A Surface Admittance Approach For Fast Calculation of the ... · Ee(p) n, and Je (p) n nth-Fourier...

A Surface Admittance Approach For Fast Calculation of the SeriesImpedance of Cables Including Skin, Proximity, and Ground Return

Effects

by

Utkarsh R. Patel

A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Science

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

© Copyright 2014 by Utkarsh R. Patel

Abstract

A Surface Admittance Approach For Fast Calculation of the Series Impedance of Cables

Including Skin, Proximity, and Ground Return Effects

Utkarsh R. Patel

Master of Applied Science

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

Engineering

University of Toronto

2014

The accurate calculation of broadband series impedance of power cables is required to predict

transients induced in power systems. Since modern power cables have complex geometries with

hundreds of tightly packed conductors, existing techniques to compute their series impedance

are either inaccurate or very slow. This thesis presents MoM-SO, a fast and accurate tech-

nique to compute the per-unit length impedance of cables made up of solid and hollow round

conductors placed inside a tunnel in a multilayer ground environment. MoM-SO employs a

surface-approach to solve for the impedance. In this approach, only fields on the boundaries

of the conductors are discretized and calculated. As shown in the examples, this discretization

scheme makes MoM-SO over 1000 times faster than existing volumetric-based methods like

finite-element tools. MoM-SO is also accurate because it includes skin, proximity, and ground

return effects which govern the behaviour of fields inside a cable.

ii

Dedicated to my parents, and brother

for their endless support!

iii

Acknowledgements

I would like to express my deepest gratitude to my supervisor Prof. Piero Triverio for giving

me an opportunity to work on this exciting and challenging research project. His support and

guidance throughout the course of this project has been invaluable. I especially thank him for

inspiring my interest in conducting research, and I look forward to working with him for my

PhD studies!

I am also thankful to Prof. Hum, Prof. Sarris, and Prof. Iravani for reading my thesis and

being on my thesis committee. Their suggestions have improved this thesis.

Finally, I am grateful to Dr. Bjørn Gustavsen, a senior scientist at SINTEF Energy Re-

search, for his technical insights on this project. I have learned a lot from him about power

transients through various discussions we have had during this project.

I acknowledge Norwegian Research Council (RENERGI programme) and a consortium of

industry partners led by SINTEF Energy Research, DONG Energy, EdF, EirGrid, Hafslund

Nett, National Grid, Nexans Norway, RTE, Siemens Wind Power, Statnett, Statkraft, and

Vestas Wind Systems for financial support through KPN project “Electromagnetic transients

in future power systems (ref. 207160/E20). I also acknowledge Ontario student assistance

program (OSAP) for awarding me Ontario graduate scholarship (OGS).

iv

Contents

1 Introduction to Power Cable Modelling 1

1.1 Trends in Power Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Electricity & Power Networks . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Future Outlook for Power Grids . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Power Grid Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.1 Technological Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.2 Cable Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.3 Existing Techniques for Impedance Calculation . . . . . . . . . . . . . . . 14

1.3.4 Existing Techniques to Include Ground Return Effect . . . . . . . . . . . 16

1.3.5 Cable Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4 Thesis Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.5 Modelling of Electronics Cables for Signal Integrity Analyses . . . . . . . . . . . 20

1.6 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Cables Surrounded by a Homogeneous Lossless Medium 22

2.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Surface Admittance Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.1 Surface Admittance Operator for a Solid Conductor . . . . . . . . . . . . 23

2.2.2 Surface Admittance Operator for a Hollow Conductor . . . . . . . . . . . 28

2.2.3 Surface Admittance Operator for Multiple Conductors . . . . . . . . . . . 31

2.3 Electric Field Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

v

2.4 Computation of p.u.l. Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.5 Computational Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.6.1 Example # 1: A Two-wire Line . . . . . . . . . . . . . . . . . . . . . . . . 40

2.6.2 Example # 2: Coaxial Cable . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.6.3 Example # 3: Three-Phase Cable . . . . . . . . . . . . . . . . . . . . . . 43

2.6.4 Example # 4: Pipe-Type Cable . . . . . . . . . . . . . . . . . . . . . . . . 47

2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3 Cables Surrounded by a Lossy and Multilayered Medium 55

3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2 Surface Admittance Representation for the Cable-Hole System . . . . . . . . . . 57

3.2.1 Surface Admittance Operator for the Conductors . . . . . . . . . . . . . . 57

3.2.2 Surface Admittance Operator for the Cable-Hole System . . . . . . . . . . 58

3.3 Inclusion of Ground Return Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.4 Computation of Per-Unit Length Parameters . . . . . . . . . . . . . . . . . . . . 67

3.5 Extension to a Multilayered Surrounding Medium . . . . . . . . . . . . . . . . . . 69

3.5.1 Green’s Function of a Multilayer Media . . . . . . . . . . . . . . . . . . . 70

3.5.2 Entries of Green’s Matrix Gg . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.6 Extension to Hollow Conductors and Multiple Holes . . . . . . . . . . . . . . . . 80

3.7 An Approximate Technique to Include the Ground Return Effect . . . . . . . . . 80

3.8 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.8.1 Example # 1: Three Single Core Cables Buried in Earth . . . . . . . . . 82

3.8.2 Example # 2: Proximity Effect in Surrounding Ground . . . . . . . . . . 88

3.8.3 Example # 3: Three Single-Core Cables Inside a Tunnel . . . . . . . . . . 89

3.8.4 Example # 4: Two Conductors in a Four-layer Medium . . . . . . . . . . 90

3.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4 Fields Computation and Adaptive Discretization 94

4.1 Fields and Current Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.1.1 Fields and Equivalent Currents on the Boundaries . . . . . . . . . . . . . 95

vi

4.1.2 Inside a Conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.1.3 Inside a Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.1.4 Inside Ground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.1.5 Validation Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.2 Automatic Basis Function Estimation . . . . . . . . . . . . . . . . . . . . . . . . 107

4.2.1 Trial & Error Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.2.2 Adaptive Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.2.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5 Conclusions 114

5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.3 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

A Post-Processing of the p.u.l. Parameters 120

A.1 Procedure to Calculate Line Impedances . . . . . . . . . . . . . . . . . . . . . . . 121

A.2 Procedure to Calculate the Total P.u.l. Impedance . . . . . . . . . . . . . . . . . 123

B Analytic Evaluation of Matrices 125

B.1 Green’s Matrix G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

B.2 Green’s Matrix Gc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

B.3 Green’s Matrix G0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

B.4 Green’s Matrix G0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

B.5 Matrix H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Bibliography 138

vii

Notation

Notation Description

Conductors (p is the index of a conductor)

µ, ε, σ Permeability, permittivity, conductivity of the conductor 1

ap Conductor’s outer radius

k Wavenumber inside the conductor

ap Conductor’s inner radius (for hollow conductors only)

(xp, yp) Conductor’s center

cp Conductor’s outer contour

cp Conductor’s inner contour (for hollow conductors only)

rp(ρp, θp) Position vector pointing to (ρp, θp) in a cylindrical coordinate system cen-

tered at (xp, yp)

E(p)z (θp), J

(p)s (θp) Electric field and equivalent current on the outer boundary of the conductor

E(p)n , J

(p)n nth Fourier coefficient of the Fourier expansion of electric field and equivalent

current on the outer boundary of the conductor.

