEC surface and the other half on the

dielectric surface. This is shown in Fig. 2.10e. For this junction, the magnetic current

coefficients are zero. The electric current coefficients in the inner and outer regions have


the same magnitude but opposite orientation in order to satisfy the continuity of the

tangential magnetic field. Therefore, the incidence matrix UR is populated similarly to

the case of interface between two regions (Case 2).

By substituting (2.69) into (2.68), we obtain

AURx = b , (2.70)

which is an overdetermined system of equations. We can eliminate the extra set of equations by

applying the PMCHWT formulation [21]. For the five boundary conditions discussed earlier,

the overdetermined system is reduced as follows [94]:

1. PEC surface: For a PEC boundary, we drop T-MFIE (2.50b) because the magnetic

current is zero.

2. Interface of two regions: The interface between two dielectric regions is treated using

the PMCHWT formulation. That is, we add T-EFIE (2.50a) for the inner region to the

T-EFIE for the outer region, and we add the T-MFIE for the inner region to the T-MFIE

for the outer region.

3. PEC surface on the interface of two regions: Since the electric current basis func-

tions on either side of a PEC interface are independent, we keep the T-EFIE for inner

and outer regions in the final system. T-MFIE for both the inner and the outer regions

are discarded.

4. Dielectric junction: Dielectric junctions are treated similarly to the interface of two

regions (case 2). Therefore, we add up the T-EFIE and T-MFIE of all intersecting regions.

5. PEC–dielectric junction: Since the magnetic current basis function for a PEC–dielectric

junction is zero, we drop the T-MFIE equations for the inner and outer regions. T-EFIE

of the inner and outer regions, on the other hand, are added together as in PMCHWT.

From an implementation viewpoint, extra equations may be eliminated using another incidence

matrix UL, which is just the transpose of UR. Therefore, the final system of equations is

ULAURx = ULb . (2.71)

After solving for x in (2.71), we can obtain the electric and magnetic current coefficients on all

regions via UR.

2.5.4 Numerical Validation

The MoM formulation to model composite structures discussed in this section was implemented

in C++ and validated against a commercial MoM solver. As an example, let us consider the


εr = 2.2

13.5 mm

8.5 mm

Figure 2.11: Patch antenna with ground plane considered in Sec. 2.5.4.

square patch antenna with a ground plane shown in Fig. 2.11. The radar cross-section of this

patch antenna was calculated with the in-house MoM solver and FEKO. For this simulation,

the structure was excited with a plane wave traveling in the −z direction. Figure 2.12 shows a

good agreement between the bi-static radar cross-section of the patch antenna calculated with

the two techniques.

2.6 Chapter Summary

The goal of this chapter was to present background on the surface integral equation method.

We first discussed how to compute electromagnetic fields due electric and magnetic current

sources in a homogeneous medium through the ~L and ~K integral operators. Next, in Sec 2.2,

we discussed how to evaluate the electromagnetic fields due to electric and magnetic currents in

an inhomogeneous medium through the application of the equivalence principle. According to

the Love’s equivalence principle, the electromagnetic fields in each region of an inhomogeneous

medium are obtained by the superposition of electromagnetic fields due to sources in that region

and equivalent electric and magnetic currents introduced on the surface enclosing the region.

Sec. 2.3 presented the MoM to compute scattering from a PEC object. Also, in this section, we

discussed properties of various basis functions used in this thesis. In Sec. 2.4, we discussed how

we can combine the equivalence principle and the MoM to compute scattering from penetrable

objects, like dielectrics. This section introduced the so-called tangential electric and magnetic

field integral equations (T-EFIE/T-MFIE). We also discussed some of the shortcomings of the

~L and ~K operators when using only RWG basis functions to expand the electric and magnetic

currents. The section concluded by presenting the PMCHWT formulation to model dielectric

objects. Numerical implementation of this formulation was validated against external tools.

Finally, in Sec. 2.5, we generalized the surface integral equation method to model composite

structures with dielectrics and PECs.


8.5 9 9.5 10 10.5

800

1000

1200

1400

1600

1800

2000

2200

Frequency [GHz]

RC

S [

mm

2]

φ = 0o, θ = 0

o

FEKO

MoM

(a) θ = 0

8.5 9 9.5 10 10.5

800

1000

1200

1400

1600

Frequency [GHz]

RC

S [

mm

2]

φ = 0o, θ = 30

o

FEKO

MoM

(b) θ = 30

8.5 9 9.5 10 10.5200

300

400

500

600

700

800

Frequency [GHz]

RC

S [

mm

2]

φ = 0o, θ = 60

o

FEKO

MoM

(c) θ = 60

8.5 9 9.5 10 10.50

100

200

300

400

500

Frequency [GHz]

RC

S [

mm

2]

φ = 0o, θ = 90

o

FEKO

MoM

(d) θ = 90

Figure 2.12: Radar cross-section of the patch antenna in Fig. 2.11 over 8.5 GHz–10.5 GHz.

Chapter 3

Transmission Line Modeling with a

2-D Surface Method

3.1 Introduction

Transmission line modeling is important to design systems such as electronic boards [100],

integrated circuits [101], microwave systems [102], and power grids [103]. For many applications,

such as signal integrity analyses, the transmission line models need to be accurate over a broad

frequency range, typically from DC to tens of gigahertz. Transmission line modeling requires

accurate per-unit length (p.u.l.) impedance and admittance parameters, which are computed

by solving the magnetostatic problem and the electrostatic problem, respectively [15].

Of the two sets of p.u.l. parameters, computation of the p.u.l. impedance parameters is very

challenging due to the skin effect inside the conductors which strongly influences the line be-

havior over a broad frequency range. In order to obtain the p.u.l. impedance of a transmission

line, one must capture the electromagnetic fields both inside the conductors (interior problem)

and outside (exterior problem). For the exterior problem, integral equations [42, 104, 105] are

commonly utilized. For the interior problem, which is responsible for capturing skin effect,

both volumetric and surface methods have been used in the past. Volumetric approaches in-

clude the FEM [39,40,106,107], conductor partitioning [38] and the volumetric integral equation

method [43]. Unfortunately, as frequency increases, volumetric methods become computation-

ally inefficient, since a very fine mesh is needed to capture the pronounced skin effect.

Surface methods solve the interior problem with unknowns only on the boundary of the

conductors. This is achieved by describing the electromagnetic behaviour of the conductor

through a surface operator that relates the electric and magnetic fields on the boundary. This

operator is responsible for modeling skin and proximity effects [108]. The surface operator can

be obtained analytically or numerically. In some works [109,110], a surface operator is derived

using the so-called surface impedance boundary conditions [24], under the approximation of

small curvature and a well-developed skin effect [111]. At low frequencies, however, in the ab-

45

Chapter 3. Transmission Line Modeling with a 2-D Surface Method 46

sence of the skin effect, the surface impedance model is inaccurate. While high-order boundary

conditions have been proposed to increase accuracy [112], this approach is mostly suitable for

conductors with smooth boundaries. Numerically, the surface operator can be obtained using

finite differences [113], finite elements [114], or the electric field integral equation [115,116]. To

calculate the surface operator, all these approaches require a volumetric discretization of the

interior problem and the calculation of several kernel matrices whose dimensions depend on

mesh size. When the skin effect inside the conductors is extremely small, the computational

cost and complexity of these methods are very high.

The differential surface admittance approach is another interesting approach to model skin

effect in conductors [29]. Using the eigenfunctions of the Helmholtz equation, the differential

surface admittance operator is obtained analytically, avoiding a discretization of the interior

problem. This key idea leads to a simple and efficient formulation. However, since eigen-

functions can realistically be obtained only for canonical geometries, this approach has been

restricted so far to circular [29, 117], tubular [118], rectangular [29], and triangular conduc-

tors [32]. In this chapter, we show that the differential surface admittance operator [29] can

be generalized to model arbitrarily-shaped conductors and dielectrics via the contour integral

method [119], which is simply the 2-D version of the surface integral equations discussed in

Sec. 2.2.1. Later in this thesis, we will generalize the concept even further to model 3-D con-

ductors and dielectrics.

In the literature, the p.u.l. admittance parameters are commonly computed by solving

the electrostatic problem with the FEM [20] or the integral equation method [120]. In the

past, a method analogous to the differential surface admittance operator has been proposed

to compute the p.u.l. admittance parameters [121]. This method models all dielectrics by an

equivalent charge density that is introduced on their boundary. However, this method also relies

on analytic solution of the Laplace’s equation inside the dielectric. Therefore, it is limited to

dielectrics with canonical shapes for their cross-sections, such as rectangles and circles [121]. In

this chapter, we also generalize this concept to dielectrics of arbitrary shape using the contour

integral equation method for Laplace’s equation.

This chapter is organized as follows. In Sec. 3.2, we define the multiconductor transmis-

sion line problem that we are trying to solve. We will then focus on the computation of the

p.u.l. impedance parameters. For this, in Sec. 3.3, we discuss the key idea of how the differen-

tial surface admittance operator can be obtained through the contour integral method from a

theoretical viewpoint. In Sec. 3.4, we discuss the numerical implementation of the differential

surface admittance operator. Then, in Sec. 3.5, the differential surface admittance operator

is used to compute the p.u.l. impedance of multiconductor transmission lines. In Sec. 3.6, we

show how the same methodology can be adapted to compute the p.u.l. capacitance of multicon-

ductor transmission lines. Finally, in Sec. 3.7, we demonstrate the accuracy, robustness, and

computational efficiency of the proposed method through a comprehensive set of examples.


3.2 Problem Definition

We consider a system composed of P conductors and D dielectrics of arbitrary shape. Conduc-

tors have a conductivity σ, permittivity ε0, and permeability µ. Dielectrics have a permittivity

ε. In general, conductors and dielectrics may be inside a multilayer substrate. For transmis-

sion line modeling, our goal is to calculate the partial p.u.l. impedance1 Z(ω) and the partial

p.u.l. admittance Y(ω) that appear in Telegrapher’s equations, [108]

∂V

∂z= −Z(ω)I , (3.1a)

∂I

∂z= −Y(ω)V . (3.1b)

In (3.1a)–(3.1b), the vector V =[V1 . . . VP

]Tcollects the potential Vp of each conduc-

tor, while I =[I1 . . . IP

]Tcollects the current Ip flowing in each conductor. The par-

tial p.u.l. impedance can be written as Z(ω) = R(ω) + jωL(ω), where R(ω) and L(ω) are

the p.u.l. resistance and inductance, respectively. Similarly, the partial p.u.l. admittance is

Y(ω) = jωC(ω) + G(ω), where C(ω) and G(ω) are the p.u.l. capacitance and conductance,

respectively. In multiconductor transmission lines with good conductors, Z(ω) is obtained by

solving the magnetostatic problem including resistive and inductive effects caused by conduc-

tors. On the other hand, Y(ω) is obtained by solving the electrostatic problem accounting for

capacitive effects due to dielectrics [122].

3.3 Differential Surface Admittance Operator

For simplicity, we discuss the proposed differential surface admittance formulation by consid-

ering a single conductor. The conductor has an arbitrary cross-section, has a simply-connected

contour γ, and is depicted in Fig. 3.1a. The position vector of an arbitrary point on γ is de-

noted by ~r. Finally, the unit vectors normal and tangential to contour γ are denoted by n and

t, respectively, as shown in Fig. 3.1a. The p.u.l. impedance parameters are computed assuming

transverse magnetic (TM) propagation. Under this assumption, each conductor is assumed to

be invariant along the z direction in which the current flows. Furthermore, the magnetic field

inside and outside the conductors is in the transverse direction. The electric field is dominant

in the transverse direction outside the conductor. However, due to the finite conductivity, the

electric field will also have a small longitudinal component inside the conductor.

1partial p.u.l. parameters are defined with respect to reference at infinity. The p.u.l. parameters can becomputed from the partial p.u.l. parameters by setting one of the conductors as the reference.


γ

Top layers

Bottom layers

n′t′

O

nt

~r ′

~r~d

σ, µ, ε

µl, εl

(a) Original configuration

Top layers

Bottom layers

γ

z

y

x

µl, εl

µl, εl

Js(~r)

(b) Equivalent configuration

Figure 3.1: (a): sample geometry of a conductor with arbitrary cross-section inside a stratified medium.(b): equivalent configuration obtained after replacing the conductor with the surrounding medium, andintroducing an equivalent current density Js(~r) on contour γ.

3.3.1 Equivalence Principle

Under the TM assumption [29], the electric field Ez(~r) inside contour γ satisfies the scalar

Helmholtz equation [89]

∇2Ez(~r) + k2Ez(~r) = 0 , (3.2)

where, k =√ωµ (ωε− jσ) is the wavenumber inside the conductor. Assuming a constant

potential across the cross-section of the conductor, the transverse electric field is zero inside

the conductor. Furthermore, the tangential magnetic field Ht(~r) along the contour γ follows

directly from Maxwell’s equations under the TM assumption [89]

Ht(~r) =1

jωµ

[∂Ez(~r)

∂n

], (3.3)

where the derivative is taken along the direction normal to contour γ, as shown in Fig. 3.1a.

We now apply the Schelkunoff equivalence principle discussed in Sec. 2.2.2. We replace the

conductor with the material of the surrounding layer [29], as shown in Fig. 3.1b. While doing

this change of material, we also enforce that the longitudinal electric field on γ is maintained to

be the same as before. From here onwards, we will call this the equivalent configuration. Due

to this modification, the electric field inside the conductor changes to Eeq,z(~r), which satisfies

the Helmholtz equation

∇2Eeq,z(~r) + k2outEeq,z(~r) = 0 , (3.4)

subject to Dirichlet boundary condition

Eeq,z(~r) = Ez(~r) ~r ∈ γ . (3.5)


In (3.4), kout = ω√µlεl is the wavenumber of the layer surrounding the conductor. In this new

configuration, the tangential magnetic field along γ is

Heq,t(~r) =1

jωµl

[∂Eeq,z(~r)

∂n

]. (3.6)

By replacing the conductor with the surrounding medium, we have modified the fields both

inside and outside γ. Hence, to restore the original fields outside γ, we invoke the equivalence

principle [83] that was discussed in Sec. 2.2.2. According to the equivalence principle, we

introduce an equivalent surface current density Js(~r) on γ [29], as shown in Fig. 3.1b. The

equivalent current density directed along z is given by

Js(~r) = Ht(~r)−Heq,t(~r) . (3.7)

It is important to note that an equivalent magnetic current density on γ is not required because

the electric field on γ in the original and equivalent configurations remain the same due to (3.5).

By substituting (3.3) and (3.6) into (3.7), we obtain [29]

Js(~r) =1

jω

[1

µ

∂Ez(~r)

∂n− 1

µl

∂Eeq,z(~r)

∂n

]. (3.8)

Equation (3.8) defines the differential surface admittance operator Ys that relates the longitu-

dinal electric field and equivalent current on γ as

Js(~r) = YsEz(~r) . (3.9)

An explicit expression for Ys can be written in terms of the eigenfunctions of the Helmhotz

equations (3.2) and (3.4), as demonstrated previously [29]. However, this approach is viable

only for canonical conductor shapes, for which eigenfunctions are known analytically [29, 32,

117, 118]. For arbitrary shapes, eigenfunctions can only be computed numerically. Since many

eigenfunctions are needed to accurately model the operator, this approach can be very time

consuming, and is typically avoided. In the next section, we show that the contour integral

method [119] provides an efficient and robust way to numerically compute such an operator for

conductors of arbitrary shapes.

3.3.2 Contour Integral Method

Equating (3.8) and (3.9), we see that in order to derive an explicit expression for the differential

surface admittance operator, we need a relation between electric field Ez and magnetic field Ht

on the boundary γ. With this goal in mind, we reconsider the way we solve (3.2). By applying

the Green’s theorem on (3.2), as discussed in Sec. 2.2.1 for three-dimensional problems, we

obtain the so-called contour integral method [119]. The contour integral method gives the


solution of the Helmholtz equation in terms of the electric field and the magnetic field on γ,

Ez(~r) =j

2

‰γ

[∂G(~r, ~r ′)

∂n′Ez(~r

′)− jωµG(~r, ~r ′)Ht(~r′)

]dr′ (3.10)

where ~r and ~r ′ are both on γ, and n′ is the unit vector normal to the contour at the point ~r ′.

The Green’s function is

G(~r, ~r ′) = C0J0(kd)− jY0(kd) (3.11)

where J0(.) and Y0(.) are the zero-th order Bessel and Neumann functions [123], respectively.

As shown in Fig. 3.1a, the distance between points ~r and ~r ′ is denoted by

~d = ~r ′ − ~r , (3.12)

and d =∣∣∣~d ∣∣∣. The Green’s function in (3.11) satisfies the Helmholtz equation for two-dimensional

problems for all complex values of C0 [119]. Commonly, C0 = 1 is used for 2-D problems. How-

ever, to avoid numerical issues at low frequencies, we had to introduce a non-zero constant

C0 [124]. In Sec. 3.4.5, we will discuss how to choose C0 at low and high frequencies to achieve

high numerical robustness. The contour integral equation (3.10) relates the longitudinal elec-

tric field and tangential magnetic field on the boundary. It can thus be used, after numerical

discretization, to derive an explicit expression for the differential surface admittance opera-

tor (3.9).

3.4 Numerical Formulation

In this section, we demonstrate how to numerically obtain the differential surface admittance

operator with the MoM. This operator will then be used to calculate the p.u.l. impedance of a

transmission line made up of conductors with an arbitrary cross-section.

3.4.1 Discretization of Fields and Currents

We first divide contour γ into N segments, and expand the longitudinal electric field in terms

of pulse basis functions

Ez(~r) =N∑n=1

enΠn(~r) , (3.13)

where Πn(~r) is the n-th pulse basis function as defined in (2.40). We also define ~rn to be the

position vector of the midpoint of the n-th segment. Similarly, we discretize the tangential

magnetic fields Ht(~r) and Heq,t(~r) that appear in (3.7) as

Ht(~r) =

N∑n=1

hnΠn(~r) , (3.14)


and

Heq,t(~r) =

N∑n=1

heq,nΠn(~r) . (3.15)

For simplicity of notation, we cast all expansion coefficients into column vectors

E =[e1 e2 . . . eN

]T, (3.16)

H =[h1 h2 . . . hN

]T, (3.17)

Heq =[heq,1 heq,2 . . . heq,N

]T. (3.18)

Similarly, we discretize the equivalent current Js(~r) along the contour γ using pulse basis

functions as

Js(~r) =

N∑n=1

jnΠn(~r) , (3.19)

and store coefficients jn into the vector

J =[j1 j2 . . . jN

]T. (3.20)

From (3.7), we have the following relation between the coefficients of equivalent current and

magnetic fields

J = H−Heq . (3.21)

Next, we relate H and Heq to the electric field coefficients E via the contour integral equation

(3.10) to obtain the differential surface admittance operator.

3.4.2 Magnetic Field in the Original Configuration

After substituting (3.13) and (3.14) into (3.10), we obtain

N∑m=1

emΠm(~r) =j

2

‰γ

[∂G(~r, ~r ′)

∂n′

N∑n=1

enΠn(~r ′)− jωµG(~r, ~r ′)N∑n=1

hnΠn(~r ′)

]dr′ . (3.22)

We test this integral equation by point-matching [105] about the midpoints ~rm of allN segments,

to obtain

em =j

2

N∑n=1

en

ˆγn

∂G(~rm, ~r′)

∂n′dr′ +

ωµ

2

N∑n=1

hn

ˆγn

G(~rm, ~r′)dr′ (3.23)

for m = 1, . . . , N . Note that in (3.23), the integration is performed only over the n-th segment

γn. All N equations in (3.23) can be compactly written in matrix form as

KσE = LσH , (3.24)


where Kσ and Lσ are square matrices with dimensions N ×N . Element (m,n) of matrix Kσ,

if m 6= n, is given by

[Kσ]m,n =jk

2

ˆγ

(p)n

~dm · n′dm

[C0J1(kdm)− jY1(kdm)] dr′ , (3.25)

where ~dm = ~r ′ − ~rm, dm =∣∣∣~dm∣∣∣, and we used the fact that the normal derivative of (3.11) can

be written as [123]

∂G(~r, ~r ′)

∂n′= −k

~dm · n′dm

[C0J1(kdm)− jY1(kdm)] . (3.26)

When n = m, the contribution of the first term on the right-hand side of (3.23) is zero because~dm and n′ are orthogonal. Hence, the diagonal entries of Kσ are given by

[Kσ]m,m = 1 . (3.27)

The (m,n)-th entry of Lσ in (3.24) is given by

[Lσ]m,n =ωµ

2

ˆγn

[C0J0(kdm)− jY0(kdm)] dr′ . (3.28)

In all numerical examples that will be presented in Sec. 3.7, the integrals in (3.25) and (3.28)

were evaluated using a 5-point Gaussian quadrature routine. When n = m, the integral in

(3.28) can be evaluated analytically as shown in Appendix B.

From (3.24), we can express the magnetic field H in terms of the electric field on the same

conductor as

H = L−1σ KσE . (3.29)

3.4.3 Magnetic Field in the Equivalent Configuration

Next, we find the magnetic field in the equivalent configuration by repeating all the steps of

Sec. 3.4.2, but with the material parameters of the conductor replaced by the parameters of

the surrounding medium. These steps lead to

Heq = L−1eq KeqE , (3.30)

which is analogous to (3.29). In the equation above, entries of Leq and Keq are calculated

with (3.28), (3.25), and (3.27) with wavenumber k replaced by kout and permeability µ replaced

by µl.


3.4.4 Differential Surface Admittance Operator

Finally, by substituting (3.29) and (3.30) into (3.21), we obtain

J = Y∆E (3.31)

where

Y∆ = L−1σ Kσ − L−1

eq Keq (3.32)

is the discretized differential surface admittance operator of the conductor. Therefore, we see

that with the proposed contour integral method approach, it is sufficient to evaluate matrices Lσ,

Leq, Kσ, and Keq to easily obtain the differential surface admittance operator for a conductor

of arbitrary shape.

For systems with P conductors, vectors E and J collect electric field and equivalent current

density coefficients for all conductors. The differential surface admittance operator Y∆, in this

case, is a block diagonal matrix where the p-th block is the differential surface admittance

operator of the p-th conductor.

3.4.5 Choice of C0

In this section, we discuss how to set the constant C0 at low and high frequencies in order to

achieve a well-conditioned algorithm.

Low Frequencies

At low frequencies, where skin effect has not yet developed, computing (3.28) requires the

evaluation of Green’s function (3.11) for very small arguments. For small arguments, the

Neumann function Y0(.) dominates the Bessel function J0(.) [123]. Hence, to ensure that effects

of both the Bessel and Neumann functions are accounted, we must set C0 to a high value [124].

The value of 106 provided accurate results for all numerical tests we performed, including the

examples of Sec. 3.7.

High Frequencies

At high frequencies, we set C0 = 1. This makes the Green’s function (3.11) become the Hankel

function of the second kind, which is well-behaved for large arguments [123].

Switching Condition

As previously discussed, two different values of C0 are appropriate at low and high frequencies.

Numerical tests demonstrated that there is a large range of intermediate frequencies where

both values of C0 provide accurate results. Therefore, the choice of the frequency where one

should switch from one C0 value to the other is not critical. In our implementation, we used


the low-frequency value of C0 on the p-th conductor when

∆p

δp≤ t , (3.33)

where ∆p and δp are, respectively, the minimum transversal dimension and the skin depth in

the p-th conductor. In all numerical tests we performed, any t value between 0.2 and 0.5 gave

good results, and t = 0.5 has been used in all numerical examples presented in Sec. 3.7. When

frequency increases and inequality (3.33) no longer holds, the high-frequency value of C0 is

used.

Discussion

The choice of C0 and of the switching condition (3.33) can be explained physically by look-

ing at the wave phenomena that develops inside the conductors at low and high frequencies.

The contour integral equation (3.10) can be interpreted using the equivalence principle [91].

In (3.10), the electric field on the contour is expressed as the superposition of the field radiated

by a magnetic current on the contour (first term on the right-hand side), and the field radiated

by an electric current (second term on the right-hand side). The fields are obtained from the

currents through multiplication by the Green’s function and its derivative, respectively. This

interpretation reveals that (3.10) expresses the field in the conductor as a superposition of

cylindrical waves originating from the points of the contour. At high frequencies, where (3.33)

is not satisfied and the skin depth is very small, the cylindrical wave emanating from a point on

the contour attenuates appreciably before reaching the other side of the conductor. As a result,

the Hankel function of second kind H(2)0 (·), which describes an outgoing cylindrical wave, is the

most appropriate Green’s function for this regime. On the other hand, when (3.33) is satisfied,

the cylindrical wave reaches the other side of the conductor without significant attenuation, giv-

ing rise to standing waves, that are best modeled using a combination of Bessel and Neumann

functions J0(·) and Y0(·).

3.5 Exterior Problem and Impedance Computation

After applying the equivalence principle, we have replaced all conductors in the multiconductor

transmission line with the surrounding medium and equivalent currents on γ(p). We can now

relate the electric field and equivalent currents on the conductors’ boundaries by the electric

field integral equation. On the contour of the p-th conductor, the electric field integral equation

reads

E(p)z (~r) = jωµl

P∑q=1

‰γ(q)

J (q)s (~r ′)G0(~r, ~r ′)dr′ − ∂Vp

∂z, (3.34)

where ~r ∈ γ(p) and the integration is performed over the contour γ(q) of each conductor (q =

1, . . . , P ). In this section, we use superscripts (p) and (q) to denote conductor numbers. The


integral kernel G0(~r, ~r ′) is the Green’s function of the surrounding medium. For a free space

background medium the Green’s function is

G0(~r, ~r ′) =1

2πln∣∣~r − ~r ′∣∣ . (3.35)

Following [29,43], we substitute (3.1a), (3.13) and (3.19) into (3.34), to obtain

N(p)∑m=1

e(p)m Π(p)

m (~r) =jωµl

P∑q=1

‰γ(q)

N(q)∑n=1

j(q)n Π(q)

n (~r ′)G0(~r, ~r ′)dr′q +P∑q=1

[Rpq(ω) + jωLpq(ω)] Iq

(3.36)

for p = 1, . . . , P . Using point-matching [83], this integral equation can be converted into the

system of algebraic equations

E = jωµlGJ + U [R(ω) + jωL(ω)] I , (3.37)

where G is a block matrix of form

G =

G1,1 G1,2 . . . G1,P

G2,1 G2,2 . . . G2,P

......

. . ....

GP,1 GP,2 . . . GP,P

. (3.38)

Block Gp,q in (3.38) captures the mutual coupling between conductors p and q. It is of size

N (p) ×N (q) with (m,n)-th entry of Gp,q given by

[Gp,q]m,n =

ˆγ

(q)n

G0

(∣∣∣~r (p)m − ~r ′

∣∣∣) dr′ , (3.39)

where vector ~r(p)m is the midpoint of m-th partition on γ(p), and the integration is performed

over the n-th partition of γ(q). In (3.37), U is a block diagonal matrix

U =

1(1)

. . .

1(P )

(3.40)

where 1(p) is a vector of size N (p)× 1 whose entries are all ones. Furthermore, the total current

inside each conductor is equal to the contour integral of the equivalent current [29]. This

relationship can be expressed in the discrete domain as

I = UTWJ . (3.41)


In (3.41), W is a block diagonal matrix of the form

W =

W(1)

. . .

W(P )

, (3.42)

and W(p) is a diagonal matrix of size of N (p) ×N (p) whose (n, n)-th entry stores the width of

the n-th segment of γ(p).

By substituting (3.31) into (3.37) and rearranging the equation we obtain

J = (1− jωµ0Y∆G)−1Y∆Q (R(ω) + jωL(ω)) I (3.43)

where 1 is the identity matrix. Next, we left-multiply (3.43) by UT and use (3.41) to obtain

I = UTW (1− jωµ0Y∆G)−1 Y∆U (R(ω) + jωL(ω)) I (3.44)

We now use the fact that the partial p.u.l. impedance parameters are independent of currents,

to obtain the partial p.u.l. resistance

R(ω) = Re[UTW (1− jωµlY∆G0)−1 Y∆U

]−1(3.45)

and partial p.u.l. inductance

L(ω) = ω−1Im[UTW (1− jωµlY∆G0)−1 Y∆U

]−1. (3.46)

The p.u.l. impedance can be obtained from the partial p.u.l. impedance by taking one of the

conductors as reference.

3.6 Calculation of the Admittance Parameters

The Telegrapher’s equation (3.1b), using the continuity equation, can be rewritten as

Q = C(ω)V (3.47)

where, C(ω) = 1jωG(ω) + C(ω) is the complex capacitance matrix and Q =

[Q1 . . . QP

]Tcollects the p.u.l. charge on each conductor. Our goal is to compute C(ω) by solving the elec-

trostatic problem. For simplicity, we will assume that all dielectrics are lossless and C(ω) = C

is constant. However, it is straightforward to generalize the presented theory to lossy dielectrics

by setting the relative permittivity of a dielectric to a complex number.