E(p)z (θp), J

(p)s (θp) Electric field and equivalent current on the inner boundary of the conductor

(for hollow conductors only)

1For the sake of simplicity, we assume in our discussion that all conductors have the same properties. Theproposed techniques can handle the most general case with trivial changes.

viii

E(p)n , and J

(p)n nth-Fourier coefficient of the Fourier expansion of electric field and equivalent

current on the inner boundary of the conductor (for hollow conductors only)

E(p), J(p) Column vector which collects the Fourier coefficients of electric fields and

currents (on inner and outer boundaries)

E(p)z (ρp, θp) Electric field inside conductor p

Y(p)n Surface admittance operator for a solid conductor

Y(p)n Surface admittance operator for a hollow conductor. This operator is a 2×2

matrix

Vp Scalar potential of conductor p

Ip Current through conductor p

Np Number of harmonics used to represent electric field and current density

N Total number of current (or electric field) coefficients (see eq. (2.36))

Global vectors

V Column vector storing the potentials of all conductors

I Column vector storing the currents in all conductors

E, J Column vectors storing E(p) and J(p), respectively, for all conductors

R(ω), L(ω) P.u.l. resistance and inductance matrices

U Matrix which relates coefficients of equivalent current with conduction cur-

rent (see eq. (2.39))

Ys Surface admittance operator for all conductors (see eq. (2.38))

G Green’s matrix for homogeneous lossless medium case considered in Ch. 2

Surrounding medium

µ0, ε0 Permittivity, permeability of the surrounding medium

Gg Green’s function of the surrounding medium

ix

Gg Green’s matrix relating J to A

L Total number of layers in the surrounding ground

layer s Layer in which cables are buried

Hole

µ, ε, σ Permeability, permittivity, and conductivity inside the hole

k Wavenumber inside the hole

(x, y) center of the hole

a radius of the hole

(ρ, θ) Coordinates of cylindrical system centered at (x, y)

r(ρ, θ) Position vector of the point (ρ, θ)

Js(θ), Az(θ) Equivalent current and vector potential on the boundary of the hole

Jn, An n-th Fourier coefficient of Js(θ) and Az(θ)

J, A Vectors containing coefficients Jn, and An

Az(ρ, θ) Vector potential inside the hole

A′z(ρ, θ) General solution of Helmholtz equation for vector potential inside the hole

A′′z(ρ, θ) Particular solution of Helmholtz equation for vector potential inside the hole

Cn Coefficients of expansion of A′z(ρ, θ)

C Vector containing Fourier coefficients Cn

D1 , D2 Diagonal matrices (see (3.15) and (3.25))

Ys Surface admittance operator of the hole

T Transformation matrix in surface admittance relationship (see eq. (3.24))

G Green’s function of homogeneous medium with material of the hole

G0 Green’s matrix relating field at the boundary of the hole to the equivalent

currents J(p)s (θp) and J

(p)s (θp)

x

G0 Green’s matrix relating derivative of field at the boundary of the hole to the

equivalent currents J(p)s (θp) and J

(p)s (θp)

Gc Green’s matrix relating field on the boundary of conductors to the equivalent


(p)s (θp)

N Number of harmonics used to represent Js(θ) or Az(θ)

Miscellaneous

x, y Unit vectors in x-direction and y-direction

ρ, θ Unit vectors of a cylindrical coordinate system

Jn(.), Bessel function of first kind of order n

Hn(.), Kn(.) Hankel functions of first kind and second kind of order n

xi

List of Tables

2.1 Characteristics of the cable in the example of Sec. 2.6.3. . . . . . . . . . . . . . . 44

2.2 Armor characteristics for the structure considered in Sec. 2.6.3. . . . . . . . . . . 45

2.3 Positive- and zero-sequence impedance of the three-phase cable of Sec. 2.6.3 at

50 Hz. MoM-SO is compared against a finite element approach [1]. . . . . . . . . 46

2.4 Computation time for MoM-SO and FEM for Three-phase cable example of

Sec. 2.6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.5 Characteristics of the cable system shown in Fig. 2.9. . . . . . . . . . . . . . . . . 48

2.6 Computation time for the proposed method (MoM-SO) and finite elements (FEM)

applied to the cable system of Sec. 2.6.4. . . . . . . . . . . . . . . . . . . . . . . . 50

3.1 Single core cables of Sec. 3.8.1: geometrical and material parameters . . . . . . . 83

3.2 Example of Sec. 3.8.1: CPU time required to compute the impedance at one

frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.1 Comparison of the trial and error (TE) and adaptive (AD) estimation schemes

for MoM-SO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

B.1 Solution of the integral in (B.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

xii

List of Figures

1.1 Three-phase circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Hierarchical power flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 North sea wind farm [source: wikipedia] . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Distributed transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Typical overhead lines [source: The Guardian [2]] . . . . . . . . . . . . . . . . . . 8

1.6 Types of underground cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.7 Skin effect inside a conductor. The plot shows current density inside the con-

ductor normalized to 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.8 Proximity effect between two conductors. The plot shows current density inside

the conductor normalized to 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.9 Three-phase system connected by three SC cables . . . . . . . . . . . . . . . . . . 13

1.10 Examples of the cable systems that will be considered in this work: a) Two

submarine cables placed on a seabed; b) Three SC cables placed inside a tunnel

which is buried in multilayer earth . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1 Sample solid and hollow cross-sections . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Application of the equivalence theorem to a solid round conductor. The con-

ductor in (a) is replaced by the surrounding medium and an equivalent current

J(p)s (θp) on its surface as in (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Application of the equivalence theorem to a hollow conductor. The actual con-

ductor, shown in (a), is replaced by the surrounding medium and equivalent


(p)s (θp) are introduced on the inner and outer surface of

the conductor as in (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

xiii

2.4 P.u.l. resistance of the two-wire line of Sec. 2.6.1 for two different wire separa-

tions: 100 mm (left panel) and 25 mm (right panel) [3]. Comparison between

the high frequency approximation (2.64), the analytic formula without proxim-

ity (2.68), the proposed method (MoM-SO) and the finite elements method [4]

is shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.5 P.u.l. inductance of the two-wire line of Sec. 2.6.1 for two different wire sepa-

rations: 100 mm (left panel) and 25 mm (right panel) [3]. Comparison between

the high frequency approximation (2.65), the analytic formula without proxim-

ity (2.68), the proposed method (MoM-SO) and the finite elements method [4]

is shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.6 P.u.l. inductance(left panel) and resistance (right panel) of a coaxial cable con-

sidered in Sec. 2.6.2. Comparison between the Analytic formulas (2.69) and the

proposed approach (MoM-SO) is shown. . . . . . . . . . . . . . . . . . . . . . . . 43

2.7 Three-phase cable considered in Sec. 2.6.3 . . . . . . . . . . . . . . . . . . . . . . 44

2.8 P.u.l. resistance (left panel) and p.u.l. inductance (right panel) of the three-

phase cable of Sec. 2.6.3, obtained with MoM-SO and FEM ©[2013] IEEE [3].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.9 Three single-core cables inside a conducting pipe (seven conductors). . . . . . . . 47

2.10 P.u.l. positive sequence resistance and inductance of the cable system, obtained

with MoM-SO and FEM. Screens are continuously grounded [5]. . . . . . . . . . 49

2.11 P.u.l. positive sequence resistance and inductance of the cable system, obtained

with MoM-SO and FEM. Screens are open [5]. . . . . . . . . . . . . . . . . . . . 49

2.12 First configuration considered in Sec. 2.6.4. A unit step voltage is applied to the

top phase conductor of the cable of Fig. 2.9. All other phases and sheaths are

grounded at one end [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.13 Phase-screen voltage at the receiving end (V7 − V8) for the configuration shown

in Fig. 2.12 [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.14 Sheath voltage at the receiving end (V8) for the configuration shown in Fig. 2.12 [5]. 51

2.15 Second configuration considered in Sec. 2.6.4. A differential voltage excitation is

applied between two screens [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

xiv

2.16 Differential voltage at the receiving end (V10 − V12) for the configuration of

Fig. 2.15 [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.1 Sample circuit configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2 Cross-section of a simple cable-hole system with two conductors used to illus-

trate the proposed method. The coordinate system used in this chapter is also

presented [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3 Left panel: cross-section of the cable in Fig. 3.2 after all conductors have been

replaced by the surrounding hole medium. Equivalent currents J(p)s (θp) are intro-

duced on their contours. Right panel: cross-section of the cable after application

of the equivalence theorem to the boundary of the hole. An equivalent current

Js(θ) is introduced on the hole boundary c. . . . . . . . . . . . . . . . . . . . . . 58

3.4 Geometry of a multilayer background medium . . . . . . . . . . . . . . . . . . . . 70

3.5 Multilayer media transmission line model . . . . . . . . . . . . . . . . . . . . . . 72

3.6 Simplified transmission line circuit to model layer s . . . . . . . . . . . . . . . . . 73

3.7 System of three single core cables used for validation in Sec. 3.8.1. Conductive

media are shown in gray while insulating media are shown in white. . . . . . . . 82

3.8 Mesh used for the three single-core cable case in Sec. 3.8.1. Top panel shows the

mesh distribution inside the SC cables, and the bottom panel shows the mesh

distribution in the surrounding ground. . . . . . . . . . . . . . . . . . . . . . . . 84

3.9 Cable system of Sec. 3.8.1: positive-sequence resistance (top panel) and induc-

tance (bottom panel) computed using FEM (), MoM-SO (·), and cable constant

(- -). Screens are continuously grounded [6]. . . . . . . . . . . . . . . . . . . . . . 85

3.10 Cable system of Sec. 3.8.1: zero-sequence resistance (top panel) and inductance

(bottom panel) computed using FEM (), MoM-SO (·), and cable constant (- -).

Screens are continuously grounded [6]. . . . . . . . . . . . . . . . . . . . . . . . . 86

3.11 Cable system of Sec. 3.8.1: positive-sequence resistance (top panel) and induc-

tance (bottom panel) computed using FEM (), MoM-SO (·), cable constant (

), and MoM-SO with approximate ground return effects [7] ( ). The screens

of the cables are open [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

xv

3.12 Cable system considered in Sec. 3.8.2: resistance (top panel) and inductance

(bottom panel) computed using FEM (), MoM-SO (·), cable constant ( ), and

MoM-SO with approximate ground return effects [7] ( ). Phase conductors are

open, and current is injected in the sheaths [6]. . . . . . . . . . . . . . . . . . . . 88

3.13 System of three single-core cables in a tunnel considered in Sec. 3.8.3. Conductive

media are shown in gray while insulating media are shown in white. . . . . . . . 89

3.14 System of three SC cables in a tunnel considered in Sec. 3.8.3: resistance (top

panel), and inductance (bottom panel) computed with FEM () and MoM-SO

(·). In order to show the effect of the tunnel, the resistance and inductance of

the cables buried directly in ground are also shown (×) [6]. . . . . . . . . . . . . 90

3.15 A two-conductor system inside a four-layer medium considered in Sec. 3.8.4 . . . 91

3.16 Loop-mode and common-mode inductance of the two-conductor system consid-

ered in Sec. 3.8.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.17 Loop-mode and common-mode resistance of a two-conductor system considered

in Sec. 3.8.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.1 Equivalent Transmission line models . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.2 The magnitudes of the electric field, magnetic field, and current density obtained

with MoM-SO (left panel) and COMSOL Multiphysics (right panel) for the two-

conductor case in Sec. 4.1.5 at 1 Hz. Conductors are excited with 1 A of currents

flowing in opposite directions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103



conductor case in Sec. 4.1.5 at 6 kHz. Conductors are excited with 1 A of currents

flowing in opposite directions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104



conductor case in Sec. 4.1.5 at 200 kHz. Conductors are excited with 1 A of

currents flowing in opposite directions. . . . . . . . . . . . . . . . . . . . . . . . . 105

xvi

4.5 The Magnitudes of magnetic field and current distribution inside the ground

due to a wire carrying a current of 1 A, as considered in Sec. 4.1.5. Fields are

compared using MoM-SO and COMSOL at 5 kHz. . . . . . . . . . . . . . . . . 106

4.6 Typical cross section of a USB 2.0 cable [8]. . . . . . . . . . . . . . . . . . . . . . 110

4.7 Inductance and resistance of the USB 2.0 cable of Fig. 4.6 computed with MoM-

SO and finite elements (FEM). The plots show the self and mutual impedance

between the signal 1 (s1), signal 2 (s2), and power (p) lines with reference to the

ground line. Mutual parameters are shown in magnitude. . . . . . . . . . . . . . 111

4.8 The number of basis functions (Np) used to expand fields and currents inside

each conductor of a USB cable in Sec. 4.2.3 at various frequency points. Each

color corresponds to a specific value of Np as indicated in the colorbar. . . . . . . 112

A.1 Example to demonstrate the post-processing procedure outlined in Appendix A.