We compute the capacitance matrix by applying the equivalence principle in a different

way that is suitable for electrostatic problems. In this equivalence, finite-sized dielectrics are


Viεi

Voεo

ρo(~r)

+++

++ +

γ

n

(a) Original setup

Viεo

Voεo

ρo(~r)

+ + + +++

++

+++++

++

+ + + +++

+

+++++

+

+++

++ +

ρeq(~r)

(b) Equivalent setup

Figure 3.2: Equivalence principle for electrostatic problems.

replaced by the surrounding background medium while maintaining the scalar potential on the

boundary of the dielectrics to be the same as the original problem. The effects of the dielectrics

are modeled by the so-called contrast charge density [31], which is analogous to the equivalent

electric current density in the impedance calculation.

3.6.1 Equivalence Principle for Electrostatic Problems

To discuss the equivalence principle for electrostatic problems, let us consider the sample setup

shown in Fig. 3.2a. This setup contains two regions: Vi and Vo. The interface between Vi and

Vo is denoted by γ. The permittivity of Vi and Vo is denoted by εi and εo, respectively. There

also exists charge distribution inside Vo that is denoted by ρo(~r), as shown in Fig. 3.2a. The

scalar potential in Vo and Vi is given by Φo(~r) and Φi(~r), respectively, and it satisfies Laplace’s

and Poisson’s equations [89] in Vo and Vi, respectively,

∇2Φo(~r) = −ρoεo, ~r ∈ Vo (3.48a)

∇2Φi(~r) = 0 , ~r ∈ Vi . (3.48b)

Finally, on the interface γ the scalar potential satisfies the electrostatic boundary conditions

∂Φi

∂t=∂Φo

∂t(3.49a)

εo∂Φo

∂n= εi

∂Φi

∂n. (3.49b)

We now introduce an equivalent problem for Vo, as shown in Fig. 3.2b. In this equivalent

problem, Vi is replaced by the surrounding medium. We also enforce that the scalar potential

on γ remains the same as the original problem. By enforcing this condition, the scalar potential

inside Vi remains the same as the original problem2. To ensure that the correct electric field is

2This is the case because scalar potentials in both the original and equivalent problems satisfy Laplace’sequation with the same boundary condition on γ.


ε1

ε2

ε3


ε1

ε2

ε3

ρ, ρρρ

ρ

(b) Equivalent configura-tion

Figure 3.3: Sample cross-section and equivalent problem for capacitance calculation

maintained in Vo, we introduce an equivalent charge density

ρeq(~r) = εo∂Φo

∂n− εo

∂Φi

∂n, (3.50)

on γ, whose value is determined by the electrostatics boundary condition. Furthermore, we

substitute (3.49b) into (3.50) to obtain

ρeq(~r) = (εi − εo)∂Φi

∂n, (3.51)

which can be interpreted as the contrast charge density [31].

3.6.2 Discretization of Scalar Potential

The concept of contrast charge density can be employed to compute the p.u.l. capacitance

matrix. Previously, this approach was used to compute the C matrix for transmission lines

containing dielectrics of canonical shapes. We will generalize the work to compute C for systems

with an arbitrarily-shaped dielectrics. To demonstrate the technique, we consider the sample

cross-section in Fig. 3.3a. To calculate C, we first discretize all boundaries of the conductors

with Nc segments, and all of the remaining boundaries of the dielectrics with Nd segments.

Boundaries of the layered background medium are not discretized because their effect can be

captured via the Green’s function. We expand the scalar potential on all boundaries using pulse

basis functions

Φ(~r) =

Nc+Nd∑n=1

φnΠn(~r) . (3.52)

The expansion coefficients in (3.52) are now collected into vector

Φ =[φ1 φ2 . . . φNc+Nd

]T. (3.53)


3.6.3 Equivalent Charge Density

We replace all conductors and dielectrics by the surrounding medium and introduce equivalent

contrast charge densities ρ(~r) and ρ(~r) on the boundary of all conductors and dielectrics, re-

spectively, as shown in Fig. 3.3b. We expand both equivalent charge densities with pulse basis

functions

ρ(~r) =

Nc∑n=1

ρnΠn(~r) , (3.54)

ρ(~r) =

Nd∑n=1

ρnΠn(~r) . (3.55)

We collect the coefficients of (3.54) and (3.55) into vectors

ρ =[ρ1 ρ2 . . . ρNc

]T, (3.56)

ρ =[ρ1 ρ2 . . . ρNd

]T. (3.57)

Conductors

When a charge is placed on a conductor with high conductivity, it moves to the conductor’s

boundary very quickly. So, contrast charge density on good conductors is actually equal to the

free charge density on the conductor surface and it is related to the total charge Qc by

Qc =

˛γc

ρ(~r ′)dc′ , (3.58)

where the integration is performed over the closed contour γc enclosing the c-th conductor. The

discrete counterpart of (3.58) for all conductors may be written as

Q = UTWρ , (3.59)

where U and W are as defined in (3.40) and (3.42).

Dielectrics

The contrast charge density on the boundary γd of the d-th dielectric is related to the potential

Φ(~r) by [121]

ρ(~r) = (ε− εb)∂Φ(~r)

∂n, (3.60)

where ε is the permittivity of the dielectric and εb is the permittivity of the background medium.

In [121], an eigenfunction expansion is used to relate Φ(~r) and ∂Φ(~r)/∂n. Instead, here we use

the contour integral method to relate the scalar potential and its normal derivative. Inside the


dielectric medium, scalar potential Φ(~r) satisfies

∇2Φ(~r) = 0 . (3.61)

Following the application of Green’s identity, we can show that the solution of (3.61) satis-

fies [119] ˛γd

[Φ(~r ′)

∂ ln(~r, ~r ′)

∂n′− ln(~r, ~r ′)

∂Φ(~r)

∂n′

]dr′ = πΦ(~r) . (3.62)

We discretize (3.62) with the MoM to obtain

Φd = HΦd , (3.63)

where Φd is a subset of Φ containing potentials on the boundary γd, and Φd is a vector that

contains the expansion coefficients of the normal derivative of scalar potential on γd. Finally,

by substituting (3.63) into the discretized form of (3.60) for all dielectrics in the system, we

obtain

ρ = YcΦ , (3.64)

where Yc is the desired surface operator, which compactly relates contrast charge density to

potentials.

3.6.4 Capacitance Calculation

The equivalent charge densities and scalar potential Φ(~r) are related by

Φ(~r) = −1

ε

ˆ∑γc

ρ(~r ′)G0(~r, ~r ′)dr′ − 1

ε

ˆ∑γd

ρ(~r ′)G0(~r, ~r ′)dr′ , (3.65)

where G0(~r, ~r ′) is the Green’s function of the background medium. We discretize (3.65) using

the method of moments obtaining

Φ =[G1 G2

] [ρρ

](3.66)

where G1 and G2 are Green’s matrices. By substituting (3.64) into (3.66), we obtain

Φ = (1−G2Yc)−1 G1ρ , (3.67)

which relates conductor potentials to equivalent charge density on conductors. In (3.67), 1 is the

identity matrix. At this point, it is straightforward to find C by applying a constant potential

on each conductor, and solving (3.67) to find the total charge induced on each conductor.


20.2

20.2

0.5

21

3.5

Figure 3.4: Cross section of the two-conductor lines analyzed in Sec. 3.7.1. All dimensions are inmillimeters.

Table 3.1. Example of Sec. 3.7.1: CPU time required to compute the p.u.l. impedance at onefrequency

Test Case Proposed MoM-SO [117] De Zutter [29]Round conductors 0.04 s 0.0006 s N/ARectangular conductors 0.16 s N/A 0.12 s

3.7 Numerical Results

In this section, we demonstrate the robustness of the proposed method, and compare its ac-

curacy and CPU times against a FEM solver and other surface admittance-based approaches

in the literature [29, 117]. We illustrate the versatility of the technique by considering a com-

prehensive set of transmission lines made by circular, rectangular, trapezoidal, V-shaped, and

conformal conductors. All computations were performed on a computer with 16 GB of mem-

ory and a 3.4 GHz processor. All techniques based on a surface admittance operator were

implemented in MATLAB.

3.7.1 Two Conductor Lines

Round Conductors

We first consider a transmission line made by two round conductors with radii a1 = 1 mm,

a2 = 2 mm, and spacing d = 3.5 mm. The line cross section is shown in the left panel of Fig. 3.4.

The conductivity of both conductors is σ = 5.8 · 107 S/m. We calculated the p.u.l. impedance

using the proposed technique and MoM-SO [117], which uses the eigenfunctions method to

derive the surface admittance operator. Figure 3.5 shows the p.u.l. resistance and inductance

obtained with the proposed approach and with MoM-SO. The two methods are in excellent

agreement. For the proposed approach, the boundaries enclosing the small and large conductors

were discretized with N (1) = 28 and N (2) = 60 pulse basis functions, respectively. As shown in

Table 3.1, for each frequency point the proposed approach took only 0.04 s. In this case, the

eigenfunctions approach is faster since it can capture, with a few Fourier basis functions, the

field distribution inside the circular conductors. The advantage of the proposed approach is

generality: it can be applied to arbitrary shapes, while MoM-SO is limited to round conductors,

either solid [29,117] or hollow [118].


100

102

104

106

108

10−2

10−1

100

Frequency (Hz)

Res

ista

nce

p.u

.l. (

Ω /

m)

MoM−SOProposed

100

102

104

106

108

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Frequency (Hz)

Ind

uct

ance

p.u

.l.

( μH

/m)

MoM−SOProposed

Figure 3.5: P.u.l. resistance and inductance of the transmission line with two round conductors consideredin Sec. 3.7.1, obtained with the proposed method and MoM-SO [117].

102

104

106

108

1010

10−1

100

101

Res

ista

nce

p.u

.l. (

Ω/m

)

Frequency (Hz)

De Zutter et al.Proposed

102

104

106

108

1010

0.2

0.22

0.24

0.26

0.28

Frequency (Hz)

Indu

ctan

ce p

.u.l

. (μ

H/m

)

De Zutter et al.Proposed

Figure 3.6: P.u.l. resistance and inductance of the transmission line with two rectangular conductorsconsidered in Sec. 3.7.1. Results obtained with [29] are labeled as “De Zutter et al”.

Rectangular Conductors

We now consider the case of two rectangular conductors presented by De Zutter [29]. Each

conductor has conductivity σ = 5.6·107 S/m and dimension 2 mm×0.2 mm, as shown in the right

panel of Fig. 3.4. Fig. 3.6 shows the p.u.l. resistance and inductance for this transmission line

computed with the proposed method and with De Zutter’s method [29]. The latter computes the

surface admittance operator analytically using sinusoidal functions, which are the eigenfunctions

of the Helmholtz equation on a rectangular domain. In order to interface the surface admittance

operator to the electric field integral equation (3.36), the sinusoidal functions have to be mapped

onto pulse basis functions [29]. From Fig. 3.6, we see that the resistance and inductance values

obtained with both approaches match very well. For the proposed approach, each conductor

was discretized with N (1) = N (2) = 106 pulse basis functions. In De Zutter’s method [29],

the number of sinusoidal harmonics were chosen to be M = 4003, and each conductor was

discretized with N (1) = N (2) = 280 pulse basis functions. De Zutter’s approach [29] required

more basis functions than the proposed approach to get accurate results at very high frequencies.

The computational time taken by both techniques is given in Table 3.1. We see that the CPU

time with the proposed approach is not far from the CPU time with [29], although the proposed

3M is the number of sinuisoidal harmonics [29]


131311

7.57.511

10

3

Figure 3.7: Valley microstrip line considered in Sec. 3.7.2. All dimensions are in micrometers.

106

108

1010

103

104

Frequency [Hz]

Res

ista

nce

p.u

.l. [Ω

/m]

COMSOL

Proposed

106

108

1010

0.22

0.23

0.24

0.25

0.26

0.27

0.28

Frequency [Hz]In

duct

ance

p.u

.l. [µ

H/m

]

COMSOL

Proposed

Figure 3.8: P.u.l. resistance and inductance of the valley microstrip line considered in Sec. 3.7.2, com-puted with the proposed method and the FEM (COMSOL Multiphysics).

method is more general, as it can handle arbitrary shapes.

3.7.2 Valley Microstrip Line

We consider a valley microstrip line from [120], having the cross-section depicted in Fig. 3.7.

This line type is used in low-loss microwave integrated circuits and it has been considered in

several previous publications [125,126]. All conductors are made of copper (σ = 5.8 · 107 S/m).

The top conductor is the signal line, while the two lower conductors form the reference line.

Figure 3.8 shows the p.u.l. inductance and resistance of the system obtained with a FEM

solver (COMSOL Multiphysics) and with the proposed approach. The results from the two

methods agree well both at very low (1 MHz) and very high (150 GHz) frequencies. This

test further validates the proposed method and shows its numerical robustness. In the FEM

simulation, we had to use a fine mesh with 3,102 boundary elements and 188,107 triangular

elements, in order to properly resolve the pronounced skin effect at high frequencies. In the

proposed technique, the boundary of the signal conductor was discretized with N (1) = 194 pulse

basis functions, while each ground conductor was discretized with N (2) = N (3) = 92 pulse basis

functions. The CPU time taken by both techniques is reported in Table 3.2 and shows the

efficiency of the proposed method, which requires only 0.52 s to extract the p.u.l. impedance

at one frequency for this non-trivial line. While in such cases one can resort to a general 2-D

FEM approach, this significantly increases the CPU time, as shown in Table 3.2.


Table 3.2. Examples of Sec. 3.7.2, 3.7.3 and 3.7.4: CPU time required to compute thep.u.l. impedance at one frequency with the proposed technique and a FEM solver (COMSOL

Multiphysics).

Section Test Case Proposed COMSOLSec. 3.7.2 Valley microstrip 0.52 s 73.8 sSec. 3.7.3 On-chip line 0.18 s 9.11 sSec. 3.7.4 Curved microstrips 0.51 s 64.36 s

3.7.3 On-chip Transmission Line with Trapezoidal Conductors

Next, we consider the four-conductor transmission line [122] shown in Fig. 3.9. All conductors

are made of aluminum (σ = 35.7 MS/m). The conductivity is assumed to be constant up

to the maximum frequency of interest (1 THz) [122], since at this frequency the effect of

relaxation time is still negligible [127]. This type of transmission line is typical for interconnects

in integrated circuits. The trapezoidal shape of the signal lines is caused, for example, by

underetching [128]. As shown in [122], approximating signal lines with perfect rectangles results

in a non-negligible error in the p.u.l. resistance and inductance. Figure 3.10 shows various

entries of the p.u.l. resistance and inductance matrices of the transmission line, computed with

the proposed technique and a FEM solver (COMSOL Multiphysics). The proposed technique

correctly captures the non-trivial impedance behaviour over frequency. The results shown

in Fig. 3.10 also demonstrate the robustness of the proposed method, at both low and high

frequencies.

In order to analyze how accuracy and CPU time scale with mesh size, we repeated the

analysis for different levels of discretization, as summarized in Table 3.3. Mesh for each of

the five test cases presented in Tab. 3.3 is shown in Fig. 3.11. In order to model skin effect

efficiently, we used the so-called boundary layer elements to mesh the region near the edges of

the conductors. In COMSOL, the boundary layer elements are defined over a thin quadrilateral

mesh. To model skin effect accurately, we have used the so-called boundary layer mesh elements

in COMSOL. The normalized error in p.u.l. resistance and inductance is shown for selected self-

and mutual-terms in Table 3.4, using the finest FEM mesh as a reference. For the FEM method,

only the mesh “FEM 4” gives reasonably accurate results. Meshes 1 to 3 are inaccurate at high

frequency and fail to accurately resolve skin effect. It can be noticed that the normalized error

in the p.u.l. impedance parameters lowers as the mesh size decreases. However, due to the usage

of boundary layer elements, it is difficult to predict the rate of convergence with respect to the

mesh settings. The proposed method, on the other hand, with only 236 pulse basis functions

gives a maximum error of 2.10%, which drops below 1% if the number of basis functions is

increased to 860. CPU times, reported in Table 3.3, show the superior computational efficiency

of the proposed method with respect to FEM.


13

1

1.5

0.75

10.5

1

#0

#1 #2#3

ε0, µ0

Figure 3.9: On-chip interconnect of Sec. 3.7.3. All dimensions are in micrometers.

107

108

109

1010

1011

1012

103

104

105

R11

R12

R23

Frequency [Hz]

Res

ista

nce

p.u

.l. [Ω

/m]

COMSOL

Proposed

107

108

109

1010

1011

1012

0

0.1

0.2

0.3

0.4

0.5

L11

L22

L12

L32

Frequency (Hz)

Indu

ctan

ce p

.u.l

. (μ

H/m

)Figure 3.10: P.u.l. resistance and inductance of the on-chip transmission line with trapezoidal conductorsconsidered in Sec. 3.7.3.

Table 3.3. Example in Sec. 3.7.3: CPU time for the FEM and the proposed method for differentmesh sizes

colorCase Domain elements Boundary elements CPU time per frequency

FEM 1 3540 393 2.33 s

FEM 2 5344 446 4.16 s

FEM 3 7604 481 4.42 s

FEM 4 27596 718 9.11 s

FEM 5 (Reference solution) 64576 1090 30.96 s

Proposed 1 236 pulse basis functions 0.18 s

Proposed 2 860 pulse basis functions 1.48 s

Table 3.4. Example of Sec. 3.7.3: normalized error (%) in selected p.u.l resistance andinductance coefficients, for the FEM and the proposed method, using different mesh sizes

color Low frequency: f = 107 Hz Mid frequency: f = 3 · 109 Hz High frequency: f = 1012 Hz

R1,1 L1,1 R1,2 L1,2 R1,1 L1,1 R1,2 L1,2 R1,1 L1,1 R1,2 L1,2

FEM 1 1.8e-6 0.12 4.8e-6 1.9e-2 1.4e-3 1.2e-1 3.4e-3 2.4e-2 5.7 1.2 1.7 1.4e-4

FEM 2 2.0e-8 1.2e-2 1.0e-6 5.7e-7 1.3e-5 1.3e-2 9.5e-4 2.5e-4 3.2 0.21 0.79 5.2e-2

FEM 3 9.7e-9 6.1e-3 3.5e-8 5.0e-9 5.6e-5 6.8e-3 1.7e-5 1.1e-4 2.0 9.2e-2 0.36 1.8e-2

FEM 4 0 0 0 0 4.8e-5 3.1e-4 1.0e-4 1.1e-4 6.9e-2 1.4e-4 5.5e-2 2.4e-4

CIM 1 5.3e-4 2.6e-4 4.3e-3 0.20 0.51 0.46 2.1 0.18 1.6 0.32 1.51 0.16

CIM 2 1.0e-4 5.3e-2 2.0e-3 3.3e-2 0.15 3.1e-2 0.94 4.2e-2 0.53 5.6e-2 0.61 0.15

3.7.4 A Curved Microstrip Line

Flexible dielectrics allow for the creation of curved interconnects conformal to a cylindrical

surface [129,130]. We analyze the configuration shown in Fig. 3.12, which features four copper

conductors (σ = 5.8 · 107 S/m). Conductors were chosen to be very thin and wide to demon-


(a) FEM 1 (b) FEM 2

(c) FEM 3 (d) FEM 4

(e) FEM 5

Figure 3.11: Mesh used to model the on-chip transmission line with the five mesh settings that aresummarized in Table 3.3.

α1

α2

α3

α4

R1

R2

h

h

h #0

#1

#2#3

Figure 3.12: Cross-section of the curved microstrips of Sec. 3.7.4. Geometrical dimensions are: α1 = 10,α2 = 20, α3 = 30, α4 = 35, R1 = 80 µm, R2 = 100 µm, h = 2 µm.

strate the robustness of the proposed approach in handling conductors with large aspect ratios.

Figure 3.13 shows selected entries of the resistance and inductance matrices, computed with the

proposed method and a FEM solver (COMSOL Multiphysics). A close match can be observed,

which validates the proposed technique. For FEM simulation, we discretized the geometry with

110, 416 triangular and 28, 360 boundary elements. In the proposed method, a total of 373

pulse basis functions were used to discretize the conductors’ boundaries. Table 3.2 shows that,

with the proposed method, the p.u.l. impedance of the conformal interconnect can be obtained

in 0.51 s per frequency point.

3.7.5 A Coated Differential Line

We consider the coated differential line in Fig. 3.14. The dimensions of the system are given

in Fig. 3.14 and were obtained from [131]. Unlike previous examples, which did not contain


106

107

108

109

1010

1011

102

103

104

R11

R12

Frequency [Hz]

Res

ista

nce

p.u

.l. [Ω

/m]

R22

COMSOL

Proposed

106

107

108

109

1010

1011

0.33

0.34

0.35

0.36

0.37

0.38

0.39L

11

Frequency(Hz)

Ind

uct

ance

p.u

.l.

[µH

/m]

COMSOL

Proposed

106

107

108

109

1010

1011

0.6

0.65

0.7

0.75

L22

Frequency(Hz)In

du

ctan

ce p

.u.l

. [µ

H/m

]

COMSOL

Proposed

Figure 3.13: Selected entries of the p.u.l. resistance and inductance matrices of the curved microstriplines of Sec. 3.7.4.

ε1, µ1

l2

w2

ε2, µ2

h

∆

l1

w1

S

Figure 3.14: Coated differential pair example considered in Sec. 3.7.5, with l1 = 166 µm, w1 = 17.4 µm,w2 = 34.8 µm, S = 292.4 µm, h = 100.9 µm.

any dielectric object, a coated differential line has two dielectric objects with different electrical

permittivity. Hence, we computed the p.u.l. capacitance matrix with the proposed surface

method. Table 3.2 shows the capacitance calculated for various values of dielectric permittivities

and geometrical parameters with the proposed method and with FEM [132]. The results are in

excellent agreement with self and mutual capacitance errors always lower than 1 pF/m. The

computational time with the proposed method was 0.80 s, as opposed to FEM which took 8 s.

Fig 3.15 shows the p.u.l. resistance and inductance parameters calculated for l2 = 525 µm.

The plot shows an excellent agreement between the proposed method and FEM [132]. Fig. 3.15

also shows that when we use C0 = 1 for all frequencies, the p.u.l. inductance at low frequencies

become inaccurate. However, the p.u.l. inductance is accurate at low frequencies when we use

C0 = 106. Impedance computation with the proposed method required a total of 266 pulse


Table 3.5. Example of Sec. 3.7.5: Capacitance values (in pF/m) calculated with the proposedtechnique and with FEM [132].

l2 Proposed FEM Errorε1 ε2 ∆ [µm] C11 C12 C11 C12 [pF/m]2.94 1 - 525 76.1 -8.6 76.0 -9.0 0.44.3 1 - 525 104.3 -9.6 104.4 -10.0 0.44.3 1 - 1050 112.1 -8.6 112.8 -8.7 0.74.3 3.2 w 1050 123.1 -12.3 123.1 -12.6 0.34.3 3.3 2w 1050 128.5 -15.0 128.7 -15.4 0.4

104

106

108

1010

100

101

102

103

Frequency [Hz]

Res

ista

nce

p.u

.l. [Ω

/m]

R11

R12

Ansys 2D

Proposed

Proposed (C0=1)

104

106

108

1010

0

100

200

300

400

Frequency [Hz]In

du

ctan

ce p

.u.l

. [n

H/m

]

L11

L12

Ansys 2D

Proposed

Proposed (C0 = 1)

Figure 3.15: Per-unit length resistance (left panel) and inductance (right panel) obtained with theproposed method and FEM [132] for the example considered in Sec 3.7.5

basis functions, as opposed to the FEM simulation which required 26,170 basis functions. The

proposed technique required 0.24 s per frequency point, as opposed FEM [132] which required

9.05 s. The proposed method led to a speed-up with respect to FEM of 10X and 38X in the

extraction of capacitance and resistance/inductance, respectively.

3.8 Chapter Summary and Contributions

This chapter presented a novel 2-D surface formulation to calculate the p.u.l. impedance and

admittance parameters of multiconductor transmission lines. For p.u.l. impedance calculation,

the main idea behind the proposed technique is to model each conductor with an admittance

operator and an equivalent electric current density that is introduced on the conductor’s bound-

ary. For p.u.l. admittance calculation, the idea is to model all dielectric and conducting objects

with an equivalent charge density introduced on objects’ boundaries. Furthermore, a capaci-

tance operator is derived to relate the equivalent charge density to the scalar potential between

the dielectric’s boundaries.

The key features and contributions of the proposed technique are the following:

• The method does not require meshing the cross-section of the conductors or dielectric,

unlike most volumetric methods and some surface methods. Hence, the proposed method

is significantly faster than other works in the literature, including commercial solvers.


• The proposed surface method for the calculation of p.u.l. impedance parameters is based

on the differential surface admittance concept that was originally introduced to model

conductor cross-sections of canonical shapes [29]: round, rectangular, and triangular. In

this chapter, we generalized the method to conductors of arbitrary shapes by computing

the differential surface admittance operator using the contour integral equation method.

• The proposed surface method for the calculation of p.u.l. admittance parameters is based

on the concept of a contrast charge density and capacitance operator that was originally

introduced to model dielectrics of canonical shapes [31]: rectangular, and triangular. In

this chapter, we generalized the technique to dielectrics of arbitrary shape by computing

the capacitance operator using the contour integral method.

Numerical results have demonstrated that the proposed method is very accurate compared to

commercial simulation tools and other surface methods in the literature.

Chapter 4

A Magneto-Quasistatic Analysis of

3-D Interconnects with a 2-D

Surface Operator

4.1 Introduction

In Chapter 3, we discussed a 2-D surface formulation to compute p.u.l. parameters that account

for various effects, including skin, proximity, and dielectric effects. Modeling interconnects with

the transmission line theory is, however, only accurate for long interconnects with a constant

cross-section along their entire length. When conductors have bends, 2-D models are not accu-

rate, and 3-D formulations are needed. One option is to resort to full-wave 3-D solvers. While

accurate, full-wave 3-D solvers can be quite expensive. Therefore, to reduce the complexity

of the problem, many researchers in the past have adopted the magneto-quasistatic (MQS)

assumption to model electrically small structures. Under this assumption, the displacement

current jωε ~E(~r) inside the conductor is neglected and field variation with respect to the lon-

gitudinal direction is set to zero. If we set the displacement current in Ampere’s Law to zero,

then we obtain that ∇ · ~J(~r) = 0. That is, the current is conserved within the conductor. As a

consequence of this, the current inside the conductor flows along the longitudinal direction of

the conductor because of the absence of charges on the conductor’s surface [46]. If the current

flows only along the longitudinal direction, then we can employ the 2-D surface formulation to

model the skin effect inside the conductor.

In the literature, the MQS assumption has been adopted in a few volumetric methods [46,

133] and surface methods [134–136] to model skin effect. In volumetric methods, the current

density along the conductor is discretized with 3-D volumetric filaments. At high frequency,

this approach leads to a very large number of unknowns. Surface formulations [134,136], on the

other hand, only require unknowns on the surface of the conductors. However, surface methods

in the literature require discretizing and solving for two or more field quantities [134, 136].

70

Chapter 4. A Magneto-Quasistatic Analysis of 3-D Interconnects 71

σ

µ0, ε0


µ0, ε0

(b) Partitioned configuration

Figure 4.1: Original configuration: example of a U-shaped conductor. Dark gray surfaces indicate theterminals of the conductor. Partitioned configuration: partitioned model used by the proposed method.

In this chapter, we explore how the 2-D differential surface admittance operator presented in

Chapter 3 can be used to solve the 3-D Maxwell’s equations under the MQS assumption. Since

the differential surface admittance operator-based approach is a single-source surface method,

it is more efficient than other volumetric and surface methods in the literature.

4.2 Problem Definition

Our goal is to calculate the impedance matrix of a multiport system using a surface formulation.

For simplicity, we present the formulation considering a single conductor whose conductivity is

σ, permittivity is ε0, and permeability is µ. The surrounding medium is assumed, for simplicity,

to be free space with permittivity ε0 and permeability µ0. The conductor has one port that is

made up of two terminals. The port voltage and current are denoted by V and I, respectively.

A sample conductor geometry is presented in Fig. 4.1a. The proposed formulation makes the

following two assumptions:

1. The cross-section of the conductor is electrically small, and therefore the potential across

it can be considered to be constant.

2. Due to the MQS assumption, the electric current flows only in the longitudinal direction

of the conductor. As a consequence, the magnetic field only has a transverse component.

In order to calculate the impedance of a 3-D conductor, we partition the conductor along the

longitudinal direction into S straight segments such that each segment’s length is much smaller

than the wavelength. In the sample configuration in Fig. 4.1a, the conductor is partitioned

into three segments as shown in Fig. 4.1b. We assume that the cross-section of each segment

is invariant along the length of the segment; however, two different segments of a conductor

may have a different cross-section. We use l to denote the unit vector along the length of

segment s. The vector Vseg =[V

(1)seg . . . V

(S)seg

]Tcollects voltage drops V

(s)seg across each

segment. Similarly, the vector Iseg =[I

(1)seg . . . I

(S)seg

]Tcollects current I

(s)seg in each segment.

All voltages and currents are related by

Vseg = ZsegIseg , (4.1)


ln

σ

µ0, ε0

(a) Original configura-tion

ln

µ0, ε0

µ0, ε0

J(s)eq


Figure 4.2: Original configuration: a conductor segment with trapezoidal cross-section. The surface ofthe segment is partitioned with panels (traced by dashed lines). Equivalent configuration: equivalentproblem with the conductor segment replaced by the surrounding medium and equivalent currents onthe surface.

where Zseg is an impedance matrix. Our goal is to compute Zseg, from which the port impedance

matrix can be computed very easily as discussed in Sec. 4.4.

4.3 Skin Effect Modeling

4.3.1 Discretization of Electric and Magnetic Fields

To solve the problem, the outer surface of segment s is discretized with Ns rectangular panels,

as shown in Fig. 4.2a. Since the length of each segment is assumed to be electrically small,

the electric field and current flowing through each segment is approximately invariant along

the l-direction. This allows us to expand the electric field in the l-direction on the surface of

segment s as

E(s)l (~r) =

Ns∑n=1

E(s)n Π(s)

n (~r) , (4.2)

where Π(s)n (~r) is the n-th panel basis function on the s-th segment as defined in (2.41). The

electric field coefficients E(s)n in (4.2) for all segments of the conductor are collected into a vector

E =[E(1) T E(2) T . . . E(S) T

]T, (4.3)

where E(s) =[E

(s)1 E

(s)2 . . . E

(s)Ns

]Tcollects coefficients of electric field on the s-th segment.

Next, we expand the tangential magnetic field on the surface of each segment using panel

basis functions as

H(s)t (~r) =

Ns∑n=1

H(s)n Π(s)

n (~r) . (4.4)

Like the electric field coefficients, we collect the coefficients of magnetic field on the conductor

and segment s into vectors H and H(s), respectively, as done in (4.3).


4.3.2 Original Problem

Inside segment s, the electric field in the l-direction satisfies the Helmholtz equation [89]

∇2E(s)l (~r) + k2E

(s)l (~r) = 0 , (4.5)

subject to the Dirichlet boundary condition in (4.2). In (4.5), k =√ωµ (ωε0 − jσ) is the

wavenumber inside the conductor. Inside segment s, the position vector ~r and the Laplacian

operator ∇2 in (4.5) can be decomposed into their longitudinal and transverse components as

~r = ll + ~ρ and ∇2 = ∇2l +∇2

t . Due to our assumption ∂E(s)l (~r)/∂l = 0, we can reduce the 3-D

Helmholtz equation (4.5) to its 2-D counterpart

∇2tE

(s)l (~ρ) + k2E

(s)l (~ρ) = 0 . (4.6)

Equation (4.6) is identical to the 2-D Helmholtz equation (3.2). Hence, we can write its solution

using the contour integral equation method [119]

El(~ρ) =j

2

˛γ(s)

[∂G(~ρ, ~ρ ′)

∂n′El(~ρ

′)−G(~ρ, ~ρ ′)∂El(~ρ

′)

∂n′

]dρ′ , (4.7)

where γ(s) is the contour of the cross-section of segment s and G(~ρ, ~ρ ′) is the Green’s function

of the Helmholtz equation in 2-D with the material properties of the conductor. By substitut-

ing (4.2) and (4.4) into (4.7) and applying point-matching [105] to the resulting equation, we

obtain the discrete equation

H(s) = Y(s)E(s) , (4.8)

where Y(s) is a discretized admittance operator for segment s that relates the longitudinal

electric field to the tangential magnetic field. This surface admittance operator is exactly the

same as the 2-D operator derived in Chapter 3.

4.3.3 Equivalent Configuration

As in Chapter 3, we now apply the equivalence principle [83] to derive the surface formulation.

For this, we introduce an equivalent problem in which we replace the conductive medium in

segment s with the surrounding medium, as shown in Fig. 4.2b. We also enforce that the

electric field on the surface of segment s remains the same as in the original problem. In

this new equivalent configuration, the tangential magnetic field on the surface of segment s is

expanded as

H(s)eq,t(~r) =

Ns∑n=1

H(s)eq,nΠ(s)

n (~r) . (4.9)

Similar to (4.3), we collect expansion coefficients in (4.9) on the surface of the conductor and

of the s-th segment into vectors Heq and H(s)eq , respectively. In the equivalent configuration, we


can obtain a relation analogous to (4.8)

H(s)eq = Y(s)

eq E(s) , (4.10)

where Y(s)eq is the admittance operator of segment s for the equivalent configuration. The Y

(s)eq

operator is obtained by substituting (4.9) and (4.2) into the contour integral equation (4.7)

with the wavenumber k0 = ω√µ0ε0 of the surrounding medium and applying the MoM.

4.3.4 Equivalent Current

By replacing the conductive material in segment s by the surrounding medium, we have altered

the fields outside segment s. According to the equivalence principle [83], we restore the fields

outside segment s by introducing an l-directed equivalent electric current density

J (s)eq (~r) =

Ns∑n=1

j(s)n Π(s)

n (~r) , (4.11)

on the surface of segment s. According to the equivalence principle [83], the equivalent current

is

J (s)eq (~r) = H

(s)t (~r)−H(s)

eq,t(~r) . (4.12)

Next, we collect all coefficients of J(s)eq (~r) into J(s). By substituting (4.9) and (4.4) into (4.12)

and using (4.8) and (4.10), we obtain the relationship

J(s) =[Y(s) −Y(s)

eq

]︸︷︷︸

Y(s)∆

E(s) , (4.13)

where Y(s)∆ is the differential surface admittance operator of segment s.

We collect equivalent currents in all segments into vector J as in (4.3). The differential

surface admittance matrix Y∆ for the entire conductor is

J = Y∆E , (4.14)

where Y∆ is a block diagonal matrix in which the s-th block is Y(s)∆ . Since the 2-D differential

surface admittance operator only depends on the cross-sectional geometry of the segment, the

operator Y(s)∆ can be reused for all segments with an identical cross-section. This feature leads

to significant computational savings as discussed in Sec. 4.5.2.

We can compute the total current flowing through each segment using [29]

I(s)seg =

˛γ(s)

J (s)eq (~r)dr . (4.15)


After discretizing (4.15), we can compactly write

Iseg = UTWJ (4.16)

where W is a diagonal matrix that stores the width of each panel in the s-th segment for

s = 1, . . . , S, and U is of form

U =

1(1)

. . .

1(Ns)

, (4.17)

where 1(s) is a vector of size Ns×1 whose entries are all “1”. Equation (4.16) is of similar form

as (3.41) in the 2-D problem.

4.4 Impedance Calculation

At this point, we have replaced all conductor segments with equivalent currents on their surface.

We now apply the electric field integral equation to relate the currents to electric field and scalar

potentials. On segment s, the electric field integral equation for the l-direction is [83]

E(s)l (~r) = jωµ0

S∑q=1

‹γ(q)

J (q)eq (~r ′)G0(~r, ~r ′)dS′ − V (s)

seg , (4.18)

where the integration is performed over the surface of the segments, and

G0(~r, ~r ′) =1

4π |~r − ~r ′| (4.19)

is the quasi-static Green’s function that neglects time retardation effects.

Next, we substitute the expansion of electric field (4.2) and current (4.11) into (4.18) and

test the resulting equation with panel basis functions on all segments of the conductor to obtain

E = jωµ0GJ−UVseg . (4.20)

By substituting (4.14), (4.1), and (4.16) into (4.20), we obtain

Iseg = −UTWY∆(1− jωµ0GY∆)−1UZsegIseg , (4.21)

where 1 is the identity matrix. Since Zseg should be independent of current in each segment,

we obtain from (4.21)

Zseg =[−UTWY∆(1− jωµ0GY∆)−1U

]−1. (4.22)


30

205

1

5

1

Figure 4.3: Geometry of the loop conductor considered in Sec. 4.5.1. All dimensions are in micrometres.

We can compute the impedance seen from the port by enforcing that

• the current in each segment is equal to the current in the conductor (due to the MQS

assumption);

• the net voltage between the two terminals of the conductor is the sum of voltage drop

V(s)

seg across all segments.

It is important to clarify that we do not rigorously enforce continuity of electric current density

between connected segments. Instead, we model each segment of a 3-D conductor with an

impedance, enforcing electric current continuity at nodes common to two connected segments.

In a single conductor system, these conditions can be enforced through incidence matrix P

such that

Iseg = PI , (4.23a)

V = PTVseg , (4.23b)

where P is of size S × 1 with all entries being “1”. By left-multiplying (4.1) by PT and

right-multiplying it by P and using (4.23a)–(4.23b) we obtain the conductor’s port impedance

V

I= Z = PTZsegP . (4.24)


4.5.1 Loop Conductor

We consider the loop conductor in Fig. 4.3 made up of copper. The resistance and inductance

of the conductor calculated with FastHenry [46] and the proposed method are shown in Fig. 4.4.

The results obtained with the proposed method match well against FastHenry with the max-

imum error of 0.3% observed in inductance at low frequencies. This validates the proposed

method.


107

108

109

1010

1011

200

300

400

500

600

700

800

900

Frequency [Hz]

Res

ista

nce

[m

Ω]

FastHenry

Proposed

107

108

109

1010

1011

28

28.5

29

29.5

30

30.5

31

31.5

32

Induct

ance

[pH

]

Frequency [Hz]

FastHenry

Proposed

Figure 4.4: Resistance and inductance of the loop conductor considered in Sec. 4.5.1.

4.5.2 Chip package

EM solvers adopting the MQS assumption are commonly used to capture impedance matrix

of a pin package because such structures are electrically small [46]. We consider the 16-pin

package shown in Fig. 4.5 to demonstrate the numerical accuracy and efficiency of the proposed

formulation. This example has been adopted from the open source code for FastHenry [46].

All package leads are made up of 40 µm × 40 µm square copper conductors. We calculated

the impedance with the proposed formulation and with FastHenry [46]. In both methods, we

discretized each conductor with 3 segments. Selected entries of the resistance and inductance

matrices are shown in Fig. 4.6. It is evident from the plots that the proposed formulation agrees

very well with FastHenry.

We now compare the computational times and memory requirements of the proposed for-

mulation and FastHenry. In the proposed method, the final system of equations was solved by


dx

yz

pins 1-8pins 9-16 40

40

Figure 4.5: 16-pin package considered in Sec. 4.5.2. All pins are identical and have square cross-section.The first pin connects the following points in space (40, 20, 0) - (70, 20, 0) - (75, 0, 0) - (95, 0, 0). Thestructure is symmetric about the yz plane. The spacing between pins on the same side is d = 10. Alldimensions are in µm.

Table 4.1. Computational time and memory requirements for the chip package considered inSec. 4.5.2

FastHenry [46] - Direct FastHenry [46] - Iterative Proposed

Matrix fill time (one time) 49 s 61.9 s 104 sSolution time (per freq.) 1263 s 752 s 7 sTotal number of unknowns 10,800 10,800 1,920Memory requirement 1.46 GB 0.48 GB 0.25 GB

LU decomposition. FastHenry, on the other hand, was solved with both a direct solver and

an iterative solver that was accelerated with the FMM. Table 4.1 shows timing and memory

comparison between the two techniques at f = 10 GHz. At f = 10 GHz, the mesh in FastHenry

needs to be very fine to capture the skin effect. However, in the proposed method, the skin

effect is accurately captured with the 2-D surface operator. This difference leads to a much

lower memory consumption in the proposed method. The matrix fill time for the proposed

method is slightly higher than FastHenry, as we use numerical integration to populate matrix

G. However, as later discovered, analytic integration formulas are available to evaluate entries

of G [21]. Note that the matrix fill time is a one-time cost for the entire simulation. The

solution time for the proposed method is significantly lower than direct and an iterative Fas-

tHenry solver. Furthermore, since all conductors in the example have the same cross-section,

the differential surface admittance operator Y(s)∆ needs to be calculated only for one segment

and can be reused for all other segments.

4.6 Chapter Summary and Contribution

Many works in the literature have adopted the MQS assumption to model electrically small

3-D interconnects. In this chapter, we presented how the 2-D differential surface admittance

operator developed in Chapter 3 can be used to simulate 3-D interconnect problems under the

MQS assumption. The main idea behind the proposed approach is to partition 3-D conductors

into straight segments with a constant cross-section. Skin effect inside each segment is then


106

107

108

109

1010

0

50

100

150

R4,4

Frequency [Hz]

Res

ista

nce

[m

Ω]

FastHenry

Proposed

106

107

108

109

1010

−100

0

100

200

300

400

500

L4,4

L4,5

L4,14

L4,18

Frequency [Hz]

Induct

ance

[pH

]

FastHenry

Proposed

Figure 4.6: Selected entries of resistance and inductance matrices of the package considered in Sec. 4.5.2obtained with the proposed method and with FastHenry [46].

modeled with the 2-D differential surface admittance operator. The coupling between multiple

segments can be captured via the electric field integral equation. The presented technique is

analogous to the FastHenry solver [46], except it does not require volumetric mesh elements.

Hence, it is more efficient. Numerical results presented in this chapter showed an excellent

agreement between FastHenry and the proposed technique when simulating a chip package and

a loop conductor.

Chapter 5

Modeling Conductors and

Dielectrics with a 3-D Surface

Method

5.1 Introduction

So far, we have discussed the 2-D differential surface admittance operator to model conduc-

tors. This operator was used to analyze 2-D interconnects in Chapter 3 and 3-D interconnects,

adopting the magneto-quasistatic assumption, in Chapter 4. In the presented surface method,

each conductor was modeled with an equivalent electric current density and the 2-D differential

surface admittance operator. In this chapter, we generalize the differential surface admittance

operator concept to 3-D conductors and dielectrics of arbitrary shape. Similar to the 2-D

problem, the 3-D differential surface admittance operator is derived using the surface integral

equation method. The proposed surface method is adopted to solve two types of problems:

computing the scattering matrix of a multi-port interconnect network and computing the radar

cross-section of a dielectric object. In order to accurately solve these two problems, electromag-

netic fields must be described inside dielectric and conducting objects (interior problem) and

outside (exterior problem).

As discussed in the introduction (Sec. 1.2.3), surface methods have been proposed in the past

to model 3-D penetrable objects [25, 26]. These methods are based on the Love’s equivalence

principle and are similar to the PMCHWT formulation [21] that was discussed in Sec. 2.4 for

dielectric objects. Therefore, these methods require solving for both the equivalent electric and

magnetic current densities on the surface of each object, which, depending on the choice of

basis functions, requires solving for two or more sets of unknowns. On the other hand, the

surface formulation proposed in this chapter is a single-source formulation that requires only

one unknown per surface edge, namely an equivalent electric current density. The presented

single-source formulation is more efficient than dual-source surface methods because it requires

80

Chapter 5. A 3-D Surface Method to Model Dielectrics and Conductors 81

nV(µ, ε)S

µ0, ε0


~Jeq

(µ0, ε0)

µ0, ε0


Figure 5.1: (a) Original configuration: a sample 3-D lossy dielectric object. (b) Equivalent configuration:the original object is replaced by the surrounding medium and an equivalent current density is introducedon S.

less time to assemble and solve the linear system of equations. In the past, single-source surface

methods have been applied to model dielectrics and conductors [25, 137]. However, unlike the

presented technique which physically eliminates the equivalent magnetic current density by

means of the equivalence principle, these techniques mathematically eliminate the additional

unknowns to achieve a single-source formulation. Therefore, in comparison to the proposed

method, these single-source methods require discretizing additional integral operators for the

exterior problem.

This chapter is organized as follows. In Sec. 5.2, we discuss the numerical procedure to com-

pute the differential surface admittance operator for 3-D conductors and dielectrics of arbitrary

shapes. In Sec. 5.3, we discuss how to use the surface operator to compute the scattering matrix

of a multiport interconnect network. In Sec. 5.4 we present numerical examples that validate

the proposed method for interconnect modeling problems. Next, in Sec. 5.5, we show how to

use the surface operator to compute scattering from dielectric objects. Finally, we validate the

proposed method for scattering problems in Sec. 5.6.

5.2 Differential Surface Admittance Operator

We consider a lossy dielectric object with an arbitrary shape as shown in Fig. 5.1a. The volume

of the object is denoted by V and its boundary is denoted by S. The dielectric object has permit-

tivity ε and permeability µ. The background medium is assumed to be free space for simplicity,

although the formulation also works for a stratified background medium as demonstrated by

the numerical example in Sec. 5.4.2. For interconnect modeling or scattering problems, we need

to properly model electromagnetic effects inside and outside S. As in the 2-D problem, we will

model the inner problem through the 3-D differential surface admittance operator.


5.2.1 Discretization of Fields and Currents

We discretize S with triangular mesh elements. The tangential electric and magnetic fields on

S are then expanded with RWG basis functions [68]

n× ~E(~r) =

Ne∑n=1

en~Λn(~r) (5.1a)

n× ~H(~r) =

Ne∑n=1

hn~Λn(~r) , (5.1b)

where n is the normal unit vector pointing into V and Ne is the number of edges on S. The

RWG basis functions are defined in Sec. 2.3.2. Expansion coefficients in (5.1a) and (5.1b) are

collected into vectors E =[e1 . . . eNe

]Tand H =

[h1 . . . hNe

]T.

5.2.2 Integral Equation for the Original Problem

For 2-D problems, we related the electric and magnetic fields on an object’s boundary through

the contour integral method, which was obtained by applying Green’s identity to the 2-D

Helmholtz equation. In this chapter, we generalize that idea to 3-D by using the Stratton-Chu

formulation that was introduced in Sec. 2.2.1. Recall that the Stratton-Chu formulation was

also derived by applying Green’s identity on the 3-D vector Helmholtz equation.

According to the Stratton-Chu formulation, the tangential electric and magnetic fields on

S are related by [21]

jωµ n× n×[~Lε(n× ~H(~r)

)](~r) + n× n×

[~Kε(−n× ~E(~r)

)](~r)

= −n× n× ~E(~r) , (5.2a)

jωεηn× n×[~Lε(n× ~E(~r)

)](~r) + n× n×

[~Kε(n× η ~H(~r)

)](~r)

=n× n× η ~H(~r) , (5.2b)

where k = ω√µε and η =

√µ/ε are, respectively, the wavenumber and wave impedance of

the medium inside the object under consideration. Integral operators ~Lε and ~Kε are defined

by (2.15a)–(2.15b) with the homogeneous Green’s function of the medium of the object. The

Stratton-Chu formulation in (5.2a)–(5.2b) can also be seen as the tangential electric field integral

equation (T-EFIE) (2.49a) and the tangential magnetic field integral equation (T-MFIE) (2.49b)

with fields as the arguments instead of equivalent currents. Both the T-EFIE and T-MFIE relate

the electric and magnetic fields on the boundary of the object.

Now, we discretize the Stratton-Chu formulation. This step requires a careful choice of

testing procedure such that both the ~Lε and ~Kε are well-tested. As such, we introduce the

combined field integral equation [21] by multiplying (5.2a) by a constant α and adding the

result to (5.2b) multiplied by 1−α [92]. The resulting equation is then tested with RWG basis


functions ~Λm(~r) for m = 1, . . . , Ne to obtain the matrix equation

KεE + LεH = 0 . (5.3)

In (5.3), the (n, n′)-th entry of Lε is

[Lε]n,n′ =jωµα⟨~Λn(~r),

[~Lε~Λn′(~r ′)

](~r)⟩

+ (1− α)η⟨~Λn(~r),

[~Kε~Λn′(~r ′)

](~r)⟩

− (1− α)η⟨n× ~Λn(~r), ~Λn′(~r)

⟩, (5.4)

and the (n, n′)-th entry of Kε is

[Kε]n,n′ =jωε(1− α)η⟨~Λn(~r),

[~Lε~Λn′(~r ′)

](~r)⟩− α

⟨~Λn(~r),

[~Kε~Λn′(~r ′)

](~r)⟩

+ α⟨n× ~Λn(~r), ~Λn′(~r)

⟩. (5.5)

For highly conductive objects, α = 0 works well since both Kε and Lε are well-conditioned.

Operator Kε is well-conditioned because the first term in (2.17) dominates the second-term.

Furthermore, even though the second term is not full rank, the first term is full rank and well-

conditioned [92]. Therefore, ~Kε is well-tested for highly-conductive objects. For a dielectric

object, however, we must choose 0 < α < 1 in order to correctly model effects of the tangential

electric and magnetic fields [21]. In all our tests, we used α = 0.5 to obtain good results.

Matrix Lε is full rank and well-conditioned, so we can express the tangential magnetic field

in (5.3) as

H = − [Lε]−1 Kε︸︷︷︸

Yin

E , (5.6)

where Yin is the surface admittance matrix of the object. Notice that (5.6) is analogous

to (3.29).

5.2.3 Equivalent Problem

Next, the equivalence principle [89] is applied to replace the object by its surrounding medium

and an equivalent electric current density ~Jeq(~r), as shown in Fig. 5.1b. The tangential electric

field on S in the equivalent configuration is enforced to be the same as in the original problem,

i.e. n × ~E(~r) on S remains the same as the original problem. The tangential magnetic field

n× ~Heq(~r) on S in the equivalent problem is expanded with RWG basis functions

n× ~Heq(~r) =

Ne∑n=1

heq,n~Λn(~r) , (5.7)

and the expansion coefficients heq,n are stored in vector Heq.

Similar to (5.2a)–(5.2b), we again relate the tangential electric and magnetic fields on S in


the equivalent problem by the Stratton-Chu formulation

jωµ0 n× n×[~L0

(n× ~Heq(~r)

)](~r) + n× n×

[~K0

(−n× ~E(~r)

)](~r)

=− n× n× ~E(~r) , (5.8)

jωε0η0n×[~L0

(n× ~E(~r)

)](~r) + n×

[~K0

(n× η0

~Heq(~r))]

(~r) = n× η0~Heq(~r) , (5.9)

where k0 = ω√µ0ε0 and η0 =

√µ0/ε0 are the wavenumber and the intrinsic impedance of

the background medium. Furthermore, integral operators ~L0 and ~K0 are evaluated with the

homogeneous Green’s function of the background medium.

In order to properly test all operators and avoid interior resonances, we again multiply (5.8)

by α and add the resulting equation to (5.9) multiplied by 1− α to obtain the combined field

integral equation for the equivalent problem. We test the combined field integral equation with

~Λm(~r) to obtain linear system of equations

K0E + L0Heq = 0 , (5.10)

where entries of L0 and K0 are similar to (5.4)–(5.5), except they are evaluated with material

properties of the exterior medium. We explicitly relate the tangential electric and magnetic

fields as

Heq = −L−10 K0︸︷︷︸

Yout

E , (5.11)

where Yout is the surface admittance operator of the object with material parameters of the

surrounding medium.

5.2.4 Equivalent Electric Current Density

According to the equivalence principle, the equivalent electric current density

~Jeq(~r) = n×[~Heq(~r)− ~H(~r)

], (5.12)

restores electromagnetic fields outside S. Since we have maintained the electric field in the

equivalent problem to be the same as the original problem, we do not have to introduce an

equivalent magnetic current density on S. We expand ~Jeq(~r) with RWG basis functions,

~Jeq(~r) =

Ne∑n=1

jn~Λn(~r) , (5.13)

and store the coefficients jn into a vector J. Since all three field quantities in (5.12) are expanded

with RWG basis functions, we can express the discretized electric current density as

J = Heq −H . (5.14)


i(2)i(1)

σS

S(2)

S(1)

µ0, ε0

(a) Original problem

i(2)i(1)

~Jeq(~r)

µ0, ε0

(b) Equivalent problem

Figure 5.2: (a) Original problem: example of a U-shaped conductor. Dark gray surfaces indicate theterminals of the conductor with terminal currents i(1) and i(2). (b) Equivalent problem: the originalconductor is replaced by the surrounding medium and an equivalent current density (shown with reddots) is introduced on the surface S.

Finally, by substituting (5.11) and (5.6) into (5.14) we obtain

J = [Yout −Yin]︸︷︷︸Y∆

E , (5.15)

where Y∆ is the differential surface admittance operator that accurately models electromagnetic

fields inside the penetrable object. Note that (5.15) is similar to (3.31) in the 2-D formulation.

This differential surface operator does not require a volumetric mesh and is applicable to objects

of arbitrary geometries. The matrix Y−1∆ can be thought of as a generalization of the surface

impedance boundary [25] condition that is accurate from DC to very high frequencies.

Evaluation of Yin and Yout requires the LU factorization of a full matrix. When simu-

lating an array of penetrable objects, the cost of LU factorization is typically much smaller

than the cost of solving the entire problem. When simulating large objects, we can partition

the object into multiple smaller objects, computing the differential surface admittance oper-

ator independently. In this approach, for accurate solution, we need to enforce continuity of

tangential electric and magnetic fields between connected partitions [138]. More recently, a

single-source formulation based on the differential surface admittance has been proposed with-

out the need for LU factorization [139]. Furthermore, this approach has been integrated with

efficient preconditioners.

In the case where the object is surrounded by a stratified medium, we replace the object with

the background layer during the application of the equivalence theorem. If the object traverses

multiple layers of the stratified medium, then it is divided into multiple smaller objects such

that each object is surrounded by a single layer. Each smaller object is then replaced by its

surrounding layer and equivalent current is introduced on the boundaries of each smaller object.

5.3 Interconnect Modeling

Let us consider the problem of computing the multiport impedance or scattering matrix of a

system composed of multiple conductors. For this problem, we need to introduce ports to our


model. For illustration, let us consider the one conductor system shown in Fig. 5.2a. This

system is composed of a single port, defined between two faces S(1) and S(2), called terminals.

Both S(1) and S(2) are assumed to be electrically small with a constant electric potential. Let us

denote the scalar potential and current through the t-th terminal by v(t) and i(t), respectively,

for t = 1, 2. Port voltage and current are denoted as v = v(2) − v(1) and i = i(2) = i(1),

respectively. Our goal is to obtain the port impedance z = v/i using the surface formulation.

To solve this problem, we first apply the equivalence principle and replace the conductor by

the surrounding free space medium and ~Jeq(~r) on S. By solving the interior problem, we obtain

the differential surface admittance operator Y∆ of the conductor to relate ~Jeq(~r) to n × ~E(~r)

on S. This equivalent problem is shown in Fig. 5.2b. For the outer problem, we combine Y∆

with the electric field integral equation and boundary conditions at the terminals.

5.3.1 Boundary Conditions on the Terminals

Since the conductor is excited by a port, in addition to ~Jeq(~r), there exists a volume current

density J(t)src(~r) flowing into the surface of each terminal S(t) [140]. The port volume current

density is expanded with panel basis functions on the t-th terminal as

J (t)src(~r) =

1

S(t)n

Nt∑n

i(t)n Π(t)n (~r) , (5.16)

where S(t)n is the area of the n-th triangle of the t-th terminal. Note that the panel basis function

Π(t)n (~r) in (5.16), which is defined over a surface, is different than the pulse basis function used

for 2-D problem in Chapter 3. Expansion coefficients in (5.16) are stored into vector Jsrc. The

total current injected into the t-th terminal can be obtained by integrating J(t)src over S(t) and

is given by

i(t) =

Nt∑n=1

i(t)n . (5.17)

Due to the presence of volume current density, the modified continuity equation on S(t)

is [140]

ρ(~r) =J

(t)src(~r)−∇ · ~Jeq(~r)

jω, (5.18)

where J(t)src(~r)/(jω) represents a source charge density that is introduced on the t-th terminal’s

surface.


5.3.2 Electric Field Integral Equation

Next, we relate the electric field on S to the ~Jeq(~r) and J(t)src(~r) by the EFIE

n× ~E(~r) = −jωµ0n×[~L0~Jeq(~r ′)

](~r)−

T∑t=1

n×∇[T0

(J

(t)src(~r ′)

jω

)], (5.19)

where T = 2 for the one conductor system show in Fig. 5.2a. Operator T0 gives the scalar

potential due to a surface charge density X(~r) [140] and is defined to be

T0 (X(~r)) =1

ε0

¨SG0(~r, ~r ′)X(~r ′)dS′ . (5.20)

Operators ~L0 and T0 use the Green’s function of the surrounding medium. Note that the

second term in (5.19) is due to source volume current density at all port terminals [140]. We

discretize the EFIE, by substituting (5.1a), (5.13), and (5.16) into (5.19), and testing the

resulting equation with orthogonal RWG basis functions n × ~Λn(~r). The resulting equations

can be written in the matrix form

DE = −BJ−TJsrc , (5.21)

where matrices D and B are of the size Ne ×Ne. The (n, n′)-th entry of D and B is given by

[D]n,n′ =⟨n× ~Λn(~r), ~Λn′(~r)

⟩(5.22a)

[B]n,n′ = jωµ0

⟨n× ~Λn(~r), n×

[~L0~Λn′(~r

′)]

(~r)⟩. (5.22b)

Matrix T in (5.21) is of the form T =[T(1) . . . T(T )

], where the t-th block T(t) is of the

size Ne × N (t). The (n, n′)-th entry of T(t) is given by[T(t)

]n,n′

=1

jω

⟨n× ~Λn(~r), n×∇T0

(Π(t)n (~r ′)

)⟩. (5.23)

Next, we substitute (5.15) into (5.21) to obtain

(D + BY∆) E + TJsrc = 0 . (5.24)

5.3.3 Potential Integral Equation

Finally, on the n-th triangle of S(t), the potential is given by [140]

v(t)n (~r) = T0

(∇′s · ~Jeq(~r ′)

−jω

)+ T0

(J

(t)src

jω

). (5.25)


Figure 5.3: Top-view of the 3-D inductor considered in Sec. 5.4.1.