Each conductor with the same color belongs to the same line. Two different

shades of gray show two different return lines . . . . . . . . . . . . . . . . . . . . 121

B.1 Lengths and angles used in (B.24). . . . . . . . . . . . . . . . . . . . . . . . . . . 131

xvii

Chapter 1

Introduction to Power Cable

Modelling

1.1 Trends in Power Grids

1.1.1 Electricity & Power Networks

Electricity is a cornerstone of modern society as we depend on it for social and economic

reasons. In 2013, residential electricity usage in Canada was 153 TWh, which translated into

4.37 MWh of electricity per capita. If industrial and commercial electricity usage is included,

per capita usage jumps up to 14.6 MWh [9]. This is a lot of energy! Moreover, these numbers

have been increasing every year. Electricity has revolutionized the world: from health industry

to transportation industry and from construction to farming industry. None of these sectors

have remained the same after electricity became widely available. It has also influenced our

personal lives, as personal computers, televisions, and radios would not be possible without

electricity. Electrical power enables internet and phones which have drastically changed the

way we communicate with one another. It also has a strong influence on the industrial sector

as well, as most of the machines and equipment depend on electricity in one form or another.

In Franklin Roosevelt’s words, “Electricity is a modern necessity of life, not a luxury.”

Such an immense impact of electricity is possible due to power grids, which reliably deliver

power from generation stations to consumers. Power from generation station is typically avail-

1

Chapter 1. Introduction to Power Cable Modelling 2

Va

neutralVbVc

Tline

Tline

Tline

ZA

ZC

ZB

Figure 1.1: Three-phase circuit

Figure 1.2: Hierarchical power flow

able in three phases at 50/60 Hz. Generation station can be modelled by three-voltage sources

with the same magnitude and a 120 phase shift. A simple three-phase circuit in Fig. 1.1 illus-

trates this concept. The example in Fig. 1.1 is a very simplistic view of power transfer from a

generator to a load. In reality, the power grid is much more complicated, with a hierarchical

structure that can be conceptually divided into multiple parts, as shown in Fig. 1.2. Three-

phase power from a generation station is transmitted to the closest transmission substations,

where a step-up transformer increases its voltage. From here, the power is transmitted via a

transmission grid to a distribution substation, where voltages are down-converted. The grid

contains lines which stretch over a long distance (up to few-hundred kilometres), and operate at

high voltages (over 100 kV). Substations are equipped with switches and splitters to distribute

the power received from the transmission grid to local consumers via distribution grids that typ-

ically operate below 50 kV [10]. Consumers’ power demands fluctuate rapidly. In order to meet


this changing power demands, power is transferred from one substation to another. To support

power transfer feature, all the substations are interconnected which increases the complexity

of the grid. Transmission interconnections not only stretch across cities or provinces/states,

but also across countries; for example, last year Canada imported 11.4 TWh and exported

58 TWh of electricity to the United States [9]. Complexity of the interconnections in power

grids is immense, and this complexity will continue to grow as the next-generation of power

grids, commonly called “smart grids” will be installed around the world.

1.1.2 Future Outlook for Power Grids

The energy plans of most countries revolve around maximization of renewable energy, and de-

ployment of smart grids to support distributed generation. Traditional steam-based generation

techniques burn coal, oil, or natural gas, emitting carbon in abundance. Pressure from envi-

ronmentalists, and to a certain extent the whole society, has forced governments to look for

ways to reduce carbon emission by promoting renewable energy generation. Countries across

the globe foresee that renewable sources will be the dominant means of energy generation in a

few decades. This trend has been outlined in various publications [11–13] from organizations

overseeing generation and transmission of power in their respective countries. Here we list some

key remarks made in these publications:

• Canada has outlined that its energy demand will grow by 27% in the next 20 years [11]. To

meet this demand, solar and wind generation will play a significant role, and are expected

to make up to 12% of the total energy generation. Currently, wind, solar, biomass, and

geothermal make up only 1.5% of generation. Renewable sources will reduce dependency

on coal-based energy generation from 15% to 3%. Hydroelectric already makes up about

two-thirds of energy generation in Canada.

• United States has outlined that its demand will grow by 25%, to 5825 TWh/year, by

2040 [12]. It also expects that renewable energy generation will make up about 30% of

its total generation capacity, compared to the current 19%.

• The European Network of Transmission System Operators for Electricity (ENTSOE)

forecasts that by 2020 renewable sources will make up 42% of its generation. Currently,


Figure 1.3: North sea wind farm [source: wikipedia]

wind, solar, biomass, and hydro make up about 34% of generation [13]. ENTSOE also aims

for 100% renewable energy generation by 2050. Development of renewable energy sources

in certain countries, such as Denmark, are already strongly influenced by government

legislation through subsidies and feed-in tariff. It is noteworthy to mention that renewable

energy sources, including solar, wind, and hydro, already make up a significant portion of

generation in several countries. For example, almost 100% of generation in Iceland, and

Norway is provided by renewable sources. Additionally, solar, wind, and biomass make

up 22% of generation in Germany, 32% in Portugal, and 42% in Denmark [14]. One of

the interesting developments is that of offshore wind farms, as shown in Fig. 1.3, most

of which have come in operation in the past ten years. Many more are under a planning

phase. This trend is particularly relevant for this work since offshore turbines deliver

energy to the mainland through sophisticated power cables, which need to be modelled

in order to design the overall system.

In order to integrate renewable energy sources, power grids will have to be upgraded to smart

grids. Traditional power grids, as discussed earlier, are hierarchical, where power flows mainly

in one direction, from large-capacity generation stations to consumers, as shown in Fig. 1.2. The

introduction of renewable and distributed energy sources is challenging this paradigm. Energy,

indeed, will be generated at many locations across the grid, including households equipped

with solar panels or small turbines. Hence, the power flow architecture will have to be modified

accordingly, as shown in Fig. 1.4, so that power can be transferred from distributed sources,

which may be on a distribution grid, to the transmission substation to the consumers. To


Figure 1.4: Distributed transmission

support this distributed power generation, substantial investment will be required to upgrade

switches, controllers, and transmission lines.