We test (5.25) with Π(t)n (~r) for t = 1, . . . , T to obtain the following system of linear equations

V = RJsrc + WJ , (5.26)

where R and W are the MoM matrices obtained by discretizing (5.25), and the vector V collects

v(t)n on all terminals. To eliminate J, we substitute (5.15) into (5.26) to obtain

V = RJsrc + WY∆E . (5.27)

5.3.4 Final System of Linear Equations

Assembling (5.24) and (5.27), we obtain the linear system[D + BY∆ T

WY∆ R

][E

Jsrc

]=

[0

V

]. (5.28)

Given port voltage v, we solve (5.28) for Jsrc and E using LU factorization. Finally, terminal

currents are calculated from (5.17) and the interconnect impedance is found as z = v/i. For a

multiport network with P ports, (5.28) is solved for P right-hand sides.

5.4 Numerical Results for Interconnect Modeling

We consider two examples to demonstrate accuracy and efficiency of the proposed surface

method compared to a commercial FEM solver.

5.4.1 3-D Inductor

We validate the proposed method for conductor modeling by considering the two-port induc-

tor [47] shown in Fig. 5.3. The inductor is surrounded by free space. Figure 5.4 shows the


5 10 15 20 25 30−30

−20

−10

0

Frequency [GHz]

|S11|[

dB

]

HFSSSIBCProposed

−5

−4

−3

−2

−1

0

|S12|[

dB

]

S11

S12

Figure 5.4: Selected entries of the scattering matrix of the 2-port inductor considered in Sec. 5.4.1calculated with the SIBC method, the proposed method, and FEM (Ansys HFSS).

Table 5.1. Simulation Statistics for the Example considered in Sec. 5.4.1

Metric HFSS SIBC Proposed

Mesh elements 61,368 756 756Degrees of freedom 388,480 1,280 1,280Memory used [GB] 3.81 0.049 0.14Simulation time [s] 205 9.04 18.49

scattering matrix of the two-port inductor calculated using three methods: surface impedance

boundary condition (SIBC) [24], FEM (Ansys HFSS), and the surface formulation described in

Sec. 5.3. Based on Fig. 5.4, we can make the following observations. First, at low frequencies

(1 GHz − 10 GHz), the proposed method matches very well against Ansys HFSS. The SIBC

method gives inaccurate results because the skin effect has not yet fully developed [140]. Sec-

ond, at high frequencies (20 GHz − 30 GHz), the proposed method follows very closely the

SIBC method, which is known to be accurate at high frequencies since the skin effect is well

developed. The HFSS result, instead, begins to deviate at high frequencies, since the volumetric

mesh cannot resolve the decreasing skin depth, which is small due to the high conductivity of

copper. Table 5.1 summarizes memory consumption, problem size and CPU time for the three

methods. In comparison to the HFSS, the proposed method is 11 times faster and requires 27

times less memory.

5.4.2 Multiconductor Bus in a Stratified Medium

Finally, let us consider the multiconductor bus shown in Fig. 5.5. The bus is composed of 22

conductors, including ten conductors with bends along their axial direction. The system is


Port 1

Port 2

7.5 mm

9.0 mm

Figure 5.5: Multiconductor bus considered in Sec 5.4.2. The bus is composed of 22 conductors. Eachconductor has cross-section of 1 mm × 1 mm. Right panel shows current distribution when port 1 isexcited

.

Table 5.2. Simulation Statistics for the Example Considered in Sec. 5.4.2

HFSS Proposed Gain

Unknowns 1.1 M 7,624Memory 2.78 GB 0.34 GB 8.2×CPU time 438 s 172 s 2.6×

inside the multilayered medium shown in Fig. 5.6. The effects of fields inside each conductor is

captured accurately through the differential surface admittance operator. Unlike the rest of the

examples in this thesis that use homogeneous Green’s function and the equivalence principle

to model the dielectric, this example captures the effects of multilayer background medium via

the multilayer Green’s function. Details on how to integrate the multilayer Green’s function

with the proposed method can be found elsewhere [141]. For this problem, we solved the final

system (5.28) iteratively with matrix-vector product accelerated using the adaptive integral

method [141].

For validation purpose, we introduced two ports to the system as shown in Fig. 5.5. Fig-

ure 5.7 shows S11 and S12 computed with the proposed method and Ansys HFSS. Accuracy of

the results validate the proposed technique. Computational memory usage and time required

to simulate the structure are summarized in Tab. 5.2. In terms of computational savings, the

proposed method required 8 times lower memory and 3 times lower computational time. Mem-

ory savings and computational savings with the proposed method would be even higher for

electrically large structures where acceleration algorithms can be further exploited to speed up

matrix vector products. For problems with a multilayer background medium, the evaluation

of the multilayer Green’s function is expensive. However, the absence of a magnetic equivalent

current density in the proposed formulation results in substantial computational savings com-

pared to other surface methods that require multilayer Green’s function for both electric and

magnetic current sources.


40 μm

50 μm

Air

PEC

εr = 2.1

εr = 9.8

ç

80 μm

εr = 12.5

ç

ç

Figure 5.6: Background medium for the example considered in Sec. 5.4.2. Dashed boxes show theposition of the top five, middle twelve, and bottom five conductors in the stratified medium.

0 10 20 30 40

−30

−20

−10

0

S11

S12

f (GHz)

dB

ProposedHFSS

Figure 5.7: S11 and S12 of the scattering matrix for the system considered in Sec. 5.4.2.

5.5 Scattering from Dielectric Objects

Let us consider scattering from the dielectric object shown in Fig. 5.8a. To solve this scatter-

ing problem, we first apply the equivalence principle and replace the dielectric object by the

surrounding free space medium and an equivalent electric current density on S. By solving

the interior problem, we obtain the differential surface admittance operator Y∆ to relate the

equivalent electric current density to the tangential electric field on S. We can obtain an ad-

ditional relationship between the tangential electric field, equivalent electric current density,

and incident electromagnetic fields by solving the outer problem. In the outer problem, the

equivalent current density relates to the tangential electric field on S by [21]

n× n× ~E(~r) = −jωµ0n× n×[~L0~Jeq(~r ′)

](~r) + n× n× ~Einc(~r) , (5.29)

and the tangential magnetic field on S by

n× ~Heq(~r) = n×[~K0~Jeq(~r ′)

](~r) + n× ~H inc(~r) , (5.30)


nV(µ, ε, σ)S

( ~Einc, ~H inc) ( ~Escat, ~Hscat)

µ0, ε0


~Jeq

(µ0, ε0)

( ~Einc, ~H inc) ( ~Escat, ~Hscat)

µ0, ε0


Figure 5.8: (a) Original configuration: a sample 3-D lossy dielectric object. (b) Equivalent configuration:the original object is replaced by the surrounding medium and an equivalent current density is introducedon S.

where ~Einc(~r) and ~H inc(~r) represent the incident electric and magnetic fields, respectively. Note

that (5.29) and (5.30) can be interpreted as the EFIE and the MFIE without the presence of the

magnetic current density, which does not exist in our single-source surface formulation. Now

we substitute (5.13), (5.7) and (5.1a) into (5.29) and (5.30), and test the resulting equation

with RWG basis functions to obtain

DE = LoutJ + Ve , (5.31a)

DHeq = KoutJ + Vm , (5.31b)

where matrices D, D, Lout, and Kout are generated by testing (5.29) and (5.30) by RWG

functions. The (n, n′)-th entry of D, D, Lout, and Kout are

[D]n,n′ =⟨~Λn(~r), n× ~Λn′(~r)

⟩(5.32a)[

D]n,n′

=⟨~Λn(~r), ~Λn′(~r)

⟩(5.32b)

[Lout]n,n′ = −jωµ0

⟨~Λn(~r), n× n×

[~L0~Λn′(~r

′)]

(~r)⟩

(5.32c)

[Kout]n,n′ =⟨~Λn(~r), n×

[~K0~Λn′(~r

′)]

(~r)⟩. (5.32d)

Vectors Ve and Vh are generated by testing the incident tangential electric and magnetic fields

by the RWG functions. Their n-th entry is given by

[Ve]n =⟨~Λn(~r), n× n× ~Einc

⟩(5.33a)

[Vh]n =⟨~Λn(~r), n× ~H inc

⟩. (5.33b)

Next, we substitute (5.15) and (5.11) into (5.31a) and (5.31b) to eliminate Heq and J to


obtain

DE = LoutY∆E + Ve (5.34a)

DYoutE = KoutY∆E + Vh . (5.34b)

In order to avoid numerical issues such as interior resonances in lossless dielectrics [23], we

combine (5.29) and (5.30) to form the combined field integral equation. In the discrete domain,

the combined field integral equation is obtained by multiplying (5.34a) by α and adding the

result to (5.34b) multiplied by 1 − α as done in (5.3), and then solve the resulting equation

for E. Typically, α is chosen to be 0.5. If the penetrable object is a thin dielectric, then

resonance is not an issue, and α can be chosen to be 1, which will eliminate the need to

use (5.34b). Consequently, in this case, we would not have to discretize the ~K operator for the

outer problem. This is one of the major advantages of the proposed single-source formulation

compared to other works in the literature [25, 137]. After solving for E, we can compute the

equivalent current density using (5.15) and the scattered field based on the value of ~Jeq(~r).


We validate the proposed surface method for scattering problems by considering two examples.

5.6.1 Dielectric Sphere

First, let us consider scattering from a dielectric sphere with relative permittivity εr = 2.25 and

radius R = 0.5 m. The sphere is discretized with 1220 triangular elements. For this problem,

we computed the radar cross-section

σRCS(θ, φ) = 4πR2FF

∣∣∣ ~Escat

∣∣∣∣∣∣ ~Einc

∣∣∣ , (5.35)

by exciting the sphere with a −z traveling plane wave with∣∣∣ ~Einc

∣∣∣ = 1 V/m. In (5.35), RFF is

the far-field distance at which ~Escat is computed. For our problem, the RCS was computed with

the analytic method based on the solution of fields in spherical coordinates [98], the PMCHWT

formulation [21], the PMCHWT formulation with Schur complement [25], and the proposed

method over the frequency range 30-400 MHz. In the proposed approach, a total of Ne = 1830

RWG basis functions were used to expand the equivalent electric current density in (5.13). The

PMCHWT formulation required discretization of both electric and magnetic current density,

each with 1830 RWG basis functions. Fig. 5.10 shows an excellent agreement between RCS

computed with all four methods, which validates the accuracy of the proposed method over a

wide frequency range. While the Schur complement approach works well for conductors [25],

it suffers from numerical resonances when applied to lossless dielectrics, as seen in Fig. 5.10 at


(a) Tangential electric field (b) Tangential electric field

Figure 5.9: Tangential electric and magnetic fields on the surface of a dielectric interface at f = 80 MHz.

Table 5.3. Simulation Statistics for the Example in Sec. 5.6.2

PMCHWT Proposed

Matrix-fill time (s) 467 531Solution time (s) 1278 168Total time (s) 1745 699Memory used 13.08 GB 3.47 GB

370 MHz. Using the combined field integral equation approach may eliminate these numerical

errors [137]. Fig. 5.9 shows the tangential electric and magnetic fields on the surface of the

sphere at f = 80 MHz. It is important to note that the proposed method does not suffer from

interior resonances because we use the combined field integral equation. If we compute the RCS

using the proposed method but with scaling factor α = 0 or α = 1 in (5.4)–(5.5), then the RCS

is inaccurate in the frequency range near the resonant frequency of the PEC sphere.

5.6.2 4 × 4 Array of Dielectric Spheres

Next, we consider a 4 × 4 array of dielectric spheres placed in the x-y plane. The array is

uniformly spaced along the x and y directions with inter-element spacing of dx = dy = 2 m.

Geometrical and material properties of each sphere are the same as in Sec. 5.6.1. The array

of dielectric spheres was discretized with 9, 872 triangular elements. The electric and magnetic


50 100 150 200 250 300 350 40010

−3

10−2

10−1

100

101

Frequency [MHz]

4 π

R2 |

Esc

at|2

Analytic

PMCHWT + Schur

PMCHWT

Proposed

Figure 5.10: Scattered electric field in the example considered in Sec. 5.6.1 calculated with analyticformulas, the PMCHWT formulation [21], the PMCHWT formulation with Schur complement reduc-tion [25], and the proposed method.

equivalent current densities in the PMCHWT formulation and the equivalent electric current

density in the proposed method were each discretized with 14, 808 RWG elements. Figure 5.11

shows the scattered electric field as a function of elevation angle for φ = 0 and φ = π/4 when the

array is excited by a −z directed plane wave at f = 200 MHz. It is evident that the proposed

method is accurate when compared against the PMCHWT method. The computational times

and memory requirements for both techniques are summarized in Table 5.3. All computations

were performed with a single thread on a system with a 2.5 GHz CPU and 16 GB of memory.

The proposed method and the PMCHWT formulation both required almost the same amount of

time to generate the system matrix. The matrix fill time includes the time required to generate

the surface admittance operator. The solution time for the proposed method, however, is almost

8 times faster than the PMCHWT approach due to the absence of a magnetic equivalent current

density in the proposed method. The proposed approach also requires four times less memory

than the PMCHWT approach, for the same reason.


In this chapter, we generalized the differential surface admittance approach to model 3-D di-

electric and conductive objects. In our approach, the dielectric and conductive objects were

modeled by an equivalent electric current density that was introduced on the object’s surface.

This equivalent current density was related to the electric field on the surface via the differential

surface admittance operator, which was obtained numerically via the Stratton-Chu formulation.

The developed surface method was successfully applied to two types of problems: scattering

problems and interconnect modeling problems. It is worth emphasizing that the proposed

method has the following features:

• The technique is applicable to model conductors and dielectric objects of arbitrary shapes.


−150 −100 −50 0 50 100 150

100

102

104

Theta [degrees]

4π

R2 |E

scat

|2 [

V]

PMCHWT

Proposed

−150 −100 −50 0 50 100 15010

1

102

103

104

Theta [degrees]

4π

R2 |E

scat

|2 [

V]

PMCHWT

Proposed

Figure 5.11: Scattered electric field in the example considered in Sec. 5.6.2 obtained with the PMCHWTformulation and the proposed method for φ = 0 (top panel) and φ = 45 (bottom panel) cuts.

It can also model objects that are embedded inside a homogeneous or a stratified medium.

• Absence of volumetric mesh makes the technique efficient compared to volumetric methods

in literature and in commercial solvers. This is especially true for conductors because the

mesh does not have to be adapted to properly model the skin.

• The proposed method is a single-source method, which models each object with only an

equivalent electric current density. Absence of an equivalent magnetic current density

from the formulation makes the technique more efficient in terms of memory usage and

computation time than other surface formulations in the literature, especially when the

object is embedded in a stratified medium. The single-source formulation is also interest-

ing from a theoretical viewpoint.

Chapter 6

A Macromodeling Approach for

Complex PEC Scatterers

6.1 Introduction

In Chapters 3–5, we discussed efficient electromagnetic simulation techniques for interconnects

and simple scatterers. Now, let us address the challenge of simulating an array of complex

PEC scatterers. An array of sample complex PEC scatterers is shown in Fig. 6.1. Electrically

large arrays of PEC scatterers with multiscale features are difficult to simulate with the existing

surface integral equation methods because their simulation requires a large amount of memory

and long computation times. Due to a small mesh size of the array elements, even acceleration

algorithms are ineffective at simulating such multiscale problems because of the long times

required to compute near-field interactions. Furthermore, multiscale features in the simulation

domain lead to a poor condition number of the linear system of equations, which degrades

the convergence of iterative solvers. In Chapters 3 and 5, we discussed single-source surface

methods to simulate 2-D and 3-D interconnects and dielectric objects. These methods can be

classified as reduced-order modeling techniques because we model electromagnetic effects inside

a volumetric object in terms of surface unknowns. These algorithms are, however, restricted to

homogeneous objects. In this chapter, we investigate how the presented surface formulation for

conductors and dielectrics can be further generalized to develop a single-source macromodeling

technique for complex PEC scatterers. The proposed macromodeling technique will allow us

to efficiently simulate an array of complex PEC scatterers.

The proposed macromodeling approach can be classified as a form of the equivalence prin-

ciple algorithm (EPA) [73–82] that was briefly discussed in Sec. 1.3.4. In the proposed method,

each element of the array is modeled by an equivalent electric current density that is introduced

on a fictitious surface enclosing the element. This equivalent electric current density is related

to the tangential electric field on the equivalent surface through a surface operator, which can

be interpreted as a macromodel for the scatterer. For simulation of an array of complex PEC

97

Chapter 6. A Macromodeling Approach for Complex PEC Scatterers 98

Top view 3-D view Equivalent setup

(a) Single Element

(b) 4× 4 Array

Figure 6.1: (a): A complex scatterer composed of two-layers of rectangular loop with meander lines ismodeled by an equivalent electric current density introduced on a fictitious (box) surface enclosing thescatterers. The figure shows triangular mesh elements needed to accurately model fields and currents onthe scatterer and the equivalent surface. (b): An array of scatterers is modeled by an equivalent electriccurrent density introduced on an array of fictitious surfaces.

scatterers, the macromodeling approach has the following advantages:

• The macromodeling approach requires solving for fewer unknowns than the original prob-

lem. In the original problem, the unknowns are coefficients of the actual electric surface

current density on the scatterer. This forces us to mesh all metallic elements. As seen in

Fig. 6.1, this approach results in a lot of mesh elements and unknowns if the scatterers

have electrically fine features. On the other hand, in the macromodeling approach, the un-

knowns are only on the equivalent surface. The equivalent surface requires coarser mesh,

as shown in Fig. 6.1. Hence, for complex scatterers with multiscale features, the number

of unknowns on the equivalent surface could be significantly lower than the number of

unknowns on the scatterer.

• In comparison to the standard MoM, the macromodeling approach improves the condi-

tion number of the final linear system because the unknowns are only on the equivalent

surface, which has no fine features. In the surface integral equation method, multiscale


features worsen conditioning. However, the macromodeling approach effectively “hides”

local features from the solution of the global problem. Better conditioning is also achieved

in part by employing special (dual-RWG) basis functions to model the tangential electric

field on the equivalent surfaces.

• The proposed macromodeling approach exploits repeatability of elements in the array.

That is, a macromodel generated for one element can be reused for other identical elements

in the array, which leads to significant computational and memory savings.

In contrast to other EPAs discussed in the literature, which require both an equivalent

electric and magnetic current density to model the scatterer, the proposed method applies

the equivalence principle in a unique way such that the scatterer can be modeled by only an

equivalent electric current density. The absence of an equivalent magnetic current density leads

to solving fewer unknowns overall. In the literature, a single-source EPA has been proposed

where half of the unknowns are eliminated mathematically from the original EPA by relating

the discretized equivalent magnetic current density to the discretized electric current density by

the generalized impedance boundary conditions [142, 143]. While this approach eliminates the

magnetic current density from the EPA in the discrete domain, the magnetic current density

still exists in the continuous domain1. In the proposed formulation, however, the equivalent

magnetic current density is not required whatsoever. As a consequence, the exterior problem

in the proposed formulation is simpler with fewer integral operators than even the single-

source EPA [142,143]. Furthermore, both the EPA and the proposed method derive a different

macroscopic quantity to model interactions between the scatterer. While the EPA derives a

scattering matrix to relate the incident tangential electric and magnetic fields to the scatterered

tangential electric and magnetic fields, the proposed method derives an admittance matrix that

relates the equivalent electric current density to the tangential electric field on the equivalent

surface.

This chapter is organized as follows. First, we provide the mathematical framework to create

a macromodel for all elements of an array in Sec. 6.2. Once all macromodels are generated,

the coupling between them is captured by the electric field integral equation, as discussed in

Sec. 6.3. To tackle electrically large problems, the proposed approach is accelerated with the

adaptive integral method (AIM), as discussed in Sec. 6.4. Finally, in Sec. 6.5, we present three

numerical examples to show the accuracy and efficiency of the proposed method.

6.2 Macromodel Generation

We consider the problem of computing scattering from an M -element array of complex PEC

scatterers in free space using a macromodeling approach. As demonstrated in Fig. 6.1, in

1To compute radiated fields from the scatterer we need to first solve for the magnetic current density by Schurcomplement, and then compute radiation due to both the magnetic and electric current densities.


Sbox,m

Spec,m


Sbox,m


Figure 6.2: (a): Original configuration: Sample unit cell made up of metallic scatterers enclosed by theequivalent surface. (b): Equivalent configuration where the metallic scatterers are removed from theequivalent surface. To restore the fields outside Sbox,m, an equivalent electric current density (shownwith blue arrows) is introduced on Sbox,m.

this approach we model each array element with a macromodel, which consists of an equiva-

lent electric current density and an admittance operator. The macromodel is generated using

the equivalence principle and it is therefore exact, except for numerical errors introduced by

discretization of fields and currents. The derived macromodel efficiently describes the elec-

tromagnetic behaviour of the original array element using fewer unknowns, reducing memory

consumption and computation time. For simplicity, we assume that the array is excited by an

externally illuminated electric field. Results in Sec. 6.5, however, show that the proposed idea

is also applicable to driven antenna elements.

6.2.1 Fields and Currents Discretization

In this section, we first derive a macromodel for the m-th element of the array. The m-th

element of the array may be composed of several PEC surfaces, which are denoted by Spec,m.

We enclose the element by a fictitious closed surface, which is denoted by Sbox,m. A sample

scatterer and the enclosing equivalent surface are shown in Fig. 6.2a.

Discretization of Spec,m

The electric surface current density on the metallic scatterer is expanded as

~Jpec,m(~r) =

Npec,m∑n=1

jpec,m,n~Λpec,m,n(~r) , (6.1)

where ~Λpec,m,n(~r) is the n-th RWG basis function [68] on the m-th element. The coefficients of

~Jpec,m(~r) in (6.1) are collected into a vector

Jpec,m =[jpec,m,1 jpec,m,2 . . . jpec,m,Npec,m

]T. (6.2)


Discretization of Sbox,m

Like the surface current density on the element, the tangential magnetic field on the equivalent

surface Sbox,m is also expanded with RWG basis functions

n× ~Hbox,m(~r) =

Nbox,m∑n=1

hbox,m,n~Λbox,m,n(~r) , (6.3)

where n is the normal vector pointing into the surface Sbox,m. The tangential electric field on

Sbox,m is, instead, expanded with dual RWG basis functions [53,93,144]

n× ~Ebox,m(~r) =

Nbox,m∑n=1

ebox,m,n~Λ′box,m,n(~r) . (6.4)

As discussed in Sec. 2.3.2, the n-th dual RWG basis function ~Λ′box,m,n is approximately orthog-

onal to the n-th RWG basis function ~Λbox,m,n. The use of both RWG functions and their duals

is necessary to achieve a well-conditioned formulation and high robustness, as will be discussed

in detail in the next sections. As in (6.2), we collect the coefficients of the tangential electric

and magnetic fields in (6.3) and (6.4) into vectors

Hbox,m =[hbox,m,1 hbox,m,2 . . . hbox,m,Nbox,m

]T(6.5)

Ebox,m =[ebox,m,1 ebox,m,2 . . . ebox,m,Nbox,m

]T. (6.6)

6.2.2 Equivalence Principle

As seen from Fig. 6.2a, some PEC scatterers can be quite complex. In order to model complex

scatterers efficiently, we apply the equivalence principle [83] to Sbox,m. As shown in Fig. 6.2b,

we replace all PECs inside the surface with free space and introduce on Sbox,m an equivalent

electric current density [83]

~Jeq,m(~r) = n×[~Heq,m(~r)− ~Hbox,m(~r)

](6.7)

and an equivalent magnetic current density [83]

~Meq,m(~r) = −n×[~Eeq,m(~r)− ~Ebox,m(~r)

]. (6.8)

In (6.7) and (6.8), ~Heq,m(~r) and ~Eeq,m(~r) are the electric and magnetic fields on Sbox,m in the

equivalent problem. According to the equivalence principle, these currents will produce the

same electric and magnetic fields outside Sbox,m as the actual currents on the PEC elements,

allowing us to compute the radiation from the array.

Most surface integral equation formulations are based on the Love’s equivalence princi-


V

Sn

~H(~r), ~E(~r)

~J(~r)µ0, ε0

Figure 6.3: Sample boundary value problem considered in Sec. 6.2.3. Electric and magnetic fields aredefined on the boundary of a closed surface S. There is also an additional electric current density ~J(~r)inside V.

ple [83], which sets ~Heq,m and ~Eeq,m to zero, and require both ~Jeq,m(~r) and ~Meq,m(~r) to restore

the electromagnetic fields outside the scatterer. However, we apply the single-source approach

that was previously developed to model 2-D and 3-D objects in Chapters 3 and 5. In this work,

we enforce that ~Eeq,m(~r) is equal to ~Ebox,m(~r) [29, 145]. Therefore, the equivalent magnetic

current density in (6.8) is zero and only an equivalent electric current density is required to

model the scatterer.

We expand the tangential magnetic field n× ~Heq,m(~r) on Sbox,m in the equivalent problem

using RWG basis functions

n× ~Heq,m(~r) =

Nbox,m∑n=1

heq,m,n~Λbox,m,n(~r) , (6.9)

and collect its expansion coefficients into a vector

Heq,m =[heq,m,1 heq,m,2 . . . heq,m,Nbox,m

]T. (6.10)

Similarly, the equivalent electric current density is also expanded with RWG basis functions as

~Jeq,m(~r) =

Nbox,m∑n=1

jeq,m,n~Λbox,m,n(~r) , (6.11)

and its expansion coefficients are collected into a vector

Jeq,m =[jeq,m,1 jeq,m,2 . . . jeq,m,Nbox,m

]T. (6.12)

Typically, Sbox,m needs to be meshed with 6-8 triangles per wavelength to accurately capture

the inter-element coupling. By substituting (6.11), (6.9) and (6.3) into (6.7), we obtain

Jeq,m = Heq,m −Hbox,m (6.13)

in the discrete domain. Next, we simplify (6.13) by applying the Stratton-Chu formulation to

two problems: the original problem and the equivalent problem.


6.2.3 Stratton-Chu Formulation

Consider the sample boundary value problem shown in Fig. 6.3. Recall that according to the

equivalence principle discussed in Sec. 2.2.1, we can relate the tangential electromagnetic fields

on S and current ~J(~r) inside V by the Stratton-Chu formulation for ~r ∈ V [146]

jωµ0n× n×([LV ~J(~r ′)

](~r) +

[LV(n× ~H(~r ′)

)](~r))

+ n× n×[KV(−n× ~E(~r ′)

)](~r) = −n× n× ~E(~r) . (6.14)

In (6.14), integral operators LV and KV are as defined in (2.15b)–(2.15b) with material prop-

erties of V. It is important to note that (6.14) is independent of material and sources outside

S.

6.2.4 Stratton-Chu Formulation Applied to the Original Problem

If the Stratton-Chu formulation (6.14) is applied to the original problem shown in Fig. 6.2a,

then we obtain

jωµ0n× n×([~L0~Jpec,m(~r ′)

](~r) +

[~L0

(n× ~Hbox,m(~r ′)

)](~r))

+ n× n×[~K0

(−n× ~Ebox,m(~r ′)

)](~r) =

−n× n× ~Ebox,m(~r) ~r ∈ Sbox,m

0 ~r ∈ Spec,m

, (6.15)

where integral operators ~L0 and ~K0 are evaluated with the Green’s function of free space. For

~r ∈ Spec,m, the right-hand side of (6.15) is zero because the tangential electric field on the

surface of a PEC is zero. We evaluate this equation twice, first assuming ~r ∈ Spec,m, and then

assuming ~r ∈ Sbox,m.

Surface Integral Equation on Spec,m

We substitute the expansion of fields and currents in (6.1), (6.3), and (6.4) into (6.15). Next, we

test the resulting equation with RWG basis functions ~Λpec,m,n on Spec,m. The resulting system

of equations is compactly written as

G(E,J)m Jpec,m + G(E,Hbox)

m Hbox,m + G(E,Ebox)m Ebox,m = 0 , (6.16)

where entry (n, n′) of matrices G(E,J)m , G

(E,Hbox)m , and G

(E,Ebox)m is given by[

G(E,J)m

](n,n′)

= jωµ0

⟨~Λpec,m,n(~r), n× n×

[~L0~Λpec,m,n′(~r

′)]

(~r)⟩

(6.17)[G(E,Hbox)m

](n,n′)

= jωµ0

⟨~Λpec,m,n(~r), n× n×

[~L0~Λbox,m,n′(~r

′)]

(~r)⟩

(6.18)[G(E,Ebox)m

](n,n′)

=⟨~Λpec,m,n(~r),−n× n×

[~K0~Λ′box,m,n′(~r

′)]

(~r)⟩. (6.19)


Surface Integral Equation on Sbox,m

Now, we re-evaluate (6.15) on Sbox,m. We again substitute (6.1), (6.3), and (6.4) into (6.15),

and test the resulting equation with the RWG basis functions on Sbox,m, i.e. ~Λbox,m,n. The

resulting equations can be compactly written as

G(Ebox,J)m Jpec,m + G(Ebox,Hbox)

m Hbox,m + G(Ebox,Ebox)m Ebox,m = 0 , (6.20)

where entry (n, n′) of G(Ebox,J)m , G

(Ebox,Hbox)m , and G

(Ebox,Ebox)m is[

G(Ebox,J)m

](n,n′)

= jωµ0

⟨~Λbox,m,n(~r), n× n×

[~L0~Λpec,m,n′(~r

′)]

(~r)⟩

(6.21)[G(Ebox,Hbox)m

](n,n′)

= jωµ0


[~L0~Λbox,m,n′(~r

′)]

(~r)⟩

(6.22)[G(Ebox,Ebox)m

](n,n′)

=⟨~Λbox,m,n(~r),−n× n×

[~K0~Λ′box,m,n′(~r

′)]

(~r)⟩

+⟨~Λbox,m,n(~r), n× ~Λ′box,m,n′(~r)

⟩. (6.23)

It is important to note that the novel usage of dual basis functions to expand n × ~Ebox,m(~r)

ensures that both ~L0 and ~K0 operators in (6.15) are well-tested because RWG and dual RWG

basis functions are approximately orthogonal [92]. Hence, all three matrices in (6.20) are well-

conditioned. In particular, if we had expanded n× ~Ebox,m(~r) with RWG basis functions, then

matrix G(Ebox,Ebox)m would have been poorly conditioned.

We want to use the discretized Stratton-Chu formulation (6.20) to eliminate Hbox,m from (6.13).

Therefore, since G(Ebox,Hbox)m is a well-conditioned matrix, we rewrite (6.20) as

Hbox,m = −[G(Ebox,Hbox)m

]−1 [G(Ebox,Ebox)m Ebox,m + G(Ebox,J)

m Jpec,m

]. (6.24)

Equation (6.24) requires the LU factorization of G(Ebox,Hbox)m , which is a dense matrix. However,

this matrix is only of size Nbox,m × Nbox,m, and thus the cost of this LU factorization will be

relatively small compared to the total time to solve the entire problem.

Equation (6.24) allows us to eliminate Hbox,m from (6.16) and obtain[G(E,J)m −G(E,Hbox)

m

[G(Ebox,Hbox)m

]−1G(Ebox,J)m

]︸︷︷︸

Am

Jpec,m+

[G(E,Ebox)m −G(E,Hbox)

m

[G(Ebox,Hbox)m

]−1G(Ebox,Ebox)m

]︸︷︷︸

Bm

Ebox,m = 0 , (6.25)

where we have introduced matrices Am and Bm. From (6.25), we have

Jpec,m = −A−1m BmEbox,m , (6.26)


which relates the current density on the PEC scatterers to the tangential electric field on Sbox,m.

6.2.5 Stratton-Chu Formulation Applied to the Equivalent Problem

We now apply the Stratton-Chu formulation (6.14) to the equivalent problem shown in Fig. 6.2b.

Since there is no electric current distribution inside Sbox,m, the Stratton-Chu formulation for

~r ∈ Sbox,m reads

jωµ0n× n×[~L0

(n× ~Heq,m(~r ′)

)](~r)

+ n× n×[~K0

(−n× ~Ebox,m(~r ′)

)](~r) = −n× n× ~Ebox,m(~r) , (6.27)

which is similar to (6.15), except there is no contribution from ~Jpec,m(~r). By testing (6.27) with

RWG basis functions on Sbox,m, we obtain

G(Ebox,Hbox)m Heq,m + G(Ebox,Ebox)

m Ebox,m = 0 , (6.28)

where entries of matrices G(Ebox,Hbox)m and G

(Ebox,Ebox)m are given in (6.22)–(6.23). Similarly

to (6.24), (6.28) can be rewritten as

Heq,m = −[G(Ebox,Hbox)m

]−1G(Ebox,Ebox)m Ebox,m , (6.29)

to obtain the tangential magnetic field on Sbox,m in the equivalent problem in terms of the

tangential electric field on Sbox,m.

6.2.6 Equivalent Current

We can now simplify the expression for the equivalent electric current density on Sbox,m in (6.13).

We substitute (6.24) and (6.29) into (6.13) and simplify the resulting equation to obtain

Jeq,m =[G(Ebox,Hbox)m

]−1 [G(Ebox,J)m

]︸︷︷︸

Tm

Jpec,m , (6.30)

where Tm is the transfer matrix that relates the electric current density on Spec,m to the

equivalent electric current density on Sbox,m. By substituting (6.26) into (6.30), we obtain

Jeq,m = YmEbox,m, where the admittance operator

Ym = −TmA−1m Bm (6.31)

is analogous to the differential surface admittance operator of 2-D and 3-D penetrable ob-

jects [147]. In proposed formulation, ~Jeq,m(~r) has a different value than the equivalent electric

current density in the EPA [73]. The equivalent current densities in the EPA are equal to

the tangential electric and magnetic fields on Sbox,m. However, in this proposed formulation,


the equivalent electric current density is equal to the difference in tangential magnetic fields

with and without the scatterer. Since the proposed formulation does not require an equivalent

magnetic current density [147], it is simpler to implement and requires testing fewer integral

operators than the EPA [73].

6.2.7 Macromodel For All Array Elements

The macromodeling procedure presented so far has been for a single element. We replace the

array of scatterers with the background medium and an array of equivalent electric current

densities. An implicit assumption made here is that the PEC traces of none of the array

elements touch one another, which is the case in many practical arrays of interest. The electric

current density coefficients on all elements are collected into a vector

Jpec =[JTpec,1 . . . JTpec,M

]T(6.32)

of size Npec × 1, where Npec =∑M

m=1Npec,m. Likewise, the coefficients of ~Jeq,m(~r) on all

equivalent surfaces are collected into a vector

Jeq =[JTeq,1 . . . JTeq,M

]T(6.33)

of size Nbox×1, where Nbox =∑M

m=1Nbox,m. The two current densities, as presented in (6.30),

are related by

Jeq = TJpec , (6.34)

where

T =

T1

T2

. . .

TM

(6.35)

is a block-diagonal transfer matrix that relates the equivalent electric current density on Sbox,m

to the current density on Spec,m for all elements. Matrix Tm in (6.35) is given in (6.30). Similar

to (6.35), we introduce matrices A and B which are block diagonal matrices made up of blocks

Am and Bm, respectively.

6.3 Exterior Problem

After applying the macromodeling technique to each element, we have simplified the original

problem to an equivalent problem composed of an array of equivalent electric current densities

in free space. We now apply the electric field integral equation to capture the coupling between

these currents, and therefore array elements.


According to the electric field integral equation, the total tangential electric field on the

m-th equivalent surface is

n× n× ~Ebox,m(~r) = −jωµ0

M∑m′=1

n× n×[~L0~Jeq,m′(~r

′)]

(~r) + n× n× ~E(inc)(~r) , (6.36)

where the right-hand side is the sum of the total scattered field produced by the equivalent

electric currents ~Jeq,m′(~r) and the incident electric field ~E(inc)(~r). In contrast to the EPA [73,

142], the exterior problem in (6.36) does not have the ~K0 integral operator because of the

absence of equivalent magnetic current density, which leads to a simpler exterior problem than

the EPA.

We substitute (6.4) and (6.11) into (6.36) and test the resulting equation with the n-th

RWG basis function on the m-th element ~Λbox,m,n for m = 1, 2, . . . ,M . The resulting equations

can be compactly written as

DEbox = −G(Ebox,Jeq)Jeq + Vbox , (6.37)

where Ebox and Vbox are vectors of electric field coefficients and excitation field coefficients,

respectively, and read

Ebox =[ET

box,1 ETbox,2 . . . ET

box,M

]T, (6.38)

Vbox =[VT

box,1 VTbox,2 . . . VT

box,M

]T. (6.39)

The vector Vbox,m in (6.39) is of size Nbox,m × 1, and its n-th entry is

[Vbox,m]n =⟨~Λbox,m,n(~r), n× n× ~E(inc)(~r)

⟩, (6.40)

which is the projection of incident electric field on the RWG basis function. In (6.37), G(Ebox,Jeq)

is a block matrix of the form

G(Ebox,Jeq) =

G

(Ebox,Jeq)1,1 G

(Ebox,Jeq)1,2 . . . G

(Ebox,Jeq)1,M

G(Ebox,Jeq)2,1 G

(Ebox,Jeq)2,2 . . . G

(Ebox,Jeq)2,M

......

. . ....

G(Ebox,Jeq)M,1 G

(Ebox,Jeq)M,2 . . . G

(Ebox,Jeq)M,M

, (6.41)

where the (n, n′) entry is[G

(Ebox,Jeq)m,m′

]n,n′

= jωµ0


[~L0~Λbox,m′,n′(~r

′)]

(~r)⟩. (6.42)


Finally, matrix D in (6.37) is block diagonal and reads

D =

D1

. . .

DM

, (6.43)

where the (n, n′) entry of Dm is given by

[Dm](n,n′) =⟨~Λbox,m,n(~r), n× ~Λ′box,m,n′(~r)

⟩. (6.44)

It is important to note that D is well-conditioned and is diagonally dominant because ~Λ′box,m,n(~r)

is approximately orthogonal to ~Λbox,m,n(~r) [93].

Next, we substitute (6.34) and (6.26) into (6.37) to obtain the final system of equations(D−G(Ebox,Jeq)TA−1B

)Ebox = Vbox , (6.45)

which is only in terms of the electric field coefficients on the surface of the equivalent box.

After solving for Ebox, the current distribution Jpec on the original scatterer, if desired, can be

computed inexpensively using (6.26) for each element.

Notice that solving the original problem with the standard MoM would have required solving

for Npec unknowns. However, (6.45) involves only Nbox unknowns. For many problems with

complex, multiscale scatterers, Nbox is much smaller than Npec, and therefore the proposed

method results in faster solution times and lower memory consumption. Furthermore, (6.45) is

usually better conditioned than the standard MoM formulation because the equivalent surface

can have a coarser mesh than the original scatterer due to the absence of any fine features in

the equivalent problem.

6.4 Acceleration with the AIM

6.4.1 Matrix-Vector Product

Even after the reduction in the number of unknowns, the computation cost of the proposed

approach can grow quickly if a direct solution method is used. Therefore, we solve linear

system (6.45) iteratively using the generalized minimal residual (GMRES) algorithm [148].

This algorithm requires an efficient way to compute(D−G(Ebox,Jeq)TA−1B

)V , (6.46)

where V is an arbitrary vector of sizeNbox×1. A trivial way to compute (6.46) is to first generate

D, G(Ebox,Jeq), T, A, and B, and then carry out the required matrix-vector multiplications and

vector additions. Since matrices T, A, and B are block-diagonal with M blocks, computing


matrix-vector products with these matrices is inexpensive. Matrix D in (6.45) is also a block-

diagonal matrix with M sparse blocks, and so its corresponding matrix-vector product is also

inexpensive. However, G(Ebox,Jeq) is a dense matrix: generating this matrix, storing its values,

and computing a matrix-vector product with it require a lot of computational time and memory.

For this reason, we accelerate the computation of G(Ebox,Jeq)V using the AIM [26,63].

6.4.2 The Adaptive Integral Method

This section describes how the standard AIM can be used to accelerate matrix-vector product

between the MoM matrix and a vector. More details on this approach can be found in the

original work on this topic [26].

In this section, we consider the matrix-vector product GV, where V is an Ne × 1 vector

and G is a Ne×Ne matrix that is obtained by discretizing ~L0(~r). The (n, n′)-th entry of G is

[G]n,n′ = jωµ0

⟨~Λn(~r),

[~L0~Λn′(~r

′)]

(~r)⟩. (6.47)

Note that G is of a similar form as G(Ebox,Jeq) in the proposed macromodeling approach, except

with a simpler notation. By substituting the definition of ~L0 (2.15a) into (6.47) and simplifying

the resulting expression, we obtain

[G]n,n′ =jωµ0

¨Tn

¨Tn′

~Λn(~r) · ~Λn′(~r ′)G(~r, ~r ′) dS dS′

− jωµ0

k2

¨Tn

¨Tn′

∇ · ~Λn(~r)G(~r, ~r ′)∇ · ~Λn′(~r ′) dS dS′ , (6.48)

where the integration is performed over the support of source and test basis function Tn′ and

Tn, respectively. In (6.48), the first term on the right hand side can be interpreted as the

magnetic vector potential tested with the n-th RWG basis function, while the second term on

the right hand side can be interpreted as the gradient of scalar potential tested with the n-th

RWG basis function.

The AIM Grid

In order to apply the AIM, we introduce a 3-D Cartesian grid in the problem domain. Grid

spacing is chosen to be λ/10 or smaller. Let us assume that we have introduced a total of NG

grid points.

Each RWG basis function in our simulation domain is assigned to a stencil. Each stencil is

made up of NO + 1 grid points in each direction, where NO is the stencil order. Furthermore,

we define the volume spanned by NNF stencils in each direction surrounding the stencil of a

basis function to be its near-field region. Figure 6.4 shows a sample 2-D AIM grid with NO = 3

and NNF = 1.


projection stencil

near-field region

test basis 1

source basis

interpolation stencil

test basis 2

Figure 6.4: A sample AIM grid with one source RWG basis function and two testing functions. Theinner product between the source function and test function 1 is calculated directly. The inner productbetween source function and test function 2 is computed via the AIM.

Near-field and Far-field Decomposition

In the AIM, G is decomposed into two parts: the near-field matrix GNF and the far-field matrix

GFF . The near-field matrix is a sparse matrix. Its (n, n′)-th entry is non-zero if the stencil of

~Λn(~r) is in the near-field of the stencil of ~Λn(~r ′). Non-zero entries are computed by (6.48) with

some pre-correction [26,63]. Matrix-vector product with GNF is inexpensive.

The (n, n′)-th entry of GFF corresponds to the electric field radiated by the n′-th RWG

basis function, tested by the n-th RWG basis function. Recall that the radiated electric field

can be expressed by the superposition of vector and scalar potentials. Hence, we can write

GFF = GFF,Ax + GFF,Ay + GFF,Az + GFF,φ , (6.49)

where the (n, n′)-th entry of GFF,Ax , GFF,Ay , GFF,Az , and GFF,φ is

[GFF,Ax ]n,n′ =jωµ0

¨Tn

¨Tn′

(~Λn(~r) · x

)G(~r, ~r ′)

(x · ~Λn′(~r ′)

)dS dS′ (6.50a)

[GFF,Ay

]n,n′

=jωµ0

¨Tn

¨Tn′

(~Λn(~r) · y

)G(~r, ~r ′)

(y · ~Λn′(~r ′)

)dS dS′ (6.50b)

[GFF,Az ]n,n′ =jωµ0

¨Tn

¨Tn′

(~Λn(~r) · z

)G(~r, ~r ′)

(z · ~Λn′(~r ′)

)dS dS′ (6.50c)

[GFF,φ]n,n′ =− jωµ0

k2

¨Tn

¨Tn′

∇ · ~Λn(~r)G(~r, ~r ′)∇ · ~Λn′(~r ′) dS dS′ (6.50d)

In the AIM, we factorize GFF,ψ for ψ ∈ Ax, Ay, Az, φ as a product of three matrices as

GFF,ψ ≈ IψHψPψ , (6.51)


where Iψ, Hψ, and Pψ are the interpolation, convolution, and projection matrices, respectively.

Dimensionality and properties of these three matrices are as follows.

Interpolation matrix is of size Ne ×NG. It is a sparse matrix. Each row of this matrix has

(NO + 1)3 nonzero entries.

Convolution matrix is of size NG × NG. It is a dense matrix, but it has a Toeplitz form.

Therefore, matrix-vector products involving this matrix can be computed with FFTs.

Projection matrix is of size NG ×Ne. It is a sparse matrix. Each column of this matrix has

(NO + 1)3 nonzero entries.

Given the decomposition in (6.51), it is important to realize that the (n, n′)-th entry of GFF,ψ

can be approximated by multiplying the n-th row of Iψ by Hψ, which is then multiplied by the

n′-th column of Pψ.

Given V of the size Ne× 1, the matrix-vector product GFF,ψV is computed in the AIM by

the three phases: projection phase, convolution phase, and interpolation phase. We describe

each phase while computing GFF,φV.

Projection Phase

In the projection phase, fields radiated by each source basis function is approximated with

equivalent charges that are introduced on grid points of its enclosing stencil. Given V, mathe-

matically, we can compute grid charges on all grid points by

V1 = PφV . (6.52)

Since Pφ is sparse, the complexity of the matrix-vector product in (6.52) is O(n).

There are many ways to generate the projection matrix. In our implementation of the AIM,

we adopted the approach proposed by Zhu, Song, and White [26]. In this approach, the Green’s

function G0(~r, ~r ′) is approximated, within each stencil, with a series of polynomials as

G0(~r, ~r ′) ≈∑i,j,l

αi,j,l(x− xc)i(y − yc)j(z − zc)l , (6.53)

where i, j, l ∈ 0, . . . , NO, αi,j,l is expansion coefficient, and xc, yc, and zc are the mid points of

the stencil under consideration. Using (6.53), approximation of the radiated field due to charge

density Qm on the m-th grid point is

~E ′(~r) ≈∑i,j,l

Qmαi,j,l(x− xc)i(y − yc)j(z − zc)l . (6.54)


Similarly, the radiated field by the n′-th basis function is

~E(~r) ≈∑i,j,l

αi,j,l

¨T ′n

(∇ · Λn′(~r ′)

)(x− xc)i(y − yc)j(z − zc)l dS′ . (6.55)

By matching moments at all (NO + 1)3 grid points of the stencils surrounding the source basis

function2, we can compute equivalent charges Qeq,m. These equivalent charge values are used

to populate columns of Pφ.

Convolution Phase

Once we have obtained grid charges, we can compute scalar potential on all grid points by using

the Green’s function G0(~r, ~r ′). Mathematically, this operation can be expressed as

V2 = HψV1 . (6.56)

The (p, q)-th entry of Hψ is given by

[Hψ]p,q =e−jk|~rp−~rq |

4π |~rp − ~rq|, (6.57)

where ~rp and ~rq are the position vectors of the p-th and q-th grid point. Since the Green’s func-

tion is translation invariant, H has a Toeplitz form. Therefore, we can use FFTs to accelerate

matrix-vector products involving H, as discussed in Appendix C.

Interpolation Phase

From the calculated grid potentials, we can interpolate the scalar potential inside each stencil

using interpolation polynomial as

φ(~r) =∑i,j,l

φi,j,l(x− xc)i(y − yc)j(z − zc)l , (6.58)

where φi,j,l are used to denote potential on grid points of the stencil under consideration. The

n-th entry of the result of GFF,φV can be obtained by testing the interpolated potential by

∇ · ~Λn(~r) as

[GFF,φV]n =∑i,j,l

φi,j,l

¨Tn

(∇ · ~Λn(~r)

)(x− xc)i(y − yc)j(z − zc)l dS . (6.59)

Mathematically, this operation can be performed through the interpolation matrix Iφ as

V3 = IφV2 . (6.60)

2This is performed by equating (6.54) to (6.55) for ~r on the grid points.


3.89 m

3.89m

9.5 cm

x

y

z

Figure 6.5: Layout of the 20 × 20 array of spherical helix antennas considered in Sec. 6.5.1. The arrayis uniformly spaced along the x− and y-directions with interelement spacing of dx = dy = 0.2 m.

−150 −100 −50 0 50 100 150

−20

0

20

θ [degrees]

Dir

ecti

vit

y[d

Bi]

AIMProposed

(a) φ = 0

−150 −100 −50 0 50 100 150

−20

0

20

θ [degrees]

Dir

ecti

vit

y[d

Bi]

AIMProposed

(b) φ = 90

Figure 6.6: Directivity of the 20×20 array of spherical helix antennas considered in Sec. 6.5.1 calculatedwith the AIM and the proposed technique.


In this section, three examples are presented to demonstrate the accuracy and performance of

the proposed method compared against an in-house standard MoM solver accelerated with the

AIM. All computational codes were developed using the PETSc [149–151] and FFTW3 [152]

libraries. The numerical tests were performed on a machine with an Intel Xeon E5-2623 v3

processor and 128 GB of RAM. All simulations were run on a single thread without exploiting

any parallelization.

6.5.1 Array of Spherical Helix Antennas

Spherical helix antennas, like many other electrically small antennas, have complex geometries

with electrically fine features [153]. Therefore, simulating an array of such antennas requires a

long computation time and large memory. In the proposed technique, a macromodel is created

for each small antenna. As demonstrated by this example, the macromodel can accurately

capture radiation from the antenna using fewer unknowns, which leads to significant savings in


Table 6.1. Simulation settings and results for the 20× 20 array of spherical helix antennasconsidered in Sec. 6.5.1

AIM Proposed

AIM Parameters

Number of stencils 120× 120× 2Interpolation order 3Number of near-field stencils 4

Memory Consumption

Total number of unknowns 260,800 62,400Memory used 19.53 GB 3.60 GB

Timing Results

Macromodel generation N/A 18 sMatrix fill time 1.12 h 336 sPreconditioner factorization 414 s 23 sIterative solver 28 s 4 sTotal computation time 1.25 h 6.35 min

computation time and memory. This example also demonstrates that the proposed macromodel

approach can be applied to fed antenna arrays.

We consider the 20× 20 array of identical spherical helix antennas shown in Fig. 6.5. The

helical antenna has three turns and is made up of PEC strips of width 2.75 mm. The spherical

helix antenna has a diameter of 9.5 cm. Each element of the array is excited with a uniform

delta-gap voltage source at the center of each helix operating at 300 MHz. With this excitation,

the array radiates the main beam in the broadside direction. We computed the radiation pattern

from the antenna array using the standard MoM and the proposed method, both accelerated

with the AIM and solved iteratively using GMRES with an ILU-2 preconditioner [150]. In the

proposed method, we first created the macromodel for a spherical helix antenna by computing

Tm, Am, and Bm using a sphere of radius 5 cm as the equivalent surface. This macromodel

was then reused for all elements of the array, as described in Sec. 6.2.7. The AIM parameters,

computational times, and memory requirements to simulate this problem are given in Tab. 6.1.

As seen from Tab. 6.1, the proposed method reduces the number of unknowns by a factor of

approximately four because Npec = 260, 800 and Nbox = 62, 400. This reduction in the number

of unknowns leads to a simulation that is 12 times faster and requires 5 times less memory

than the AIM-accelerated MoM solver. As evident from Tab. 6.1, the proposed method is

faster than the AIM because it takes less time to assemble all the matrices and factorize the

preconditioner, since it has fewer unknowns than the standard MoM formulation. Figure 6.6

shows the radiation pattern of the array in the two principal plane cuts. The results in Fig. 6.6

confirm that the proposed macromodeling approach provides an excellent accuracy compared

to the standard MoM code.


w

w

h

h

(a) Original (b) Equivalent

Figure 6.7: (a): Original unit cell of the two-layer reflectarray considered in Sec. 6.5.2 with w = 3.75 mmand h = 0.76 mm. (b): Equivalent unit cell obtained after applying the image theorem.

Table 6.2. Simulation settings and results for the 21× 21 reflectarray considered in Sec. 6.5.2

AIM ADF Proposed

AIM Parameters

Number of stencils in x dir. 126 - 84Number of stencils in y dir. 126 - 84Number of stencils in z dir. 2 - 2Interpolation order 3 - 3Number of near-field stencils 4 - 4

Memory Consumption

Total number of unknowns 324,420 324,420 111,132Memory used 40 GB 37 GB 15 GB

Timing Results

Macromodel generation N/A N/A 0.054 hMatrix fill time 1.48 h 2.26 h 0.42 hPreconditioner factorization 1.25 h 1.02 h 0.31 hIterative solver 0.22 h 0.11 h 2.80 minTotal computation time 3.30 h 3.40 h 0.82 h

6.5.2 Two-layer Reflectarray with Jerusalem Cross Elements

Design and Simulation Setup

Next, we consider a two-layer dual-polarized reflectarray with 21 × 21 elements made up of

Jerusalem crosses [154]. Since the proposed method only supports PEC scatterers, the reflec-

tarray was designed using phase curve obtained from a periodic simulation of the structure in

the absence of dielectric substrate. The unit cell of the reflectarray is shown in Fig. 6.7a. In

this example, the reflectarray is electrically large with dimensions of 7.875λ×7.875λ at 30 GHz.

This 441 element array has eight unique elements. It also includes sub-wavelength features and

strong mutual coupling between the unit cells. In practice, due to simulation difficulties, reflec-

tarrays of this size and complexity are rarely simulated with full-wave electromagnetic solvers.


78.75 mm = 7.875λ

78.75

mm

x

y

z

Figure 6.8: Top view of the 21× 21 reflectarray considered in Sec. 6.5.2. Top and bottom layers of thereflectarray are shown in blue and red, respectively. The reflectarray is uniformly spaced along the x-and y-directions with interelement spacing of 3.75 mm (0.375λ).

One of the motivations of this work is to enable an efficient simulation of such problems.

Since the proposed method currently does not support multilayer dielectrics, we assume

that all layers of the reflectarray have permittivity ε0 and permeability µ0. Furthermore, we

apply image theory [83] to model the ground plane at the bottom of the reflectarray. According

to image theory, the two Jerusalem crosses in each unit cell are duplicated below the image

plane as shown in Fig. 6.7b. Thus, each unit cell has effectively four Jerusalem crosses.

In this example, the reflectarray is designed to produce the main beam of the scattered field

in the broadside direction when the reflectarray is excited by a dipole feed antenna operating

at 30 GHz3. The feed is placed 40 mm along the axis of the reflectarray at the prime focus

position, so that the focal length to diameter ratio is 0.51. The top view of the final reflectarray

design is shown in Fig. 6.8. The final design contains 441 total elements with eight unique

elements used to discretize the reflectarray phase curve.

Current Distribution and Scattered Field

We simulated the reflectarray using three tools: an in-house AIM accelerated MoM code, the

Antenna Design Framework (ADF) [155] – an AIM-accelerated commercial SIE solver, and

the proposed technique. For the proposed method, we enclosed each unique element with an

equivalent box of dimensions 3.60 mm × 3.60 mm × 4.00 mm. We generated macromodels for

3A dipole feed antenna was chosen due to its simplicity, although it is not an optimal feed model for reflec-tarrays.


Figure 6.9: Current distribution on the reflectarray considered in Sec. 6.5.2 obtained with the proposedmacromodeling technique.

the array by first computing Tm, Am, and Bm for the eight unique elements in the array, and

then generating T, A, and B.

The current distribution on the reflectarray is shown in Fig. 6.9. Figure 6.10 shows the

directivity in three planes generated with the proposed method, the in-house MoM code, and

the ADF solver. An excellent match between all three methods validates the accuracy of

the proposed method. In particular, despite the unit cells being very close to one another,

the macromodel approach accurately predicts the mutual coupling between them. Simulation

settings, memory consumption, and timing results of the simulations run with the in-house

MoM code, ADF, and the proposed method are given in Tab. 6.2. It is seen that the in-

house AIM-accelerated MoM code performs on-par with the commercial AIM-accelerated MoM

code. For this simulation, the proposed method requires 4 times less computational time and

2.7 times less memory, which is a substantial savings. In this example, the number of basis

functions to expand the surface current density was Npec = 324, 420 and the number of basis

functions to expand the equivalent electric current density was Nbox = 111, 132. In general,

for reflectarray examples, the proposed method converges monotonically with the number of

iterations. Furthermore, the number of iterations required for a simulation to converge increases

approximately linearly with the number of array elements.