In addition to supporting renewable generation, smart grids will also support electrical

vehicles, which will aid in reducing carbon emission. Furthermore, they will support demand-

response protocol that will allow loads and generators to interact automatically regarding power

demand. These capabilities will improve the network performance, but will require deployment

of additional controllers and communication systems. Canada forecasts that an investment of

$15 billion/year until 2030 will be required in order to upgrade the power grid. This amount is

over 50% more than annual maintenance cost of the network. Out of $15 billion, 33% will be

allocated towards distribution and transmission, and the rest will be allocated for upgrading

aging power generation plants [11].

In entirety, smart grids make power transmission efficient, reliable, and support green energy.

The drawback, of course, is the added complexity to the network.


1.2 Power Grid Simulation

In order to design and maintain next-generation power grids, engineers will have to rely heav-

ily upon analyses such as power-flow analysis, stability analysis, short-circuit analysis, and

transient analysis [10].

Power-flow analysis

In day-to-day operations, power-flow analysis is performed using tools such as PowerWorld

Simulator [15]. Power-flow analysis calculates node voltage magnitude and phase, and real and

reactive power flowing in each line under steady-state sinusoidal operation at 50 or 60 Hz. This

information aids transmission operator to answer questions [10] such as whether or not enough

power is generated or traded-in to meet the consumers’ demand, and whether transmission lines

and transformers are operating under their rated limits. This information allows engineers to

make informed decisions regarding operation of the grid. Through power-flow analysis engineers

can also plan future expansion of the network. For example, this analysis can help answer

whether or not the grid can support a new solar farm.

Stability analysis

Ensuring that power grids remain stable under all operating conditions is clearly a major

requirement. Stability can be assessed by looking at the eigenvalues of the power grid. Stability

analysis detects whether the synchronous generators and machines remain synchronous follow-

ing disturbances from faults or lightning. Power-flow tools such as PowerWorld Simulator [15]

have a built-in stability program.

Transient analysis

Even though transmission grid is operational more often than not, whenever it does fail

on a large-scale, it creates havoc. Examples of such large-scale failure are: the blackout that

occurred in northeast United States, and Ontario, Canada, on August 14, 2003, triggered by

the failure of a power flow controller [16], and more recently the power outage in central and

eastern Canada in December 2013, caused when many overhead lines were uprooted due to

an ice storm [17]. Such power failures may be triggered by events like lightning, faults, and


breakers operation. All such events induce transients in the power network that may cause

cascaded failures. It was reported that during a 14-year period, 26% of power outages on a

230-kV circuit, and 65% of power outages on a 345-kV circuit were due to lightning strikes [10].

Power flow tools perform a steady-state analysis and so they cannot predict transient effects,

which involve a large bandwidth. For example, switching transient analysis requires accurate

models from 50 Hz to 20 kHz, lightning over-voltages, on the other hand, emit transients

with frequency ranging from 10 kHz to 3 MHz [18]. To analyse transients, one must resort

to transient simulation tools such as ATP [19], EMTP-RV [20], and PSCAD [21] in order to

predict shape and magnitude of voltage waveform for a given transient scenario. Knowledge

of transient studies helps with the design and operation of equipment and controllers in order

to mitigate failures. For example, short-circuit faults may be prevented by better insulation

between the conductors.

Power network component modelling

Simulation tools require models for each component in the network, such as transformers,

rotating machines, transmission lines, and loads. Power-flow analysis tools require models ac-

curate at 50/60 Hz, since these tools perform a steady-state analysis. However, broadband

models, accurate at DC to few MHz, must be used to analyze transients in the network. Broad-

band models are derived from a) widely-accepted theoretical formulas, b) experimental data, or

c) simulation results. Broadband models can be created from experimental or simulation data

by employing a rational fitting algorithm like Vector fitting [22].

1.3 Cables

In this section, we narrow the focus to power cables, which are the main subject of this thesis.

After summarizing their technological characteristics, we review the state of the art on cable

modelling.

Cables make up a large portion of a power network. There exist an estimated 300,000 km of

high-voltage overhead lines and cables, and as much as 5 million kilometres of low- and medium-

voltage cabling in Europe alone [23]. Moreover, cables are the most-demanded components


Figure 1.5: Typical overhead lines [source: The Guardian [2]]

among transmission and distribution equipment. According to Energy Market Research Expert

(NRGExpert) [24], their annual demand in 2011 equalled to US $34 billion, which is 23% of

the total demand for transmission and distribution equipment. This demand is far greater than

the demand for power (generation) systems (16%), substations (13%), transformers (15%),

switches (12%), and insulators (5%). For such an important component of the network, it is

important to have an accurate model for the purpose of transient and power-flow analyses.

Unfortunately, all transient simulation tools use analytic formulas to model cables, and these

formulas are inaccurate in some scenarios. The objective of this thesis is to develop a more

accurate technique to compute the series impedance of cables.

In the next section, we provide technological overview of cables typically found in power

grids. Following which, we discuss the state of the art in the modelling of power cables.

1.3.1 Technological Overview

Overhead lines

Transmission lines may be distinguished between overhead lines and underground cables. High

voltage overhead lines, as shown in Fig. 1.5, usually consist of three phase lines (typically

two conductors per phase), each carrying current with same magnitude but out-of-phase by

120. They also contain shield wires that are periodically grounded, so that current due to

lightning strikes can be safely discharged to the ground. Simplicity of overhead lines allows one

to characterize them using analytic formulas with adequate accuracy.


Underground and Submarine Cables

Underground cables have higher installation cost than overhead lines. However, they are far less

likely to be affected by severe weather conditions, or human or animal interference. Additionally,

underground cables do not obstruct view and are more accepted by local communities. Use of

underground cabling has been limited in North America, with only about 1% of total cables in

the United States buried under ground, mostly in large cities [10]. However, in Europe many

countries, such as Denmark, have taken incentive to bury their high voltage cabling. In addition

to underground cable, submarine power cables1 (or subsea cables) are equally important to

transport power to an island or offshore oil or gas platform. With the development of offshore

wind farms, such as the North Sea offshore grid shown in Fig. 1.3, such submarine power cables

are also required to transmit power from the offshore wind farm to the distribution station

on land. Subsea high-voltage DC (HVDC) lines are also popular for long-range underwater

transmission in order to interconnect grids from countries that are separated by sea. Even

though HVDC lines carry DC power, the transient phenomena remain broadband.