This example was inspired by the reflectarray structure that was designed, fabricated, and

tested within our research group [154]. However, the actual structure includes a dielectric


−80 −60 −40 −20 0 20 40 60 80

−20

0

20

θ [degrees]

Dir

ecti

vit

y[d

Bi]

ADFAIMProposed

(a) φ = 0

−80 −60 −40 −20 0 20 40 60 80

−20

0

20

θ [degrees]

Dir

ecti

vit

y[d

Bi]

ADFAIMProposed

(b) φ = 90

−80 −60 −40 −20 0 20 40 60 80

−20

0

20

θ [degrees]

Dir

ecti

vit

y[d

Bi]

ADFAIMProposed

(c) φ = 45

Figure 6.10: Directivity of the 21 × 21 two-layer reflectarray considered in Sec. 6.5.2 calculated withADF, AIM, and the proposed technique.

substrate, which is not treated in the formulation presented in this chapter. Furthermore,

the actual structure is quite large and cannot be simulated in its entirety with the current

computational code. Therefore, we have not yet validated this example against experimental

results.

6.5.3 Reflectarray Composed of Elements with Fine Features

As a final example, we consider a single-layer reflectarray with very fine meander-line fea-

tures [156]. The top view of this reflectarray is shown in Fig. 6.11. The mesh size of such

elements is electrically very small, which causes conditioning issues with the standard MoM.

However, with the proposed approach the structure can be simulated faster due to fewer un-

knowns and better conditioning of the equations to be solved.

All elements in this example are suspended in free space 0.75 mm above a PEC ground

plane. As in Sec. 6.5.2, we use the image theory to model the ground plane. The reflectarray

has a total of 11×11 elements, out of which eight elements are unique. The structure is designed

to scatter fields with the main beam in the broadside direction. The reflectarray is excited by a

dipole antenna operating at f = 14 GHz that is placed 43 mm along the axis of the reflectarray

so that focal length to diameter ratio is 0.65. We simulated the problem with the in-house


66 mm = 3.1λ

66 mm

x

y

z

Figure 6.11: Top view of the 11× 11 reflectarray considered in Sec. 6.5.3. The array is uniformly spacedalong the x− and y-directions with interelement spacing of dx = dy = 6 mm (0.28 λ).

MoM code and the proposed macromodeling technique, both accelerated with the AIM. In the

proposed method, an equivalent box of dimensions 5.5 mm× 5.5 mm× 3.0 mm was introduced

to enclose each element.

Fig. 6.12 shows the current distribution obtained with the proposed technique. The current

distribution obtained with the proposed macromodeling technique matches very closely with the

current distribution obtained with the AIM technique. The directivity of the reflectarray calcu-

lated with both techniques is shown in Fig. 6.13. The agreement between the results obtained

with the proposed method and the standard MoM code confirms that proposed method can

accurately capture the strong coupling between the elements and fine features of the reflectarray

unit cell.

Tab. 6.3 shows AIM parameters, storage statistics, and timing statistics to simulate this

problem. As seen from Tab. 6.3, the proposed method is 24 times faster and consumes 12

times less memory than the standard MoM solver. The proposed method solves the problem in

1/2 h as opposed to 11 h required with the standard MoM formulation, which is a significant

savings. The proposed method is faster because it requires solving a problem with 9 times

less number of unknowns with Npec = 262, 616 and Nbox = 27, 588. Furthermore, even though

this array structure has 121 elements, a macromodel only needs to be generated for eight

unique elements. This is because identical meshes are used to discretize array elements with

the same geometry. Therefore, the macromodel equations are exactly the same for identical

array elements. Furthermore, the proposed formulation converges significantly faster than the

standard MoM formulation due to a significantly smaller condition number. We computed, in

PETSc, the condition number of the proposed formulation and the standard MoM formulation

for a smaller sized reflectarray with 3 × 3 elements4. It was found that the condition number

4Computing the condition number of the 11 × 11 reflectarray was not feasible due to its prohibitive compu-tational cost.


Figure 6.12: Current distribution on 11 × 11 reflectarray considered in Sec. 6.5.3 computed with themacromodeling approach.

of the proposed formulation is 129.31, which is significantly smaller than the condition number

of the standard MoM equations which is 1.013× 105.

This example demonstrates that the proposed method can be very efficient, in terms of

computation time and memory consumption, to simulate arrays with complex elements.


In this chapter, we developed a macromodeling technique to simulate an array of complex PEC

scatterers. The macromodeling technique can be considered as a further generalization of the

single-source surface formulation presented in Chapters 3 and 5. In the proposed technique,

a complex PEC scatterer is accurately modeled by an equivalent electric current density in-

troduced on a fictitious surface enclosing the scatterer and by an admittance operator. The

crux of the technique is to solve for unknowns on the equivalent surface instead of unknowns

on the scatterer. When simulating complex scatterers, this approach leads to a reduction in

the unknowns count, resulting in lower solution time and matrix fill time. Furthermore, after

eliminating unknowns associated with multiscale features of the scatterer, the final linear sys-

tem in the proposed technique is better conditioned than the standard MoM, leading to faster

convergence with iterative solvers. The proposed technique also exploits the repeatability of

elements in large arrays. Due to this, macromodels only need to be created for the unique

elements in the array.


−80 −60 −40 −20 0 20 40 60 80−40

−20

0

20

θ [degrees]

Dir

ecti

vit

y[d

Bi]

AIMProposed

(a) φ = 0

−80 −60 −40 −20 0 20 40 60 80−40

−20

0

20

θ [degrees]

Dir

ecti

vit

y[d

Bi]

AIMProposed

(b) φ = 90

−80 −60 −40 −20 0 20 40 60 80−40

−20

0

20

θ [degrees]

Dir

ecti

vit

y[d

Bi]

AIMProposed

(c) φ = 45

Figure 6.13: Directivity of the 11× 11 reflectarray considered in Sec. 6.5.3 calculated with AIM and theproposed technique.

The presented macromodeling technique has a few key differences in comparison to the

EPA in the literature. First, the macromodeling technique is a single-source technique and

only requires an equivalent electric current density to accurately model the scatterer. Absence

of an equivalent magnetic current density results in fewer unknowns and integral operators.

Second, the EPA is based on the Love’s equivalence principle and derives scattering matrix to

relate the tangential electric and magnetic fields on the fictitious surface. Instead, the proposed

macromodeling approach applies the Schelkunoff equivalence principle to derive an admittance

relationship between the equivalent electric current density and the tangential electric field. As

such, the proposed macromodeling technique is analogous to the Norton equivalent circuit model

in classical circuit theory, which can model a complex electrical network with an equivalent

current and an equivalent source admittance. Although in this chapter we only investigated

array problems, the proposed macromodeling method is quite general and can be potentially

applied to several other multiscale electromagnetic problems.


Table 6.3. Simulation settings and results for the 11× 11 reflectarray considered in Sec. 6.5.3

AIM Proposed

AIM Parameters

Number of stencils in x dir. 126 84Number of stencils in y dir. 126 84Number of stencils in z dir. 2 2Interpolation order 3 3Number of near-field stencils 4 4

Memory Consumption

Total number of unknowns 262,616 27,588Memory used 67 GB 5.4 GB

Timing Results

Macromodel generation N/A 14.2 minMatrix fill time 5.31 h 7.8 minPreconditioner factorization 3.83 h 4.15 minIterative solver 2.04 h 26 sTotal computation time 11.18 h 27 min

Chapter 7

Simulation of Complex

Electromagnetic Structures using a

Macromodeling Technique for

Composite Scatterers

7.1 Introduction

In Chapter 6, we discussed a single-source macromodeling approach to efficiently simulate

an array of complex PEC scatterers. In the proposed macromodeling approach, each PEC

scatterer was modeled more efficiently by an admittance operator and an equivalent electric

current density that was introduced on a fictitious surface enclosing the scatterer. In this

chapter, we develop the macromodeling approach for scatterers composed of PEC and dielectric

objects. Our main goal is to develop a macromodeling technique to efficiently simulate complex

electromagnetic structures, such as reflectarrays and metasurfaces, composed of PEC traces

on a multilayer dielectric substrate. As shown in Fig. 7.1, most electromagnetic structures

have complex unit cells. Furthermore, these structures may be composed of multiple layers of

dielectric substrates with PEC traces on the interface of each layer, as shown in Fig. 7.1. In

order to compute electromagnetic fields scattered by such structures, we need to solve for electric

and magnetic current densities on all surfaces of the structure. Doing so requires solving a very

large number of unknowns, which is computationally expensive. This makes the simulation of

such structures challenging even with advanced commercial EM solvers.

In our approach, we create a macromodel for each unit cell in the array by invoking the

equivalence principle. The macromodel accurately captures the electromagnetic behavior of

each element. In the literature, the equivalence principle algorithm (EPA) [73, 78] and linear

embedding via Green’s operator (LEGO) [82] have been proposed to efficiently model composite

scatterers using the equivalence principle. In these algorithms, the scatterer is enclosed with

123

Chapter 7. Macromodeling Approach for Composite Scatterers 124

Figure 7.1: A sample two-layer array composed of 64 elements.

a fictitious surface. The Love’s equivalence principle and the surface integral equation method

is then applied to derive a scattering operator that relates incident and scattered electric and

magnetic fields on the fictitious surface. In these methods, inter-element coupling is captured by

the so-called translation operator. So far, these techniques have only been applied to simulate

arrays of composite objects with some gap between each element, i.e. equivalent surfaces of

adjacent elements are not touching. Applying the equivalence principle-based approach to

simulate complex electromagnetic structures poses the following challenges:

• All elements in a reflectarray or a metasurface are “connected” through a dielectric sub-

strate. Hence, it is not possible to introduce an equivalent surface to enclose each element

of the array, as we did in Chapter 6, without the equivalent surface for adjacent elements

touching one another. Therefore, we need to explicitly enforce continuity of the tangential

electric and magnetic fields on the adjacent elements.

• The equivalent surface cuts through the dielectric interface in the array, introducing addi-

tional junctions. Enforcing the correct electromagnetic field boundary conditions on these

junctions is not trivial. In the literature, the so-called tap basis functions were proposed

to model simple junctions encountered when modeling PEC scatterers [73]. However, to

the best of our knowledge, they have not been applied to model dielectric substrates.

Furthermore, when enclosing each unit cell of the array with an equivalent surface, some

portions of the dielectrics coincide with the equivalent surface. The EPA has not been

applied to model such cases.

• In some electromagnetic structures, such as reflectarrays, the dielectric substrate is backed

by a PEC ground plane. Modeling this finite-sized ground plane with the EPA is non-

trivial because some parts of the equivalent surface coincide with a PEC ground plane.

None of the previous works on the EPA discuss how to properly model PEC surfaces that

coincide with the equivalent surface.


In this chapter, we propose a dual-source macromodeling approach for composite scatter-

ers to compute scattering from complex electromagnetic structures, such as the one shown

in Fig. 7.1. In order to simulate large arrays, we propose a rigorous FFT-based acceleration

method. After instantiating a macromodel for each element of the array, we can effectively

model an array of inhomogeneous unit cells with an array of identical equivalent surfaces (cur-

rents). This allows us to exploit the Toeplitz property of the discretized integral equation

matrices to compute inter-element coupling accurately and efficiently via FFTs. In comparison

to the AIM, which approximates far interactions with a series of polynomials, the proposed

acceleration method does not make any approximations. It is also more efficient than the AIM

because it does not require computing pre-corrections. Furthermore, computation of far fields

with the AIM requires decomposing field quantities via vector and scalar potentials. However,

in the proposed Toeplitz acceleration, far interactions can be computed directly. This step re-

sults in lower overhead cost and a more efficient algorithm overall. Previously, this acceleration

approach was proposed to simulate an array of identical antenna elements that are separated

by a finite distance in an FEM-IE hybrid solver [81]. However, it has not been applied to

non-homogeneous array problems.

This chapter is organized as follows. In Sec. 7.2, we discuss how to generate a macromodel

for a single element of the array. Then, in Sec. 7.3, we discuss how to simulate an entire

electromagnetic structure using an array of macromodels. To simulate large arrays, we present

an FFT-based acceleration algorithm in Sec. 7.4. Finally, in Sec. 7.5, we present numerical

examples to validate the proposed technique and evaluate its numerical performance.

7.2 Macromodel Generation

We consider the problem of computing scattering from a complex electromagnetic structure,

such as the one shown in Fig. 7.1. Complex electromagnetic structures are inhomogeneous with

distinct PEC traces on each unit cell, i.e. non-uniform arrays. We assume that PEC traces on

two adjacent unit cells are unconnected, but allow the presence of a finite ground plane.

7.2.1 Discretization

In order to generate a macromodel, we enclose each unit cell with a fictitious (equivalent)

surface Seq such that all PEC traces and dielectric layers are inside Seq. Since we are simulating

electromagnetic structures with uniform spacing between all elements, it is convenient to use a

rectangular box as the fictitious surface. In this case, each Seq enclosing a unit cell will have the

same dimensions. If the electromagnetic structure is backed by a PEC ground plane, then the

bottom face of Seq is set to coincide with the ground plane. Fig. 7.2a shows this step for two of

the elements in Fig. 7.1. After introducing these equivalent surfaces, there are R = 4 regions

inside each box, in this case two dielectric and two air regions. Next, we mesh all surfaces

with triangular elements. Since multiple equivalent boxes are connected in array problems,


(a) Original setup (b) Equivalent setup

Figure 7.2: (a): Original configuration: two unit cells of the sample array in Fig. 7.1 are enclosed byequivalent boxes (shown in red and blue). (b): Equivalent configuration: unit cells are modeled byequivalent electric and magnetic currents that are introduced on equivalent boxes.

identical meshes must be introduced on opposite side faces of the box in order to properly

enforce electromagnetic boundary conditions. Most meshing tools, such as Gmsh [157], provide

this feature.

Throughout the rest of this section, we focus on creating a macromodel for a single element

of the array. To explain macromodel generation, let us consider the cross-section of a sample

unit cell with a PEC ground plane shown in Fig. 7.3. This unit cell has R = 3 interior regions.

The r-th region is denoted by Vr. The surface enclosing the r-th region is denoted by Sr. Surface

Sr may be composed of a union of P subsurfaces that are denoted by Spr for p = 1, . . . , P , as

shown in Fig. 7.3. We expand the tangential electric and magnetic fields on all surfaces inside

Seq with RWG basis functions. On Sr, the tangential electric and magnetic fields are expanded

as

nr × ~Hr(~r) = ~Jr(~r) =

Nr∑n=1

jr,n~Λr,n(~r) (7.1a)

−nr × ~Er(~r) = ~Mr(~r) =

N ′r∑n=1

mr,n~Λr,n(~r) , (7.1b)

where nr is the unit normal vector pointing into the r-th region, Nr is the number RWG

functions used to expand tangential magnetic field, andN ′r is the number of RWG basis functions

used to expand the tangential electric field. Throughout the rest of this section, we will omit

the subscript in nr for brevity. Furthermore, since our formulation is based on the Love’s

equivalence principle (Stratton-Chu formulation), we have equated the tangential magnetic

and electric fields to electric and magnetic current density, respectively, in (7.1a)–(7.1b). If

any subsurface Spr is a PEC, then the tangential magnetic field on it is the same as the surface

electric current density. Furthermore, the tangential electric field is not discretized on a PEC

surface because it is zero. Hence, Nr is always greater than or equal to N ′r. All tangential

magnetic field coefficients on the surface enclosing the r-th region are collected into a vector

Jr =[jr,1 . . . jr,Nr

]Tand all tangential electric field coefficients are collected into a vector


V1

V2

V3

S13 S2

3 S33

S12 S2

2 S32

j1,nj1,njeq,n′

j1,n

jeq,n′

2

jeq,n′j2,n

j3,n

1

3jeq,n′

4

Figure 7.3: Side view of a sample unit cell. Equivalent surface Seq is drawn in red. Rest of the surfaces(in black) are interior surfaces. Regions V1 and V2 are dielectric regions. V3 is an air region that isintroduce in order to ensure that all PEC surfaces (except for the ground plane) are strictly inside thebox surface. Special junctions are shown in red, blue, green, and cyan colors and labeled 1 , . . ., 4 .

Mr =[mr,1 . . . mr,N ′r

]T.

We also expand the tangential electric and magnetic fields on Seq with RWG basis functions

n× ~Heq(~r) = ~Jeq(~r) =

Neq∑n=1

jeq,n~Λeq,n(~r) (7.2a)

−n× ~Eeq(~r) = ~Meq(~r) =

N ′eq∑n=1

meq,n~Λeq,n(~r) , (7.2b)

where the unit normal vector n points in the outer region (free space), Neq is the number of

RWG basis functions used to discretize the tangential magnetic field, and N ′eq is the number of

RWG basis functions used to discretize the tangential electric field. The tangential magnetic and

electric field coefficients in (7.2a)–(7.2b) are collected into vectors Jeq =[jeq,n . . . jeq,Neq

]Tand Meq =

[meq,n . . . meq,N ′eq

]T, respectively. Furthermore, we collect electric and magnetic

field coefficients for the Seq into a vector Xeq =[JTeq MT

eq

]T.

7.2.2 Surface Integral Equations

We now apply the surface integral equation method to generate a macromodel for the m-th

element of the array. Forming such a macromodel is interesting from a theoretical perspective

and efficient from a numerical perspective. It will ultimately allow us to model an array of

complex scatterers, such as the one shown in Fig. 7.2a, with an array of equivalent current

densities on fictitious surfaces, as shown in Fig. 7.2b.

According to the Love’s equivalence principle [91], we can relate tangential electric and


magnetic fields on Sr through the T-EFIE and T-MFIE

−n× ~Mr(~r) = −jωµ0n× n×[~Lr ~Jr(~r ′)

](~r)− n× n×

[~Kr ~Mr(~r

′)]

(~r) , (7.3a)

n× ~Jr(~r) = −jωε0n× n×[~Lr ~Mr(~r

′)]

(~r) + n× n×[~Kr ~Jr(~r ′)

](~r) , (7.3b)

where operators ~Lr and ~Kr are given in (2.15a)–(2.15b) and are evaluated with the Green’s

function of the r-th region of the m-th element. If the r-th region is anistropic, then the

Green’s function of an anisotropic medium needs to be used. Note that the T-MFIE is only

used for non-PEC surfaces.

Next, we substitute (7.1a)–(7.1b) into (7.3a)–(7.3b) and test the resulting integral equa-

tions with RWG basis functions. For the r-th region, we obtain the following linear system of

equations [LEr KE

r

KHr LHr

][Jr

Mr

]=

[0

0

], (7.4)

where entries of matrices LEr , KEr , KH

r , and LHr are as defined in (2.57a)–(2.57d), but computed

with the Green’s function of the r-th region. We can collect (7.4) for all R regions into a larger

system of linear equations

[LE1 KE

1

KH1 LH1

]0 0

0. . . 0

0 0

[LER KE

R

KHR LHR

]

︸︷︷︸

Z

[J1

M1

]...[

JR

MR

]

︸︷︷︸

X

=

[0

0

]...[0

0

]

. (7.5)

To simplify the presentation of subsequent sections, we will denote the block-diagonal matrix

in (7.5) with Z and the vector of field coefficients with X. Note that the right-hand side of (7.5)

is zero due to an absence of any sources inside the Seq. If there are internal sources within the

region due to lumped ports, then the right-hand side will be non-zero and the formulation needs

to be updated to reflect this change.

7.2.3 Enforcement of Boundary Conditions

Discretizing tangential electric and magnetic fields inside Seq creates duplicated basis functions

for some surfaces. For example, we discretized fields for S12 and S1

3 for the unit cell shown in

Fig. 7.3. However, S12 and S1

3 are on two different sides of the same interface. Therefore, the

field quantities used to expand S12 and S1

3 are related by electromagnetic boundary conditions.

Now, let us enforce boundary conditions on electromagnetic fields and eliminate unnecessary

unknowns in the m-th element. As explained in Sec. 2.5.3, we will enforce boundary condi-

tions using incidence matrices. Following the removal of redundant unknowns, the final set of


unknowns for the m-th element are collected into a vector

X =[XT

eq XTint

]T, (7.6)

where Xeq collects unknown field coefficients on Seq that appear in (7.2a)–(7.2b) and Xint

collects the rest of the unknowns associated with field quantities inside Seq. We relate X to X

by

X = UX , (7.7)

where U is a sparse matrix with a few entries per row, which may be ±1. This matrix serves

two purposes. First, it eliminates redundant unknowns by explicitly enforcing continuity of

tangential electric and magnetic fields on the interfaces between two or more regions. Second,

it rearranges the list of unknowns in order to group unknowns on Seq and unknowns inside Seq.

To discuss how to enforce all boundary conditions, we reconsider the sample unit cell shown

in Fig. 7.3. Let us first discuss boundary conditions that need to be enforced on fields inside

Seq. These boundary conditions were discussed in Sec. 2.5.3 and are summarized below:

1. Interface of dielectric regions: The tangential electric and magnetic fields are con-

tinuous on the interface of two dielectric regions. Hence, expansion coefficients jr,n and

mr,n in (7.1a)–(7.1b) on the two sides of the interface are equal, assuming the RWG basis

functions are oriented in opposite directions. Therefore, they can be expressed in terms

of a single unknown that is collected into Xint. For example, tangential electric and field

coefficients on S33 and S3

2 in Fig. 7.3 are equal.

2. PEC surface at the interface of two regions: The tangential magnetic fields on the

two sides of a PEC surface are independent. For example, in Fig. 7.3, the tangential mag-

netic field coefficients on S23 are independent of the tangential magnetic field coefficients

on S22 . Hence, we have two independent unknowns that are both collected into Xint.

3. PEC-dielectric junctions: A PEC-dielectric junction is defined on the edge of a PEC

surface, where half of the RWG basis function is on a PEC surface and the other half

is on a dielectric interface. Due to continuity of electromagnetic fields on the interface,

the coefficients for tangential magnetic fields are continuous on the interface and can

be related to a unique unknown, which is collected into vector Xint. The tangential

electric field coefficient on this junction, on the other hand, is zero. An example of a

PEC-dielectric junction is on the edges of S23 and S1

3 in Fig. 7.3.

As discussed above, after enforcing the above boundary conditions, all unique unknowns will

appear in Xint. The following boundary conditions need to be enforced on Seq:

1. Interface between the equivalent box and an interior region: We consider a

sample interface on Seq between the exterior medium and an interior region V1 that is


shown in red and labeled 1 in Fig. 7.3. On this interface, due to continuity of the

tangential magnetic field, j1,n is equal to jeq,n′ for some values of n and n′, assuming that

basis functions ~Λ1,n and ~Λeq,n′ are oriented in directions shown in Fig. 7.3. This boundary

condition can be enforced by setting 1 in entries (qj1,n , qjeq,n′ ) and (qjeq,n′ , qjeq,n′ ) of U,

where qj1,n and qjeq,n′ are, respectively, the index of j1,n and jeq,n′ in X. Likewise, qjeq,n′

is the index of jeq,n′ in the vector Xeq. Continuity of the tangential electric field can be

enforced similarly.

2. Junction on Seq between two interior regions and the outer region: We consider

a sample junction on Seq that is shown in blue and labeled with 2 in Fig. 7.3. On this

junction, we need to enforce continuity of the tangential electric and magnetic fields1. We

can enforce the tangential magnetic field continuity by setting the two electric current

coefficients j2,n and j3,n that are inside V2 and V3, respectively, to be equal to jeq,n′ on

Seq for some values of n, n′, and n. We can enforce this boundary condition by setting 1

in entries (qj2,n , qjeq,n′ ), (qjeq,n′ , qjeq,n′ ), and (qj3,n , qjeq,n′ ), where qj2,n and qj3,n are indices

of j2,n and j3,n in X. We can enforce continuity of the tangential electric field similarly.

3. PEC ground plane: We consider the PEC ground plane junction shown in green and

labeled with 3 in Fig. 7.3. On this junction, the tangential electric field is zero. The sur-

face electric current densities on the two sides of the interface are independent. Therefore,

two unique unknown coefficients are required to properly model this boundary condition.

One of these unknowns is inside Seq and is collected in Xint. The other is outside Seq and

is, therefore, collected in Xeq. To enforce this boundary condition, we need to insert 1 in

entry (qj1,n , qj1,n) of U, where qj1,n is the index of j1,n in X. We do not need an entry for

the jeq,n′ since it does not appear in X.

4. Junction at edges of a PEC ground plane: Let us consider the junction at an edge

of a PEC ground plane shown in cyan and labeled with 4 in Fig. 7.3. The tangential

electric field on this junction is zero. Furthermore, the tangential magnetic fields on the

two sides of the interface may or may not be independent depending on whether or not

this unit cell is connected to other array unit cells. If the element is connected to another

array element, then the current on two sides of the interface will be independent, needing

two unknowns. These two unknowns are collected into Xeq. We can implement this

condition by setting 1 in the entry (qj1,n , qj1,n) of U. If the element is not connected to

another array element, then the tangential magnetic fields on two sides of the interface

are equal. This condition will be enforced in Sec. 7.3.3.

By substituting (7.7) into (7.5), we obtain

ZUX = 0 (7.8)

1Recall that we had considered junction boundary conditions in Sec. 2.5.3, as illustrated in Fig. 2.10d.


which is an over-determined linear system of equations. As discussed in Sec. 2.5.3, we can

eliminate additional equations in such cases by simply left-multiplying the equation by UT .

This multiplication eliminates additional equations according to the PMCHWT formulation by

adding the discretized T-EFIE and T-MFIE for the regions that share an interface or a junction.

7.2.4 Macromodel Generation

After performing the matrix multiplication in (7.8), we obtain a linear system of the form

UTZUX =

[Zeq,eq Zeq,int

Zint,eq Zint,int

][Xeq

Xint

]=

[0

0

]. (7.9)

Next, we eliminate Xint from our formulation by Schur complement and obtain[Zeq,eq − Zeq,intZ

−1int,intZint,eq

]︸︷︷︸

Zeq,eq

Xeq = 0 . (7.10)

Notice that after eliminating Xint from (7.10), we have unknowns only on Seq. If the unit cell

has many electrically fine features, then all the unknowns associated with these features are

eliminated in (7.10). This is advantageous when simulating complex electromagnetic structures

that typically have PEC traces with fine features. The proposed technique can lead to savings

even when simulating an array of square patch antennas, which need to be meshed with very

small triangular elements in order to capture edge singularities in the current distribution [158].

The behavior of the scatterer inside Seq is accurately captured by Zeq,eq in (7.10), which can

be considered as a macromodel matrix. As evident from (7.10), macromodeling requires an LU

factorization in order to eliminate interior unknowns. For a very complex unit cell, this step can

be expensive. However, since we operate on a single unit cell, the relative complexity of this step

is low compared to solving the entire array. Another advantage of the proposed macromodeling

approach is that we only need to generate macromodels for unique elements. Therefore, even

a complete electromagnetic surface with thousands of array elements will typically require

generation of only 20–30 macromodels.

7.3 Simulation of Electromagnetic Structures

7.3.1 Array of Macromodels

We now consider simulation of large electromagnetic structures with M unit cells. Throughout

the rest of this section, we will append the notation for all quantities defined earlier with

superscript (m) to denote element number in the array. To simulate electromagnetic structures,

we first create a macromodel for each unique element in the array. Then, (7.10) for all elements


can be written as Z

(1)eq,eq

. . .

Z(M)eq,eq

X(1)eq

...

X(M)eq

=

0...

0

(7.11)

where Z(m)eq,eq and X

(m)eq are used to denote, respectively, the macromodel matrix and the list of

field coefficients on S(m)eq , the equivalent surface for the m-th element.

7.3.2 Inter-Element Coupling

Next, we apply the Love’s equivalence principle for the exterior medium (free space). We

remove the scatterers inside S(m)eq and model their effects through the equivalent electric current

density ~J(m)

eq (~r) = n× ~H(~r) and the equivalent magnetic current density ~M(m)eq (~r) = −n× ~E(~r)

introduced on S(m)eq . Furthermore, equivalent electric and magnetic current densities on S(m)

eq

are related by the T-EFIE and T-MFIE

n× ~M (m)eq (~r) =n× n× ~Einc(~r)− jωµ0n× n×

(M∑n=1

[~Lo ~J (n)

eq (~r ′)]

(~r)

)

+ n× n×(

M∑n=1

[~Ko ~M (n)

eq (~r ′)]

(~r)

)(7.12a)

−n× ~J (m)eq (~r) =n× n× ~H inc(~r) + jωε0n× n×

(M∑n=1

[~Lo ~M (n)

eq (~r ′)]

(~r)

)

− n× n×(

M∑n=1

[~Ko ~J (n)

eq (~r ′)]

(~r)

), (7.12b)

where the ~Lo and ~Ko operators are computed with material properties of free space (background

medium), and ~Einc and ~H inc are the incident electric and magnetic fields due to the excitation

source.

We discretize the integral equations in (7.12a)–(7.12b) by testing them with RWG basis

functions. Discretized (7.12a)–(7.12b) can be compactly written asZ

(1,1)o Z

(1,2)o . . . Z

(1,M)o

Z(2,1)o Z

(2,2)o . . . Z

(2,M)o

...... . . .

...

Z(M,1)o Z

(M,2)o . . . Z

(M,M)o

X(1)eq

X(2)eq

...

X(M)eq

=

V(1)

V(2)

...