Underground cables tend to have a more complex geometry when compared to overhead lines

because the phase conductors are tightly packed, and the cables require additional conductors

for shielding. Underground cables may be classified into three groups: single-core (SC) cables,

pipe-type cables, and submarine cables, all of which are shown in Fig. 1.6. A typical single-core

cable contains the following elements [18]:

Core conductor that carries the power. In each SC cable of a three-phase system, the core

conductor carries current with same magnitude and 120 phase shift.

Metallic sheath is used to shield the fields. Metallic sheaths may be grounded at both ends

(most likely), at a single end, or cross-bonded2.

Insulation insulates the metallic core from the sheath. Outer insulation is also important to

isolate the metallic sheath from the surrounding ground. Its other purpose is to protect the

cable from corrosion and humidity. Insulation influences the capacitance of the conductor;

1In this thesis, the term “underground cable” will be used for both underwater and underground cables, unlessexplicitly stated.

2sheaths are cross-connected periodically after a fixed distance (for example 1 km) in order to reduce coupling.


Core conductor

Inner semiconductor

Inner insulation

Outer semiconductor

Metallic sheath

Outer insulation

(a) Single-core cable

Armor

(b) Pipe-type cable

(c) Submarine three-phase cable [source: Nexans cable]

Figure 1.6: Types of underground cables

however, it has no noticeable effect on the series impedance of the cable for frequencies

up to the low MHz range.

Inner/Outer semiconductor layers provide a smooth transition between low electric field

region (conductor) and high electric field region (insulation). Semiconductors have mini-

mal effect on impedance calculation because conduction losses in semiconductors are much

lower than the losses in conductors.

Pipe-type cables contain three SC cables asymmetrically located inside a metallic armor. In

some cases, sheaths and armors are made up of thin wires, and modelling them as tubular

conductors as in Fig. 1.6 can cause loss of accuracy in transient simulations. Submarine cables

may contain additional control signals in addition to three-phase SC cable, as shown in Fig. 1.6.


1.3.2 Cable Modelling

Cable modelling may be broken down into two stages: i) calculating transmission line parame-

ters, and ii) creating a model that is compatible with time-domain transient simulations. Cable

models are required because at high frequencies, where the length of the cable is comparable

to the wavelength, voltages and currents cannot be solved using the circuit theory, instead, the

transmission line theory must be used. In transmission line theory, voltages and currents are

given by the Telegrapher’s equations [25]

∂V

∂z= − [R(ω) + jωL(ω)] I (1.1a)

∂I

∂z= − [G(ω) + jωC(ω)] V , (1.1b)

where V and I represent voltages and currents in the conductors of the cable. The solution

of differential equations (1.1a)-(1.1b) depends on the per-unit-length (p.u.l.) resistance matrix

R(ω), p.u.l. inductance matrix L(ω), p.u.l. conductance matrix G(ω), and p.u.l. capacitance

matrix C(ω). These transmission line parameters depend on the geometry and material prop-

erties of the cable and of the surrounding medium, and in general are frequency-dependent.

However, in a frequency range from DC to a few-MHz the capacitance matrix can be con-

sidered constant because at these frequencies a charge placed on a conductor disperses to its

edge instantaneously (in a time of the order of 10−19 s) [26]. Since most of the power propa-

gates longitudinally along the cable, resistive losses are dominant compared to dielectric losses.

Therefore, G(ω) is set to zero and dielectric losses are neglected in transient simulations involv-

ing a cable. Contrary to the capacitance matrix, the p.u.l. resistance and inductance matrices

are frequency-dependent quantities due to the presence of skin, proximity and ground return

effects. We will now briefly discuss these three properties.

Skin effect

Skin effect causes current to crowd at the edges of a conductor as frequency increases,

as shown in Fig. 1.7. This phenomenon is caused by induced eddy current at the center of

the conductor, which opposes the time-varying magnetic fields due to the AC current flowing

inside the conductor. Eddy current flows in the direction opposite to the injected current in


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

Figure 1.7: Skin effect inside a conductor. The plot shows current density inside the conductornormalized to 1.

the conductor. Hence, the net current density, which is the sum of injected current and eddy

current, will vanish at the center of the conductor and crowd near its edges.

Proximity effect

Proximity effect causes current to be distributed asymmetrically inside conductors, as shown

in Fig. 1.8, when another current-carrying conductor is nearby. This phenomenon is caused by

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

Figure 1.8: Proximity effect between two conductors. The plot shows current density inside theconductor normalized to 1.

induced eddy currents in a conductor, which opposes the changing magnetic fields created by

the current in another conductor.

Ground return effect

Inclusion of ground return effect is important because currents flow inside the ground during

unbalanced operations and fault scenarios. Ground is conductive with conductivity ranging

from 10−4 S/m for dry rock up to 5 S/m for salt water. Unless explicitly stated, we will use

the word “ground” to indicate both soil and water present, for example, around a submarine

cable. Due to the conductive nature of ground, there will be induced currents inside ground,


which will result in ohmic losses.

Figure 1.9: Three-phase system connected by three SC cables

Furthermore, ground may act as a neutral conductor in a three-phase circuit. Consider

Fig. 1.9, where a three-phase load (ZA, ZB, ZC) is connected to a three-phase generator (Va,

Vb, Vc) by three SC cables. The cores of the three SC cables are connected to the generator, so

each core carries voltage with the same magnitude, but with a phase shift of 120. Sheaths are

grounded at both ends. During normal operations, neutral current flows in a three-phase circuit

when the three-phase load is unbalanced (ZA 6= ZB 6= ZC). In the circuit shown in Fig. 1.9,

neutral current will divide itself between sheaths and ground because there is a parallel con-

nection between them. The magnitude of the current in each branch depends on the resistance

of sheaths and earth. Sheaths around phase conductors are not always present, as for example

in overhead lines. In such cases, earth is the only path through which the unbalanced current

can flow. Hence, it is important to include earth in an impedance computation. Moreover,

ground is important to include during fault analysis because a large current could potentially

flow in ground. A fault may short a phase-conductor to the earth, which will unbalance the

three-phase circuit causing large current to flow in earth. Another scenario is a fault involving

sheaths, where a sheath may be shorted to earth driving all its current in ground.

Ground resistance depends on the material properties of the surrounding soil. For cables

buried deep inside earth, ground resistance depends on conductivity of the soil. For cables

buried shallowly from the air-ground interface, ground resistance depends on conductivity of


soil as well as the depth of the cable. Similarly, the ground return impedance of a subsea cable

depends on the conductivity and the depth of each layer surrounding the cable, such as water,

wet seabed, and soil beneath the seabed.

1.3.3 Existing Techniques for Impedance Calculation

We now review existing techniques to compute cable impedance. In this subsection we will

ignore the ground return effect, which will be the focus of Sec. 1.3.4 .

Analytic formulas

For low frequencies, analytical expressions to compute p.u.l. impedance and admittance of

cables with round conductors may be easily derived from electrostatic or magnetostatic princi-

ples, as shown in [25]. These formulas neglect skin and proximity effect, and so they are only

accurate at low frequencies for cables with no proximity effect (e.g. single SC cable). A more

accurate analytic method for impedance calculation is given in [27]. This method is based on

the surface impedance of a round conductor. This analytic method was generalized in [28],

to compute the impedance matrix of a power cable made up of SC cables. Even though this

method captures skin effect, proximity effect is still neglected. Proximity effect is minimal for

overhead lines, where phase-conductors have wide separation. However, for underground cables

proximity effect can be significant. To account for proximity effect, one must resort to full

electromagnetic simulations.

Volumetric methods: FEM, conductor partitioning

Volumetric simulation techniques, such as the finite element method (FEM) [1,29,30], success-

fully account for proximity effect. However, these techniques have a high computational cost

because the entire cross-section of the cable must be partitioned. Furthermore, the compu-

tational cost for volumetric methods increase with frequency because the mesh size must be

refined at high frequencies in order to successfully capture the skin effect. Generally, mesh

size at the edges of the conductor must be 2-3 times smaller than the skin depth inside the

conductor. This implies that at high frequencies many triangles are required to mesh the con-

ductors. General-purpose FEM tools such as COMSOL Multiphysics [4] provide the so-called


‘boundary-layer elements’ which can be used to model skin effect and reduce the computational

cost slightly. However, such features are complicated to set up. Conductor partitioning (also

called subconductor method) [28,31–34] is another set of techniques that account for proximity

and skin effect. These techniques model each conductor with many smaller round conductors.

In these techniques, the fields are solved by assuming that the current in each of the smaller

round conductors is uniform. Conductor partitioning techniques, like FEM, require discretiza-

tion of entire cross-section of the cable, which makes them computationally expensive. For

example, it took 3.5 hours on a 2.5 GHz computer with 16 GB memory to compute impedance

of a pipe-type cable in Fig 1.6b using FEM [5]3.

Surface methods

Due to the high computational cost associated with FEM tools, analytical formulas are com-

monly used by the power industry. Analytical formulas provide good accuracy in several cases

of practical interest, when proximity effect is not significant. However, with development of

offshore wind farms and underwater transmission grids, the submarine cables are becoming in-

creasingly complicated. Proximity effect inside these cables can no longer be neglected. Surface

methods provide an efficient alternative to compute p.u.l. parameters, since they are both ac-

curate and faster than FEM or other conductor partitioning techniques. Surface methods rely

on solving fields on the boundary of each conductor, rather than solving them across the whole

cross-section. This means surface methods require fewer unknowns and thus they are faster than

a traditional FEM approach. Additionally, surface methods do not encounter meshing prob-

lems at high frequencies. In the past, surface methods have been applied by many researchers

to compute the p.u.l. parameters of electronic interconnects [35–38]. These papers focus on

rectangular and triangular geometries because any arbitrary cross-section can be decomposed

into triangles and rectangles. Cables are made up of solid and hollow round conductors. Even

though few discussions on the application of the surface methods to solid round conductors are

found in the literature, for example in [38], no one has fully developed an efficient technique to

calculate impedance of solid and hollow round conductors using surface approach. Moreover,

surface methods have never been applied to include ground return effects of a cable.

3impedance is calculated at 120 frequency points logarithmically spaced from 1 Hz to 1 MHz.


1.3.4 Existing Techniques to Include Ground Return Effect

There are two ways to include ground return effect: analytic formulas and computational tools.

Analytic formulas

An analytic formula for ground return impedance was derived by Carson in 1926 for a thin wire

above a homogeneous lossy ground [39]. The solution provided by Carson involves an infinite

integral, which is approximated by a finite series. Carson’s formula neglects displacement

current, and thus the formulation is valid only at low frequencies and for low earth resistivity.

Several authors thereafter have tried to extend this formulation to higher frequencies [40, 41].

An alternative approach to Carson’s formulation to compute impedance at low frequencies was

presented in [42]. This formulation is based on the image theory, where an image plane replaces

ground. This image plane is displaced at a complex distance away from the original ground.

The distance is related to the skin depth in the ground. The most popular formula for an

underground configuration was developed by Pollaczek [43] in 1926; the approach poses similar

challenges as the Carson’s formula because the solution involves infinite integrals which are

difficult to solve analytically.

Transient programs such as ATP and PSCAD use analytical formulas to compute p.u.l.

parameters. To account for ground return effect, these programs use Carson and Pollaczek

approximations [44, 45]. The earth return impedance computed with analytical formulas in-

correctly assumes that the cable acts as a wire with constant current distribution (thin wire

approximation), but this assumption fails at high frequencies due to skin effect. Additionally,

these formulas assume that the field inside the ground is radially symmetric, and that is not

true because of the presence of proximity effect inside a highly conductive ground (e.g. salt

water). Several authors have tried to derive analytical expressions for overhead lines above the

multilayered earth, with the most popular being [46] and [45]. The expression provided in [46]

for overhead lines has been implemented in electromagnetic transient programs like ATP and

PSCAD. However, no analytic formula exists to compute the ground return impedance of an

underground cable buried inside a stratified medium.


Computational techniques: FEM

As an alternative to analytic formulas, computational techniques based on FEM and conductor

partitioning may be extended to include ground return effect as done in [29]. These techniques

provide accurate results since, unlike analytic formulas, they do not assume a radially symmetric

distribution of current inside the cable or the ground. However, setting up a FEM simulation

to include ground surrounding the cable is complicated. This complication arises because the

computational domain must be at least three times larger than the skin-depth in soil. Since

soil is a poor conductor, at low frequencies this translates into a very large mesh size (up to

5 kilometres at 1 Hz for soil with conductivity of 0.01 S/m), which makes the simulation very

time consuming. For example, FEM took 6 hours on a 3.40 GHz computer with 16 GB memory

to compute impedance parameters of a power cable made up of three SC cables buried in earth4.

The main problem with these tools is that we have to simulate a cable, which is typically few

centimetres wide, inside a domain that is few kilometres large. To circumvent this problem,

there exists special mesh elements in commercial FEM tools, such as the infinite element domain

in COMSOL Multiphysics, that may mitigate the complexity. However, as mentioned earlier,

these tools are not popular in power industry for cable modelling, due to setup complexities,

and high computational time.