V(M)

, (7.13)

where

Z(m,n)o =

[LE,(m,n)o K

E,(m,n)o

KH,(m,n)o L

H,(m,n)o

](7.14)

stores the discretized T-EFIE and T-MFIE when source basis functions are on S(n)eq and test


j(m)eq,nj

(m)eq,n′

j(m)eq,n

3

1

2j

(m)eq,n

∆→ 0

j(m′)eq,n′

j(m′)eq,n′

Figure 7.4: A sample array of two equivalent surfaces that are touching one another. In the graphics,we have shown some space between the boxes to draw current directions. Three boundary conditionsthat need to be enforced when equivalent surfaces are connected are shown in red, blue, and cyan.

basis functions are on S(m)eq . Matrix Z

(m,n)o captures the mutual coupling between macromodels

of the m-th and n-th unit cells of the array. Entries of LE,(m,n)o , L

H,(m,n)o , K

E,(m,n)o , and K

H,(m,n)o

can be evaluated as (2.57a)–(2.57d) with the free space Green’s function. In (7.13), Vm is the

excitation vector that is generated by testing the incident electric and magnetic fields with

RWG basis functions on S(m)eq .

7.3.3 Boundary Conditions

Now, we have two sets of equations. The first set of equations is the macromodel equation (7.10),

which captures electromagnetic behavior of the scatterer inside S(m)eq . The second set of equa-

tions is the discretized T-EFIE and T-MFIE for the exterior problem (7.13), which captures

the coupling between macromodels of all unit cells. Hence, this is an overdetermined problem.

As in the PMCHWT formulation, we will add up the two sets of equations to form a full rank,

well-conditioned system of linear equations of the form

Z

(1)eq,eq

. . .

Z(M)eq,eq

︸︷︷︸

Zeq

+

Z

(1,1)o Z

(1,2)o . . . Z

(1,M)o

Z(2,1)o Z

(2,2)o . . . Z

(2,M)o

...... . . .

...

Z(M,1)o Z

(M,2)o . . . Z

(M,M)o

︸︷︷︸

Zo

X

(1)eq

X(2)eq

...

X(M)eq

︸︷︷︸

I

=

V(1)

V(2)

...

V(M)

︸︷︷︸

V

, (7.15)

which is compactly written as

(Zeq + Zo) I = V . (7.16)


7.3.4 Modeling Connected Equivalent Surfaces

When simulating planar electromagnetic structures, S(m)eq for the m-th unit cell touches adjacent

equivalent surfaces. Therefore, tangential electric and magnetic fields on a common surface may

have been expanded with a duplicated set of basis functions, resulting in redundant unknowns.

We eliminate redundant unknowns and enforce proper boundary conditions through another

sparse matrix Uo. Matrix Uo relates I to a vector of unique unknowns IR by

I = UoIR . (7.17)

To discuss boundary conditions, we consider a sample array with two equivalent surfaces, as

shown in Fig. 7.4. Matrix Uo is generated by applying the following boundary conditions:

1. Fields on a surface common to two equivalent surfaces: On a surface that is

shared by two equivalent surfaces, the tangential electric and magnetic fields are equal,

as shown in red and labeled with 1 in Fig. 7.4. Therefore, the electric current density

coefficient j(m)eq,n on S(m)

eq is equal to j(m′)eq,n′ on S(m′)

eq , as shown in Fig. 7.4.

2. PEC ground plane connection: As discussed in Sec. 7.2.3, the electric current density

on the two sides of a ground plane on the edge of a unit cell are independent. When S(m)eq

and S(m′)eq are connected, we need to correctly model continuity of the electric current

density on both sides of the ground plane. Therefore, on both sides of the ground plane,

we equate the electric current density coefficients j(m)eq,n on S(m)

eq and j(m′)eq,n′ on S(m′)

eq , as

shown in blue and red (labeled with 2 ) in Fig. 7.4.

3. Edges of a PEC ground plane: We consider an edge of a PEC ground plane shown in

cyan and labeled with 3 in Fig. 7.4. On this edge, the electric current coefficients j(m)eq,n

and j(m)eq,n′ on the two sides of the ground plane are equal in order to satisfy the continuity

of the tangential magnetic field.

We substitute (7.17) into (7.16), and left-multiply the resulting equation by UTo to eliminate

additional equations through the PMCHWT formulation. The final equation is given by

UTo [Zeq + Zo] UoIR = UT

o V︸︷︷︸VR

, (7.18)

where VR is the excitation vector obtained after enforcing boundary conditions. We can

solve (7.18) using a direct or an iterative solver to obtain field coefficients on S(m)eq for m =

1, . . . ,M . Once we have computed the tangential electric and magnetic fields on the equivalent

surface of each unit cell, we can compute fields scattered from the electromagnetic structure

through ~Lo and ~Ko operators (2.15a)–(2.15b) with the free space Green’s function. Note that

the final set of unknowns in (7.18) correspond to unknowns that are only on S(m)eq , and not on the

scatterer inside S(m)eq . Therefore, we need to solve for fewer unknowns, which is computationally

more efficient.


7.4 Acceleration for Large Arrays

7.4.1 Iterative solver

When simulating large electromagnetic structures with many unit cells, solving the linear system

of equations in (7.18) with LU factorization is computationally expensive. Therefore, we solve

the linear system of equations iteratively with GMRES [150]. For this step, we need an efficient

preconditioner and a fast technique to compute matrix-vector products.

In our formulation, we use the preconditioner matrix

P = UTo

[ZNF

eq + ZNFo

]Uo , (7.19)

where ZNFeq and ZNF

o collect near-field entries of Zeq and Zo, respectively. That is, the (p, q)-th

entry of ZNFeq and ZNF

eq is non-zero, and is equal to the (p, q)-th entry of Zeq and Zo, respectively,

if the distance between basis functions associated with the p-th and q-th unknowns is less than

∆NF. In our simulations, we use ∆NF to be between λ0/10 and λ0/6. We apply the right-

preconditioner, solving

UTo [Zeq + Zo] UoP

−1 PIR︸︷︷︸IR

= VR , (7.20)

for IR. To solve (7.20), we need to evaluate P−1x, given some vector x. Since P is very sparse,

we use LU factorization to compute P−1x.

An iterative solver requires the computation of

UTo [Zeq + Zo] Uox , (7.21)

given x. In this matrix-vector multiplication, x1 = Uox is very cheap to compute because Uo

is a highly sparse matrix. Furthermore, since Zeq is a block-diagonal matrix, matrix-vector

product Zeqx1 is also inexpensive. However, Zox1 is expensive to compute because Zo is a

dense matrix. Furthermore, if we store Zo in a dense format, then it requires a lot of memory.

Therefore, we need to apply an acceleration algorithm to compute Zox1. We can apply the

MLFMM or AIM to accelerate the computation of Zox1. However, in our method, we used an

FFT-based algorithm that exploits the Toeplitz-like structure of Zo. Unlike the MLFMM [50]

or even the FFT-based AIM technique [26], which requires some approximations, the presented

technique is accurate without any approximations. Previously, this technique was applied to

analyze arrays of identical PEC scatterers [81]. However, in conjunction with the proposed

macromodel approach, this algorithm will allow simulation of an array of distinct elements,

which makes it ideal to simulate planar electromagnetic structures.


1, a

2, b

3, a

4, b

5, a

6, b

Figure 7.5: Top view of a sample array of three equivalent surfaces (boxes) with identical meshes.Each surface has two identical basis functions. Each basis function has a local and a global identificationnumber. Local identification numbers are denoted by a and b. Global identification numbers are denotedby 1, . . . , 6.

7.4.2 Evaluation of Matrix-Vector Product with the FFT

To discuss how to accelerate the computation of Zox with the fast Fourier transform [152], we

consider the sample array of equivalent surfaces shown in Fig. 7.5. This array has three equiv-

alent surfaces that are uniformly spaced and have identical meshes. For the sake of simplicity,

let us assume that electric and magnetic current densities on each surface are expanded with

only two basis functions2. For simplicity, we also assume that there is no ground plane at the

bottom of the equivalent surfaces. Each basis function in the array is assigned a local and

a global identification number. Local identification numbers are denoted by a and b. Global

identification numbers are denoted by 1, . . . , 6.

We can see from (7.14) that Zo is generated by discretizing the ~Lo and ~Ko operators.

Therefore, Zox can be evaluated by multiplying the discretized ~Lo and ~Ko operators, scaled by

appropriate constants, with a subset of the column vector x. Since equivalent surfaces have

identical meshes and are uniformly spaced, and the free space Green’s function is translation-

invariant, both the discretized ~Lo and ~Ko operators can be cast into Toeplitz matrices. A short

review on Toeplitz and circulant matrices is presented in Appendix C. For the example shown

in Fig. 7.5, the matrix-vector product with the discretized ~Lo operator can be written as

Laa11 Lab12 Laa13 Lab14 Laa15 Lab16

Lba21 Lbb22 Lba23 Lbb24 Lba25 Lbb26





Ja1Jb2Ja3Jb4Ja5Jb6

, (7.22)

where Lijmn denotes the reaction term due to the n-th source basis function (j-th local basis

function) and the m-th test basis function (i-th local basis function). Similarly, J in is the n-th

source coefficient.

Now let us rearrange the matrix and the excitation vector by grouping together identical

2Typically, tangential electric and magnetic fields are discretized with 300-500 RWG basis functions


basis functions on the equivalent surfaces. By doing this, we obtain

Laa11 Laa13 Laa15 Lab12 Lab14 Lab16



Lba21 Lba23 Lba25 Lbb22 Lbb24 Lbb26



Ja1Ja3Ja5

Jb2Jb4Jb6

, (7.23)

where the matrix is subdivided into 4 blocks, each of which collects reaction integrals between

a pair of local basis functions. We can compactly write (7.23) as[Laa Lab

Lba Lbb

][Ja

Jb

], (7.24)

where Laa, Lab, Lba, and Lbb are 3× 3 matrices. We can also rewrite (7.24) as[Laa Lab

Lba Lbb

][Ja

Jb

]=

[LaaJa + LabJb

LbaJa + LbbJb

]. (7.25)

Since Laa, Lab, Lba, and Lbb have Toeplitz structure, we can use FFT to compute the matrix-

vector product. Here, we demonstrate how to use FFT to compute LaaJa, other matrix-vector

products can be computed similarly. To compute LaaJa, we augment Laa to form a circulant

matrix

L′aa =

Laa11 Laa13 Laa15 Laa51 Laa31





, (7.26)

where we have color coded the entries that have the same value. Using the circulant matrix

L′aa, the matrix-vector product is

[LaaJa

][∗∗

] =






Ja1Ja3Ja50

0

, (7.27)

where we have used ∗ in the left-hand side vector to indicate values that are not useful for us.


As discussed in Appendix C, the right-hand side of (7.27) can be calculated as[LaaJa

][∗∗

] = F−1[Laa · Ja

], (7.28)

where F−1[.] is the inverse fast Fourier transform operator, “·” denotes point-wise multiplica-

tion, and vectors Laa and Ja are

Laa = F[Laa11 Laa13 Laa15 Laa51 Laa31

](7.29a)

Ja = F[Ja1 Ja3 Ja5 0 0

]. (7.29b)

F [.] is the fast Fourier transform operator. The matrix-vector products involving the discretized

~Ko operator can be accelerated using a similar procedure. Note that, while we only demon-

strated the matrix-vector product acceleration for a 1-D array, the technique can be generalized

to 2-D and 3-D arrays using higher-dimensional FFTs.

7.4.3 Discussion

Recall that in our formulation there are M equivalent surfaces, and the tangential electric and

magnetic fields on each surface are discretized with Neq RWG basis functions. Therefore, the

computational cost of the matrix-vector product Zox with the direct approach is O(N2eqM

2).

The direct approach also requires storing 4N2eqM

2 complex numbers for Zo. In comparison,

the computational cost of evaluating the matrix-vector product with the proposed Toeplitz

method is O(N2

eqM log2M), which scales roughly linearly with the number of array elements.

Furthermore, the Toeplitz approach requires storing only 2d+1N2eqM complex numbers, where

d is the dimensionality of the array. For a 30×30 array, this translates to memory savings of 15

times compared to the direct approach. The proposed Toeplitz acceleration method also avoids

a lot of overhead costs associated with other acceleration methods, like the MLFMM and AIM.

For example, the AIM requires computing and storing projection and interpolation matrices,

which are not needed with the proposed method. Memory cost to store the near-field matrix

entries in the AIM and MLFMM is roughly the same as the Toeplitz-based approach.


In this section, we validate the proposed macromodeling approach for composite structures by

considering two reflectarray examples.


(a) 16× 16 Array (b) 30× 30 Array

Figure 7.6: Top view of reflectarrays considered in Sec. 7.5.1

7.5.1 Single-Layer Reflectarray with Square Elements

Let us first consider a single-layer reflectarray made up of square patch elements. This test

case was previously presented in another paper [158]. The reflectarray dielectric substrate

has relative permittivity of εr = 3.66 and thickness of 0.762 mm. The dielectric substrate

is backed by a PEC ground plane. The reflectarray is designed to steer the main beam in

the (φ = 180, θ = 30) direction. We obtained the design of this reflectarray from TICRA to

compare the results obtained from the macromodeling solver and TICRA’s GRASP solver [158].

We simulated two different sizes of this reflectarray: 16× 16 and 30× 30. Each unit cell of the

reflectarray is 13.5 mm× 13.5 mm. The 16× 16 reflectarray is a subset of the 30× 30 array, i.e.

it is created using patch sizes of the middle 256 elements of the 30×30 reflectarray. Patch sizes

vary between 5.4 mm and 10 mm. The reflectarray is excited by a linearly-polarized corrugated

horn antenna operating at f = 9.6 GHz. In our simulation, we modeled the horn antenna

with spherical wave expansion derived from a measured horn antenna [158]. The reflectarray is

placed in the xy plane and is centred about the z axis. It is offset fed by a horn antenna that

is centered at (0.30 m, 0, 0.51962 m) and points towards the center of the reflectarray, i.e. the

horn antenna is 30 off the axis of the reflectarray.

16× 16 Array

This 256-element array was designed with nine distinct square patch sizes. A top view of this

reflectarray is shown in Fig. 7.6a. We simulated this reflectarray with FEKO and the proposed

macromodeling technique. Accurate modeling of singular current distribution on patch antennas

required a very fine mesh. Therefore, we meshed patch antennas with a characteristic mesh

length of 0.80 mm. Using a coarser mesh for patch antennas did not properly model the singular

behaviour of the current distribution on the edges, affecting the directivity calculations. This


−150 −100 −50 0 50 100 150

−20

−10

0

10

20

Theta [deg]

Dir

ecti

vit

y [

dB

i]

Proposed

FEKO

(a) φ = 0

−150 −100 −50 0 50 100 150

−30

−20

−10

0

10

20

Theta [deg]

Dir

ecti

vit

y [

dB

i]

Proposed

FEKO

(b) φ = 90

−150 −100 −50 0 50 100 150

−20

−10

0

10

20

Theta [deg]

Dir

ecti

vit

y [

dB

i]

Proposed

FEKO

(c) φ = 45

Figure 7.7: Directivity of the 16 × 16 reflectarray considered in Sec. 7.5.1 calculated with FEKO andwith the proposed technique.


problem is well known in the literature and many papers have proposed higher-order basis

functions to tackle this problem [158]. While the higher-order basis functions may work well

with square patch elements, they may not be ideal for other unit cell geometries that have many

fine features. The characteristic mesh length along the dielectric substrate was chosen to be

1.75 mm in both techniques. In the macromodeling approach, each unit cell was enclosed by

a box of size 13.5 mm× 13.5 mm× 2 mm, with the top region set to have material properties

of air. The bottom surface of each box coincided with the PEC ground plane. The equivalent

surface was discretized with the characteristic mesh length of 2.5 mm. With the chosen mesh

lengths, patch antennas and substrate-air interfaces had more than twice as many triangles as

the equivalent surfaces. Therefore, the proposed macromodeling technique was anticipated to

aid significantly in reducing the computational complexity of this problem.

Figure 7.7 shows the directivity of the reflectarray in the φ = 0, φ = 45, and φ =

90 cuts. Results obtained with the macromodeling approach and FEKO match very well,

validating the proposed technique. Breakdown of computational time and memory required

to solve this problem with the macromodeling approach and FEKO is presented in Tab. 7.1.

All computations were performed with a single thread and double precision. We can observe

that the macromodel approach is 14 times faster and requires 8 times lower memory than

FEKO, which uses the MLFMM to simulate the problem. Since the MLFMM requires accurate

computation of near-field interactions3, it is not suitable for multiscale problems with fine mesh

size because the cost to compute and store near-field interactions is extremely high. In our

approach, near-field interactions need to be computed accurately for a single unit cell, regardless

of the size of the unit cell, which makes the approach efficient for multiscale problems.

Table 7.1. Simulation Statistics for the 16× 16 reflectarray considered in Sec. 7.5.1

FEKO Proposed

Total number of unknowns 479,562 177,924Memory used 307.9 GB 37.1 GBMacromodel generation N/A 10.9 minMatrix fill time 5.0 h 14.9 minPreconditioner factorization 8.7 h 23.8 minIterative solver 1.4 h 10.8 minTotal computation time 15.2 h 65 min

30× 30 Array

Now, let us consider the reflectarray of size 30×30. This reflectarray is composed of 32 distinct

unit cells and was previously analyzed in the literature [158]. For the proposed macromodeling

3Typically, source and test basis functions within λ0/2 of each other need to be evaluated accurately in theMLFMM.


−150 −100 −50 0 50 100 150−30

−20

−10

0

10

20

30

Theta [deg]

Dir

ecti

vit

y [

dB

i]

TICRA

Measurement

Proposed

Figure 7.8: Directivity of the 30 × 30 reflectarray considered in Sec. 7.5.1 in the φ = 0 cut calculatedwith measurements, TICRA MLFMM solver [159], and the proposed technique.

approach, we used the same mesh settings as the 16× 16 case. However, it was not possible to

simulate the structure with the same mesh settings in FEKO due to insufficient memory. The

directivity obtained with the proposed macromodeling approach, TICRA MLFMM solver, and

measurements [158] is presented in Fig. 7.8. Directivity obtained with the proposed method

agrees very well against the results from TICRA MLFMM solver and experimental results,

accurately capturing nulls and peak directivity. The solver also correctly predicts null fillings

due to a strong mutual coupling between array elements.

Simulation statistics with the proposed macromodeling approach for this test case are sum-

marized in Tab. 7.2. Since FEKO results are inaccurate, we have not reported timings and

memory usage of that simulation. As summarized in Tab. 7.2, the proposed macromodeling

approach took only 5.9 h to simulate this structure on a single thread. Simulation of this struc-

ture required memory usage of 144 GB. Simulating this structure accurately (with fine mesh)

in FEKO would have required an estimated 1230 GB of memory. The ability of the proposed

technique to accurately simulate such a large structure highlights the potential of the proposed

macromodeling technique.

7.5.2 Two-Layer Reflectarray with Jerusalem Cross Elements

We now consider a 20×20 two-layer dual-polarized reflectarray made up of Jerusalem crosses [154].

The top view of this reflectarray is shown in Fig. 7.10. This example was chosen to demonstrate

that the proposed macromodeling approach can simulate electromagnetic surfaces with multiple

layers and complex unit cells. The reflectarray is composed of 11 distinct unit cells. Each unit

cell has dimensions of 10 mm×10 mm. A sample unit cell is shown in Fig. 7.9. The reflectarray

substrate has two layers, each with a thickness of 0.762 mm. The relative permittivity of the


w

w

h

h

Figure 7.9: Unit cell of the two-layer reflectarray considered in Sec. 7.5.2 with w = 10 mm, h = 0.762 mm.Reflectarray has a two-layer dielectric substrate (shown in green and yellow). The top layer has relativepermittivity of εr = 3.0 and the bottom layer has relative permittivity of εr = 2.2. The reflectarray isbacked by a PEC ground plane.

bottom and top layers of the substrate is εr = 2.2 and εr = 3.0, respectively. The reflectarray

is center fed by a horn antenna operating at f = 10 GHz, which is modeled with a spherical

wave expansion of a measured horn. The horn antenna is placed 0.4 m along the axis of the

reflectarray.

This reflectarray is multiscale. It has dimensions of 6.66λ0×6.66λ0, where λ0 is wavelength

in free space, while each of its unit cell is only λ0/3× λ0/3. Furthermore, the size of Jerusalem

crosses in each unit cell is between λ0/5 to λ0/4, while their widths are approximately λ0/15.

Simulation of reflectarray of this size and complexity is difficult, if at all possible, with existing

integral equation solvers. As such, due to insufficient memory, we could not simulate even a 5×5

reflectarray with similar unit cells in FEKO using the MLFMM solver on a 256 GB machine.

However, the proposed macromodeling solver was able to simulate this 20×20 reflectarray using

129.3 GB memory. This was only possible because, in the proposed method, only 553 unknowns

were required for each unit cell, as opposed to the 6, 847 unknowns (on average) required with

the traditional surface integral equation method based on the PMCHWT formulation. Overall,

this meant that a total of 329, 368 unknowns had to be solved with the proposed macromodeling

technique, instead of an estimated 3, 080, 000 unknowns needed to be solved with the surface

integral equations based on the PMCHWT formulation. Simulation of this reflectarray took

10.3 h on a single thread with the proposed macromodeling approach.

The reflectarray was designed to radiate the main beam in the broadside direction. Fig-

Table 7.2. Simulation Statistics for the 30× 30 reflectarray considered in Sec. 7.5.1

Proposed

Total number of unknowns 702,468Memory used 143.8 GBMacromodel generation 21.2 minMatrix fill time 54.63 minPreconditioner factorization 2.71 hIterative solver 0.95 hTotal computation time 5.1 h


Figure 7.10: Top view of patch reflectarrays of various sizes considered in Sec. 7.5.2

ure 7.11 shows the scattered field directivity of the reflectarray in the φ = 0 and φ = 90

cuts. Directivity obtained for this reflectarray could not be validated against FEKO, which

required a lot of memory. However, we validated the results against array factor theory using

the reflection coefficient of each element obtained from a periodic EM solver [11]. As shown

in Fig. 7.11, the side lobe levels and null locations obtained with the proposed solver and the

array factor theory are off. This difference can be attributed to the following two limitations of

array factor theory:

• Reflection coefficients used for the array factor analysis are obtained assuming perfect

periodicity. This assumption is violated in the designed reflectarray.

• Array factor theory neglects edge effects.

The ability to analyze such a complex reflectarray demonstrates the potential of the proposed

macromodeling technique.

7.6 Summary and Contributions

In this chapter, we presented a macromodeling approach to efficiently simulate complex elec-

tromagnetic structures such as reflectarrays and metasurfaces. The proposed approach is based

on the equivalence principle. In this approach, each element of an electromagnetic structure

is modeled with equivalent electric and magnetic current densities that are introduced on an

equivalent surface enclosing the element. The main idea is to solve for unknowns on equivalent

surfaces instead of unknowns on the scatterers. For this to work, we apply the Stratton-Chu for-

mulation to generate a macromodel matrix that captures the electromagnetic behavior of each


−80 −60 −40 −20 0 20 40 60 80−20

−10

0

10

20

30

Theta [deg]

Dir

ecti

vit

y [

dB

i]

Proposed

Array factor

(a) φ = 0

−80 −60 −40 −20 0 20 40 60 80−20

−10

0

10

20

30

Theta [deg]

Dir

ecti

vit

y [

dB

i]

Proposed

Array factor

(b) φ = 90

Figure 7.11: Directivity of the 20× 20 reflectarray considered in Sec. 7.5.2.

element in terms of the fields on its enclosing equivalent surface. For electromagnetic structures

with complex geometries, the proposed approach leads to significant savings in terms of memory

and computational times compared to the traditional surface integral equation method based

on the PMCHWT formulation. To solve electrically large structures, we also introduced a fast

FFT-based acceleration algorithm to compute matrix-vector products. The main contributions

of this chapter include:

• In the proposed technique, we addressed the three modeling challenges of applying the

equivalence principle algorithm to simulate electromagnetic structures that were discussed

in Sec. 7.1. We solved these challenges by carefully discretizing the mesh for the simu-

lation, by rigorously enforcing electromagnetic boundary conditions when equivalent sur-

faces are connected, and by using the Schur complement. It is important to emphasize

that the usage of sparse incidence matrices simplified the technique from a programming

perspective. To the best of our knowledge, no prior works have applied the equivalence

principle algorithm to simulate complex electromagnetic structures such as reflectarrays

and metasurfaces.

• An FFT-based acceleration algorithm that exploits the Toeplitz structure of the dis-

cretized surface integral equations has never been applied to simulate non-homogeneous

arrays. In conjunction with the macromodeling approach, we use this acceleration method

to efficiently simulate electrically large electromagnetic structures with distinct unit cells.

Results presented in Sec. 7.5 demonstrated that the proposed method is accurate and can

efficiently simulate complex planar electromagnetic structures.

Chapter 8

Conclusion

The main goal of this thesis was to address simulation challenges for two types of problems:

interconnect networks and complex electromagnetic surfaces, such as reflectarrays and metasur-

faces. Because these problems are multiscale, their full-wave simulation with existing methods

requires prohibitively large memory and computation times. In this thesis, we demonstrated

how the equivalence principle and linearity of Maxwell’s equations can be exploited to develop

reduced-order modeling techniques to simulate these two problems more efficiently than existing

methods.

8.1 Research Achievements: Interconnect Network Modeling

Electromagnetic analysis of interconnects at system, board, and chip levels are important due

to increasing clock speeds and complexity of electronic systems. Interconnects are modeled

with either a 2-D approach, that is based on the transmission line theory, or a full-wave 3-D

approach. In both these approaches, the main modeling challenge is to accurately capture

the skin effect that develops inside conductors at high frequencies. All commercial simulation

tools model skin effect using a volumetric approach. In this approach, the current distribution

inside a conductor is modeled with volumetric filaments. Therefore, when the skin effect is

very pronounced, volumetric techniques require solving for a large number of unknowns, which

leads to poor performance and scalability. Therefore, many researchers have been working on

reduced-order modeling techniques in which the electromagnetic effects inside the conductors are

captured via surface unknowns. However, as discussed in Chapter 1, many of these techniques

are restricted in terms of efficiency, generality, or robustness.

In this thesis, we proposed a surface method based on the differential surface admittance

concept that was previously applied to model 2-D conductors of canonical shapes. In this

thesis, we made two new contributions. First, we demonstrated that we can generalize the

differential surface admittance approach to arbitrarily-shaped conductors by applying the con-

tour integral method and the MoM. Second, we generalized the concept of the 2-D differential

surface admittance operator to 3-D problems using the surface integral equation method. These

146

Chapter 8. Conclusion 147

generalizations allowed us to simulate a wide range of interconnect problems such as on-chip

interconnects with trapezoidal conductors, valley-shaped interconnects, curved microstrip lines,

multiconductor bus in a stratified medium, and a 3-D inductor. Furthermore, since the proposed

method only requires surface unknowns, it is also more efficient than volumetric methods such

as the FEM. The surface formulation used to model conductors resulted in between 10 to 100

times speedups and memory savings compared to the FEM when simulating the aforementioned

interconnect problems.

From a theoretical perspective, we discovered that by applying the equivalence principle

in a way different than other works in the literature, we can eliminate the magnetic current

density from the formulation. This single-source formulation is interesting from a theoretical

viewpoint and efficient from a numerical viewpoint. The single-source formulation, however,

requires LU factorization of a matrix whose size is equal to the number of basis functions used

to expand the tangential electric and magnetic fields on the object’s surface. For large objects,

this inversion could be expensive, and, in future, it is worth looking into ways to avoid this LU

factorization.

Theoretically, the proposed approach is rigorous and does not make any assumptions. How-

ever, we encountered several numerical implementation challenges. While the differential surface

admittance operator was robust at high frequencies, we discovered that the method suffered

from numerical artifacts at low frequencies. In fact, we learned that many integral equation

formulations are unstable at low-frequencies. For 2-D problems, we overcame this numerical

issue by using two different Green’s functions at low and high frequencies. While logical, the

way we treated the low-frequency issues was not rigorous, and it would be interesting to im-

prove the formulation in the future. For 3-D problems, low-frequency issues were addressed

by introducing charge density as an additional set of unknowns to the formulation [141]. The

choice of basis and testing functions was another implementation challenge that we encountered

while deriving the differential surface admittance operator for 3-D problems. We learned that if

RWG basis functions are used to expand both the tangential electric and magnetic fields, then

the test functions in the MoM need to be chosen carefully in order to obtain a well-conditioned

differential surface admittance operator. We believe that, in the future, the testing procedure

can be simplified by adopting dual basis functions for the tangential electric field. The surface

formulation can also be made more efficient by adopting higher-order basis functions to expand

the tangential electric and magnetic fields on objects’ surface.

8.2 Research Achievement: Complex Electromagnetic Struc-

ture Modeling

In 2-D and 3-D interconnect modeling problems, we achieved a reduction by turning a volumet-

ric problem into a surface problem. However, many surface problems can also be very challeng-

ing to simulate. For example, many reflectarrays and metasurfaces cannot be simulated with


full-wave solvers because they contain multiscale features and resonant elements that require

a large number of mesh elements and unknowns. Multiscale features also cause the condition

number of the final system of linear equations to be very high, which degrades the performance

of iterative solvers. During our investigation, we also learned that even popular acceleration

algorithms, such as the MLFMM, cannot tackle multiscale features adequately. For such prob-

lems, we investigated whether it was possible to model a complex scatterer with a macromodel

that requires fewer unknowns and “hides” the local behavior of a scatterer when solving the

global problem. This question was in part motivated by the Norton equivalent model in circuit

theory, which models a complex network of electrical components with a Norton equivalent

current source and an admittance.

We discovered that through a proper application of the equivalence principle and the sur-

face integral equation method it was possible to rigorously derive a macromodel for a complex

scatterer. In this thesis, we presented two main contributions on the macromodeling approach.

First, we invented a single-source macromodeling approach for PEC scatterers by generalizing

the theory that was applied to interconnect problems. Second, we invented a dual-source macro-

modeling approach to model composite scatterers made up of PECs and dielectric objects. The

dual-source approach was especially geared towards the simulation of complex electromagnetic

surfaces. The main idea behind both approaches is to solve for unknowns on a fictitious surface

enclosing the scatterer, instead of directly solving for unknowns on the scatterer. This proposed

approach requires solving fewer unknowns than the traditional surface integral equation method

when the scatterer contains multiscale features. From an accuracy viewpoint, the macromodel

approach predicted electromagnetic behavior of a complex scatterer through equivalent currents

extremely well, often with up to 10 times fewer unknowns.

In the single-source macromodeling approach, a complex PEC scatterer is modeled with

an admittance operator and an equivalent electric current density that is introduced on a

fictitious surface enclosing the scatterer. We learned that to obtain a robust macromodel, we

had to use an orthogonal set of basis functions to expand tangential electric and magnetic

fields on the equivalent surface. The single-source macromodeling approach was applied to

compute scattering from an array of complex PEC scatterers. When simulating array problems,

the macromodeling technique is computationally efficient than the traditional surface integral

equation method because of three reasons. First, it reduces the number of unknowns needed

to model each array element. Second, it exploits element repeatability that is common in large

array problems, which are typically composed of a few distinct elements. Finally, it improves

numerical properties, such as the condition number, of the linear system matrix, which results in

faster convergence of an iterative solver. Since the proposed macromodeling technique is single-

source, needing only an equivalent electric current density to model the scatterer, it is more

efficient than other equivalence principle-based techniques in the literature. For electrically large

arrays, we also integrated the macromodeling technique with an iterative solver and accelerated

matrix-vector products with the adaptive integral method.


We further generalized the macromodeling idea to simulate complex planar electromagnetic

structures that are composed of PEC traces on a dielectric substrate. For this, we developed

a macromodeling technique for composite structures, in which we model each unit cell of a

complex electromagnetic structure with equivalent electric and magnetic current densities in-

troduced on a fictitious surface enclosing the unit cell. To simulate planar electromagnetic

structures, we addressed many practical challenges from an implementation perspective. These

challenges include: how to properly enforce boundary conditions on electromagnetic fields when

adjacent equivalent surfaces are connected, how to model PEC ground plane, and how to model

equivalent surfaces that cut through a dielectric substrate. To simulate electrically large struc-

tures, we integrated the technique with an iterative solver and accelerated the computation of

matrix-vector products with the fast Fourier transform. Numerical results demonstrated that

the macromodeling technique can simulate complex reflectarrays that are tens of wavelengths

large with electrically fine features in each unit cell. Many of these structures could not be

simulated accurately even with some commercial integral equation solvers.

8.3 Contributions

Research presented in this thesis has resulted in several scientific publications. These publica-

tions are listed below.

Journal Papers

1. U. R. Patel and P. Triverio, “Skin Effect Modeling in Conductors of Arbitrary Shape

Through a Surface Admittance Operator and the Contour Integral Method,” IEEE Trans-

actions on Microwave Theory and Techniques, vol. 64, no. 9, pp. 2708–2717, Sep. 2016.

2. U. R. Patel, P. Triverio, and S. V. Hum, “A Novel Single-Source Surface Integral Method

to Compute Scattering from Dielectric Objects,” IEEE Antennas and Wireless Propaga-

tion Letters, vol. 16, pp. 1715–1718, 2017.

3. S. Sharma, U. R. Patel, S. V. Hum, and P. Triverio, “A Complete Surface Integral

Method for Broadband Modeling of 3D Interconnects in Stratified Media,” IEEE Trans.

Compon., Packag., Manuf. Technol., pp. 1–11, 2019, (submitted).

4. U. R. Patel, P. Triverio, and S.V. Hum, “A Macromodeling Approach to Efficiently

Compute Scattering from Large Arrays of Complex Scatterers,” IEEE Transactions on

Antennas and Propagation, vol. 66, no. 11, pp. 6158–6169, Nov. 2018.

Conference Papers

1. U. R. Patel, P. Triverio, and S. V. Hum, “Analysis of Radiating Microstrip Structures

Using the Contour Integral Method,” in IEEE Symposium on Antennas and Propagation


and North American Radio Science Meeting (APS), Fajardo, Puerto Rico, June 2016.

2. U. R. Patel and P. Triverio, “A Fast Surface Method to Model Skin Effect in Trans-

mission Lines with Conductors of Arbitrary Shape or Rough Profile,” in 2016 IEEE

International Conference on Signal and Power Integrity (SIPI 2016), Ottawa, Canada,

July 2016, (Finalist for Best Student Paper Award).

3. U. R. Patel, P. Triverio, and S. V. Hum, “Fast Parameter Extraction for Transmis-

sion Lines with Arbitrary-Shaped Conductors and Dielectrics Using the Contour Integral

Method,” in 25th Conference on Electrical Performance of Electronic Packaging and Sys-

tems (EPEPS 2016), San Diego, CA, USA, Oct. 2016, (Finalist for Best Student

Paper Award).

4. U. R. Patel, P. Triverio, and S. V. Hum, “A Single-Source Surface Integral Equation

Formulation for Composite Dielectric Objects,” in 2017 IEEE AP-S Symposium on An-

tennas and Propagation and USNC-URSI Radio Science Meeting, San Diego, CA, USA,

July 2017, (Honorable Mention for Best Student Paper Award).

5. U. R. Patel, S. V. Hum, and P. Triverio, “A Magneto-Quasi-Static Surface Formulation

to Calculate the Impedance of 3D Interconnects with Arbitrary Cross-section,” in 21st

Workshop on Signal and Power Integrity, Lake Maggiore (Baveno), Italy, May 2017,

(Best Student Paper Award).

6. U. R. Patel, S. Sharma, S. Yang, S. V. Hum, and P. Triverio, “Full-Wave Electro-

magnetic Characterization of 3D Interconnects Using a Surface Integral Formulation,” in

26th IEEE Conference on Electrical Performance of Electronic Packaging and Systems

(EPEPS), San Jose, CA, Oct. 2017, (Best Paper Award)

7. U. R. Patel, P. Triverio, and S. V. Hum, “A rigorous macromodeling approach to

efficiently simulate large arrays of complex scatterers,” in IEEE Symposium on Antennas

and Propagation and North American Radio Science Meeting (APS), Boston, MA, 2018.

8. U. R. Patel, P. Triverio, and S. V. Hum, “A Fast Macromodeling Approach to Simulate

Complex Electromagnetic Surfaces,” in IEEE Symposium on Antennas and Propagation

and North American Radio Science Meeting (APS), Atlanta, GA, 2019. (submitted)

Note that the contribution made in Chapter 7 on the macromodeling technique for composite

structures will soon be submitted for publication.

8.4 Closing Remarks

Despite many advances in the field of computational EM over the past few decades, EM simula-

tion of large and complex interconnect networks and electromagnetic surfaces is still a significant


challenge with the practical limitations on computational resources. In this thesis, we focused on

inventing reduced-order representations of complicated systems. Reduced-order representations

allow us to abstract away from the complexity of the problem, while still accurately capturing

their behaviour. In the case of interconnect modeling, we proposed reduced-order models to

capture the physics inside each conductor of an interconnect network accurately and efficiently.

In the case of electromagnetic surfaces, we proposed reduced-order models to compactly model

the intricate features of individual elements in large EM arrays. We demonstrated that reduc-

ing the problem complexity, undoubtedly, improves the efficiency of the overall simulation, as

evident from the various numerical examples presented in this thesis. A natural progression of

this work is to find ways to further reduce the solution times of proposed techniques. There

are many ways to do this in the future. One way is to apply higher-order basis functions to

expand equivalent currents in our formulations. In comparison to RWG basis functions that are

employed in our techniques, higher-order basis functions can drastically reduce the unknowns

count, leading to faster simulations. We can also parallelize the proposed techniques to solve

problems on computing clusters. Currently, all our codes have been designed to run on a single-

thread. However, most of the algorithms presented in this thesis can be parallelized efficiently.

Another way we can improve the performance of proposed techniques is by adopting a more

sophisticated preconditioner. Another possible direction for this work is to further validate

the computational codes against experimental results. All these strategies, coupled with the

proposed reduced-order models, will reduce time needed to simulate interconnects, complex

electromagnetic surfaces, and other EM problems, which will ultimately allow improvements in

the design of electronic and communication systems.

Appendix A

Integration Routines for the Method

of Moments

Types of Reaction Integrals

In this Appendix, we discuss how to efficiently evaluate two types of reaction integrals that

are commonly needed in a method of moments solver for 3-D problems. Evaluation of reaction

integrals are needed to fill entries of matrices: LEo and KEo . The (m,n)-th entry of these

matrices are given by

[LEo]m,n

= jωµ0

⟨~Λm(~r), n× n×

[~Lo~Λn(~r ′)

](~r)⟩

(A.1)[KEo

]m,n

=⟨~Λm(~r), n× n×

[~Ko~Λn(~r ′)

](~r)⟩, (A.2)

where ~Λm(~r) and ~Λn(~r) are test and source RWG basis functions. Integral operators are com-

puted with the 3-D Green’s function of free space medium with wavenumber k = ω√µ0ε0.

Inner products in (A.1)–(A.2) can be evaluated using a combination of numerical and analytic

integration techniques. In this Appendix, we develop integration routines that are independent

of free vertices of source and test RWG basis functions. Such routines allow us to fill the method

of moments matrices triangle-by-triangle, instead of edge-by-edge. This is approximately nine

times faster than filling matrices edge-by-edge because each triangle contributes to nine RWG

edges. By substituting (2.15a)-(2.15b) into (A.1)–(A.2), we obtain

[LEo]m,n

=− jωµ0

¨Tm

¨Tn

~Λm(~r) · ~Λn(~r ′)G(~r, ~r ′)dSdS′

+jωµ0

k2

¨Tm

¨Tn

∇ · ~Λm(~r)∇′ · ~Λn(~r ′)G(~r, ~r ′)dSdS′ (A.3)

[KEo

]m,n

=

¨Tm

¨Tn

~Λm(~r) ·[~Λn(~r ′)×∇G(~r, ~r ′)

]dSdS′ . (A.4)

152

Appendix A. Integration Routines for the Method of Moments 153

Evaluation of Reaction Integrals for[LEo

]First, we consider how to evaluate the (m,n)-th entry of

[LEo]m,n

=− jωµ0

¨Tm

¨Tn

~Λm(~r) · ~Λn(~r ′)G(~r, ~r ′)dSdS′

+jωµ0

k2

¨Tm

¨Tn

∇ · ~Λm(~r)∇′ · ~Λn(~r ′)G(~r, ~r ′)dSdS′ . (A.5)

For simplicity, we only consider integration over the “plus” triangles Tn and Tm of source and

test basis functions, respectively. It is straightforward to evaluate the integrals over “minus”

triangles using the same results with a few minor changes. Recall that on the “plus” triangles

~Λm(~r) and ~Λn(~r) are RWG basis functions that are defined to be

~Λm(~r) =lm

2Am

(~r − ~Qm

)~r ∈ Tm (A.6a)

~Λn(~r ′) =ln

2An

(~r ′ − ~Qn

)~r ′ ∈ Tn , (A.6b)

where lm and Am are edgelength and triangle area of the m-th RWG basis function. We

substitute (A.6a) and (A.6b) into (A.5) to obtain

[LEo]m,n

=− jωµ0lmln

4AmAn

¨Tm

¨Tn

(~r − ~Qm

)·(~r ′ − ~Qn

)G(~r, ~r ′)dSdS′

+jωµ0

k2

1

AmAn

¨Tm

¨Tn

G(~r, ~r ′) dSdS′ , (A.7)

The Green’s function can be decomposed into a singular Green’s function Gs(~r, ~r′), whose

integral is evaluated analytically, and a non-singular Green’s function Gn(~r, ~r ′), whose integral

is evaluated numerically, as

G(~r, ~r ′) = Gn(~r, ~r ′) +Gs(~r, ~r′)

=1

4πR(e−jkR − 1) +

1

4πR, (A.8)

where R = |~r − ~r ′| is the distance between source and test position vectors ~r and ~r ′. First, we

treat the numerical integration portion

[LEo]num

m,n=− jωµ0

lmln4AmAn

¨Tm

¨Tn

(~r − ~Qm

)·(~r ′ − ~Qn

)Gn(~r, ~r ′) dSdS′

+jωµ0

k2

1

Am

1

An

¨Tm

¨Tn

G(~r, ~r ′) dSdS′

=− jωµ0lmln

4AmAn

¨Tm

¨Tn

~r · ~r ′Gn(~r, ~r ′) dSdS′︸︷︷︸L1


+ jωµ0lmln

4AmAn~Qn ·¨Tm

¨Tn

~rGn(~r, ~r ′) dSdS′︸︷︷︸~L2

+ jωµ0lmln

4AmAn~Qm ·

¨Tm

¨Tn

~r ′Gn(~r, ~r ′) dSdS′︸︷︷︸~L3

− jωµ0lmln

4AmAn~Qm · ~Qn

¨Tm

¨Tn

Gn(~r, ~r ′) dSdS′︸︷︷︸L4

+jωµ0

k2

1

Am

1

An

¨Tm

¨Tn

G(~r, ~r ′)dSdS′︸︷︷︸L4

=− jωµ0lmln

4AmAn

[L1 − ~Qn · ~L2 − ~Qm · ~L3 + ~Qm · ~QnL4 −

4

lmlnk2L4

]. (A.9)

Integrals L1, ~L2, ~L3, and L4 in (A.9) can be evaluated with Gaussian quadrature. Next, we

treat the singular term in the integration

[LEo]sing

m,n=− jωµ0

lmln4AmAn

¨Tm

¨Tn

(~r − ~Qm

)·(~r ′ − ~Qn

) 1

4πRdS′dS

+jωµ0

k2

1

Am

1

An

¨Tm

¨Tn

1

4πRdSdS′ (A.10)

=− jωµ0lmln

4AmAn

¨Tm

(~r − ~Qm

)·¨Tn

(~ρ ′ − ~ρn

) 1

4πRdS′ dS

+jωµ0

k2

1

Am

1

An

¨Tm

¨Tn

1

4πRdSdS′ (A.11)

=− jωµ0lmln

4AmAn

¨Tm

(~r − ~Qm

)·[¨

Tn

(~ρ ′ − ~ρ

) 1

4πRdS′ +

¨Tn

(~ρ− ~ρn)1

4πRdS′]dS

+jωµ0

k2

1

Am

1

An

¨Tm

¨Tn

1

4πRdSdS′

=− jωµ0lmln

4AmAn

¨Tm

(~r − ~Qm

)·[¨

Tn

(~ρ ′ − ~ρ

) 1

4πRdS′ + (~ρ− ~ρn)

¨Tn

1

4πRdS′]dS

+jωµ0

k2

1

Am

1

An

¨Tm

¨Tn

1

4πRdSdS′

=− jωµ0lmln

4AmAn

¨Tm

(~r − ~Qm

)·[~I1 + (~ρ− ~ρn) I2

]dS +

jωµ0

k2

1

Am

1

An

¨Tm

I2dS

(A.12)

=− jωµ0lmln

4AmAn

¨Tm

~r · ~I1 dS︸︷︷︸L5

−jωµ0lmln

4AmAn

¨Tm

~r · ~ρI2 dS︸︷︷︸L6

+ jωµ0lmln

4AmAn~ρn ·¨Tm

~rI2 dS︸︷︷︸~L7

+jωµ0lmln

4AmAn~Qm ·

¨Tm

~I1 dS︸︷︷︸~L8


+ jωµ0lmln

4AmAn~Qm ·

¨Tm

I2~ρ dS︸︷︷︸~L9

−jωµ0lmln

4AmAn~Qm · ~ρn

¨Tm

I2 dS︸︷︷︸L10

+jωµ0

k2

1

AmAn

¨Tm

I2dS︸︷︷︸L10

=− jωµ0lmln

4AmAn

[L5 − L6 + ~ρn · ~L7 + ~Qm · ~L8 + ~Qm · ~L9

+ ~Qm · ~ρnL10 −4

k2

1

lmlnL10

]. (A.13)

In (A.11), we introduced ~ρ, ~ρ ′, and ~ρn to denote projection of ~r, ~r ′, and ~Qn on the plane of

triangle Tn. In (A.11), we denoted singular integrals with

I1 =

¨Tn

(~ρ− ~ρn)1

4πRdS′ (A.14)

I2 =

¨Tn

1

4πRdS′dS . (A.15)

These integrals can be evaluated analytical as discussed in several works [21].

Evaluation of Reaction Integrals for[KE

o

]The (m,n)-th entry of KE

o is

[KEo

]m,n

=

¨Tm

¨Tn

~Λm(~r) ·[~Λn(~r ′)×∇G(~r, ~r ′)

]dS′dS , (A.16)

where,

∇G(~r, ~r ′) = −~r − ~r′

4πR3(jkR+ 1)e−jkR . (A.17)

Substituting (A.17), (A.6a) and (A.6b) into (A.16) we obtain

[KEo

]m,n

=

¨Tm

¨Tn

(~r − ~Qm

)·[~r − ~r ′ ×

(~r ′ − ~Qn

)] 1

4πR3(jkR+ 1)e−jkR dS′dS

=

¨Tm

(~r − ~Qm

)·[(~r − ~Qn

)×¨Tn

(~r ′ − ~Qn

) 1

4πR3(jkR+ 1)e−jkR dS′

]dS

=

¨Tm

(~r − ~Qm

)·[(~r − ~Qn

)×¨Tn

(~r ′ − ~Qn

)Gn(~r, ~r ′) dS′

]dS

+

¨Tm

(~r − ~Qm

)·[(~r − ~Qn

)×¨Tn

(~ρ ′ − ~ρn

)( 1

4πR3+

k2

8πR

)dS′]dS , (A.18)

where,

Gn(~r, ~r ′) =(1 + jkR)e−jkR − (1 + 0.5k2R2)

R3, (A.19)


is the Green’s function after subtracting the singularity. The first term in (A.18) is evaluated

numerically as

[KEo

]num

m,n=

¨Tm

(~r − ~Qm

)·[(~r − ~Qn

)×¨Tn

(~r ′ − ~Qn

)Gn(~r, ~r ′) dS′

]dS

=

¨Tm

¨Tn

(~r − ~Qm

)·[(~r × ~r ′

)−(~r × ~Qn

)−(~Qn × ~r ′

)]Gn(~r, ~r ′) dS′dS

=

¨Tm

¨Tn

~r · (~r × ~r ′)︸︷︷︸=0

−~r ·(~r × ~Qn

)︸︷︷︸

=0

−~r ·(~Qn × ~r ′

) Gn(~r, ~r ′) dS′dS

+

¨Tm

¨Tn

[− ~Qm ·

(~r × ~r ′

)+ ~Qm ·

(~r × ~Qn

)+ ~Qm ·

(~Qn × ~r ′

)]Gn(~r, ~r ′) dS′dS

=(~Qn − ~Qm

)·¨Tm

¨Tn

(~r × ~r ′

)Gn(~r, ~r ′) dS′dS︸︷︷︸

~K1

+(~Qn × ~Qm

)·¨Tm

¨Tn

~r Gn(~r, ~r ′) dS′dS︸︷︷︸~K2

−(~Qn × ~Qm

)·¨Tm

¨Tn

~r ′ Gn(~r, ~r ′) dS′dS︸︷︷︸~K3

= ( ~Qn − ~Qm) · ~K1 + ( ~Qn × ~Qm) · ( ~K2 − ~K3) (A.20)

The second term in (A.18) has singularities, and thus need to be evaluated semi-analytically as

[KEo

]sing

m,n=

¨Tm

(~r − ~Qm

)·[(~r − ~Qn

)×¨Tn

(~ρ ′ − ~ρn

)( 1

4πR3+

k2

8πR

)dS′]dS , (A.21)

=

¨Tm

(~r − ~Qm

)·[(~r − ~Qn

)×[¨

Tn

(~ρ ′ − ~ρ

)( 1

4πR3+

k2

8πR

)dS′

+(~ρ− ~ρn)

¨Tn

(1

4πR3+

k2

8πR

)dS′]]

dS

=

¨Tm

dS(~r − ~Qm

)·[(~r − ~Qn

)×[~I3 + (~ρ− ~ρn)I4

]]=

¨Tm

dS(~r − ~Qm

)×(~r − ~Qn

)·[~I3 +

k2

2~I1 + (~ρ− ~ρn)

(I4 +

k2

2I2

)]=

¨Tm

dS[~Qn × ~r − ~Qm × ~r + ~Qm × ~Qn

]·[~I3 +

k2

2~I1 + (~ρ− ~ρn)

(I4 +

k2

2I2

)]=

¨Tm

dS[(~Qn − ~Qm

)× ~r + ~Qm × ~Qn

]·[~I3 +

k2

2~I1 + (~ρ− ~ρn)

(I4 +

k2

2I2

)]=(~Qn − ~Qm

)·¨Tm

dS ~r × ~I3︸︷︷︸~K4

+(~Qm × ~Qn

)·¨Tm

~I3dS︸︷︷︸~K5


+(~Qn − ~Qm

)·¨Tm

dS (~r × ~ρ)I4︸︷︷︸~K6

+(~Qm × ~Qn

)·¨Tm

dS ~ρI4︸︷︷︸~K7

+(~Qn − ~Qm

)× ~ρn ·

¨Tm

dS ~rI4︸︷︷︸~K8

+(~Qn × ~Qm

)· ~ρn¨Tm

I4 dS︸︷︷︸K9

+k2

2

(~Qn − ~Qm

)·¨Tm

dS ~r × ~I1︸︷︷︸~K10

+k2

2

(~Qm × ~Qn

)·¨Tm

~I1dS︸︷︷︸~K11

+k2

2

(~Qn − ~Qm

)·¨Tm

dS (~r × ~ρ)I2︸︷︷︸~K12

+k2

2

(~Qm × ~Qn

)·¨Tm

dS ~ρI2︸︷︷︸~K13

+k2

2

(~Qn − ~Qm

)× ~ρn ·

¨Tm

dS ~rI2︸︷︷︸~K14

+k2

2

(~Qn × ~Qm

)· ~ρn¨Tm

I2 dS︸︷︷︸K15

(A.22)

=( ~Qn − ~Qm) · ~K4 − ( ~Qn × ~Qm) · ~K5 + ( ~Qn − ~Qm) · ~K6 − ( ~Qn × ~Qm) · ~K7

+ ( ~Qn − ~Qm)× ~ρn · ~K8 + ( ~Qn × ~Qm) · ~ρnK9

+k2

2( ~Qn − ~Qm) · ~K10 −

k2

2( ~Qn × ~Qm) · ~K11 +

k2

2( ~Qn − ~Qm) · ~K12

− k2

2( ~Qn × ~Qm) · ~K13 + ( ~Qn − ~Qm)× ~ρn · ~K14 + ( ~Qn × ~Qm) · ~ρnK15 . (A.23)

We can compute Ki for i = 1, . . . , 15 in (A.20)–(A.23) numerically, using Gaussian quadrature.

In (A.23), we introduced singular integrals we denoted singular integrals with

I3 =

¨Tn

(~ρ− ~ρn)1

4πR3dS′ (A.24)

I4 =

¨Tn

1

4πR3dS′dS , (A.25)

which can be evaluated analytically [21].

Appendix B

Evaluation of Singular Integrals in

2-D Problems

For the diagonal entries of P, the integral in (3.28) reads

[P]m,m = ωµ

ˆ wm2

0

[C0J0(kr′)− jY0(kr′)

]dr′ , (B.1)

where wm is the width of m-th pulse basis function. The Neumann function Y0(kr′) has a

singularity when its argument is equal to zero. Therefore, we must evaluate the above integral

using the low argument approximation of the Bessel and the Neumann functions [123], as shown

in [119]. For low and medium frequency, the small argument approximation holds for all values

of r′ in the integral of (B.1), which can be evaluated analytically yielding

[P]m,m =C0ωµwm

2− ωµwm

π

(ln

(kwm

4

)− 1 + γ

), (B.2)

where γ is Euler’s constant [123] . At high frequency, the integral in (B.1) is broken down

into two integrals, corresponding to small and large values of k′r. For kr′ < 0.1, the small-

argument approximation of J0(kr′) and Y0(kr′) still holds, and a formula analogous to (B.2)

can be used. For kr′ > 0.1, the integral has to be evaluated numerically using, for example,

Gaussian quadrature formulas, as done in the numerical tests in this thesis.

158

Appendix C

Properties of Circulant and Toeplitz

Matrices

A Circulant Matrix

Matrix G is a circulant matrix if it has the form of [160]

G =

G0 G−1 G−2 . . . G−(N−1)

G−(N−1) G0 G−1 . . . G−(N−2)

G−(N−2) G−(N−1) G0 . . . G−(N−3)...

......

......

G−1 G−2 G−3 . . . G0

, (C.1)

where we have color-coded entries that have the same value. A circulant matrix is diagonalized

by discrete Fourier transform. Therefore, we can compute the matrix-vector product Gx, for

some vector x =[x1 x2 . . . xN

]T, using FFTs as

Gx = F−1F[G0 G−1 . . . G−(N−1)

]· F[x1 x2 . . . xN

]. (C.2)

In (C.2), F [·] and F−1 [·] are used to denote, respectively, the fast Fourier transform and the

inverse fast Fourier transform. The matrix vector product computed in (C.2) has computa-

tional complexity of O(n log n), which is significantly more efficient than a direct matrix-vector

product whose complexity is O(n2).

159

Appendix C. Properties of Circulant and Toeplitz Matrices 160

Toeplitz Matrix

Matrix T is a Toeplitz matrix if it has the form of [160]

T =

T0 T−1 T−2 . . . T−(N−1)

T1 T0 T−1 . . . T−(N−2)

T2 T1 T0 . . . T−(N−3)...

......

......

T(N−1) TN−2 TN−3 . . . T0

. (C.3)

We can efficiently perform the matrix-vector product involving a T by augmenting T to form

a circulant matrix Tc of the form

Tc =

T0 T−1 T−2 . . . T−(N−1) T(N−2) . . . T1

T1 . . . . . . . . . . . . . . . . . . . . .

T2 . . . . . . . . . . . . . . . . . . . . ....

......

......

......

...

TN−1 . . . . . . . . . . . . . . . . . . . . .

T−(N−1) . . . . . . . . . . . . . . . . . . . . .

T−(N−2) . . . . . . . . . . . . . . . . . . . . ....

......

......

......

...

T−1 . . . . . . . . . . . . . . . . . . . . .

.

(C.4)

Note that we have not filled in all entries of Tc because its entries are uniquely defined by the

first row and the first column. All other rows can be obtained by shifting the previous row,

i.e. the (m,n)-th entry for m > 1, n > 1 is [Tc]m,n = [Tc]m−1,n−1. The matrix-vector product

y = Tx can be computed efficiently using Tc as

T0 T−1 T−2 . . . T−(N−1) T(N−2) . . . T1

T1 . . . . . . . . . . . . . . . . . . . . .

T2 . . . . . . . . . . . . . . . . . . . . ....

......

......

......

...

TN−1 . . . . . . . . . . . . . . . . . . . . .

T−(N−1) . . . . . . . . . . . . . . . . . . . . .

T−(N−2) . . . . . . . . . . . . . . . . . . . . ....

......

......

......

...

T−1 . . . . . . . . . . . . . . . . . . . . .

x1

x2

x3

...

xN

0

0...

0

=

y1

y2

y3

...

yN

∗∗...

∗

, (C.5)

where we have augmented x with a vector of zeros and “∗” in the right-hand side vector is used

to denote entries that are not useful for us. Since Tc is a circulant matrix, we can compute

Appendix C. Properties of Circulant and Toeplitz Matrices 161

the matrix-vector product in (C.5) with FFTs as shown in (C.2) to achieve the computational

complexity of O(n log n).

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