Tunnels

Cables are often placed inside a tunnel, however, there is no adequate technique to handle this

scenario. Often in such scenarios the tunnel is either neglected, or the material properties of the

cable are changed so that the tunnel effects are approximately accounted [18]. FEM can handle

this scenario as well, but due to its setup challenges and computational cost it is unpopular.

1.3.5 Cable Models

Once the p.u.l. parameters are computed for broadband frequencies, typically from DC to

few MHz, frequency domain cable models are created using these p.u.l. parameters. Com-

mercial electromagnetic transient programs support several different cable models, such as

4at 60 frequency points


Universal Line Model (ULM) [47, 48]. Most of the cable models are based on the travelling

wave method [18], which requires the computation of the characteristic impedance and the

propagation constant of the line. These parameters can be calculated from the p.u.l. series

impedance and shunt admittance of the line. Once calculated, the characteristic impedance

and the propagation constant are passed on to a fitting algorithm such as Vector fitting [22],

which fits the sampled frequency response to a rational function. The time-domain response

for a transient scenario is efficiently retrieved from this fitted function through techniques like

recursive convolution [47].

1.4 Thesis Goal

Air (ε0, µ0)

Sea (ε0, µ0, σ1)

Seabed (ε0, µ0, σ2)

Soil (ε0, µ0, σ3)

(a)

Air (ε0, µ0)

Soil layer 1 (ε0, µ0, σ1)

Tunnel (ε0, µ0)

Soil layer 2 (ε0, µ0, σ2)

(b)

Figure 1.10: Examples of the cable systems that will be considered in this work: a) Twosubmarine cables placed on a seabed; b) Three SC cables placed inside a tunnel which is buriedin multilayer earth

The goal of this thesis is to develop a computational method that can accurately compute

the p.u.l. impedance of cables:

a) made up of solid and hollow round conductors placed in an arbitrary fashion,

b) possibly placed inside a tunnel or hole,

c) and buried inside a multilayer earth.


Attainment of this goal will allow us to compute accurately the p.u.l. parameters of complex

submarine and underground cables with great speed. Our tool will address power industry’s

need for a fast and accurate tool to model next-generation underground and underwater power

cables. Figure 1.10a shows examples of practical cable systems which can be studied using the

proposed approach. We will also investigate the influence of skin, proximity and ground return

effects on transient simulations. In this thesis, we do not account for “end-effects” of the cable,

nor do we account for twisting of the cables. Both these effects may play an important role on

the transient simulations, and may be pursued in future works. Our main focus in this work is to

account for proximity effect, which can potentially cause very large deviations in the transient

simulations as shown in Sec. 2.6.4. It is also very difficult to predict when proximity effect have

to be accounted and when analytic tools are sufficiently accurate, which further motivates the

development of a technique that accounts for proximity at a very low computational cost. Our

technique will achieve this task.

To develop the technique, we will exploit the following concepts: surface admittance, method

of moments, and electric field integral equation. Surface admittance concept will be used to

turn a volumetric problem into a surface problem through an application of the equivalence

theorem. Surface admittance for solid round conductors is available in [38]. However, we will

develop surface admittance relations for hollow conductors and a cable-hole system, which are

novel and are not found in the literature. Once we have formulated a surface problem, the

method of moments will be applied to the electric field integral equation to solve for fields and

impedance quantities. Method of moments matrices will be evaluated analytically, which will

make the proposed approach extremely fast. Our technique will also address the issue on the

inclusion of tunnels, which has never been properly investigated in the literature, even though

a lot of underground cables are placed inside a tunnel. Tunnels will be included by proposing a

novel surface operator that compactly describes the tunnel and the cables placed inside. Finally,

ground return effects will be accurately included for multilayer ground structure. Commonly

used analytic formulas can only include the effect of a two-layer ground.


1.5 Modelling of Electronics Cables for Signal Integrity Anal-

yses

Before presenting our technique, we briefly discuss the modelling of electronics cables for signal

integrity analyses.

The frequency of operation of electronic interconnects has been steadily growing with the

increase in clock speed [49]. At these higher frequencies, the electrical dimensions of intercon-

nects are larger, and so they can adversely affect the quality of the transmitted signals. In

order to understand and minimize such phenomena, the signal propagation in the interconnects

must be analyzed through Maxwell’s equations. Complex electronic cables like USB cables, or

multiconductor busses, are major sources of signal integrity issues such as cross-talk. These

cables contain many conductors that carry different signals. Due to the close proximity of cable

conductors, signals interfere with each other as a consequence of electromagnetic induction.

In signal integrity analyses, one is usually interested in predicting how a signal propagates

along an interconnect for variety of different excitations. It is computationally expensive to run

a full-wave simulation for many combinations of excitation signals. Hence, the transmission

line theory is usually invoked for signal integrity analyses. In order to apply the transmission

line theory, the p.u.l. impedance and admittance parameters that are accurate from DC to

GHz are required. Even though the frequency range of interest for electronic cables differs from

the power cables, the electrical dimensions of the cable remain the same. Hence, the same set

of tools can be used to calculate the impedance parameters of both types of cables. Once the

impedance parameters are obtained, the Telegrapher’s equation is solved to predict the signal

waveforms for variety of different excitations.

1.6 Thesis Outline

The remainder of this thesis is organized as follows. In Chapter 2, we will formulate the problem

of computing the p.u.l. impedance of cables made up of solid and hollow round conductors. This

chapter will introduce how a surface admittance operator can be derived to replace complex

cable structures with equivalent current sources via the equivalence principle. The electric field


integral equation will then be applied to this equivalent problem to compute fields and p.u.l.

impedance of the cable. In Chapter 3, we will develop a technique that calculates impedance of

a cable placed inside a hole in a multilayered earth. To develop this technique, we will derive

the surface admittance operator of a cable-hole system that will be used to replace the hole,

and all the conductors inside it, by the surrounding medium and a single equivalent current

source on the boundary of the hole. Thereafter, Green’s function of a multilayered medium

will be used to include ground return effect. In both Chapter 2 and Chapter 3, results will

be provided to show the accuracy of the proposed method, as well as the impact of accurate

impedance parameters on transient simulations. In Chapter 4, we will show how electric field,

magnetic field, and current densities can be computed inside the cables and ground. The final

chapter will summarize and conclude this research work.

Chapter 2

Cables Surrounded by a

Homogeneous Lossless Medium

2.1 Problem Statement

In this chapter, we consider a cable oriented along the z-direction and surrounded by a lossless

medium that has permittivity ε0, and permeability µ0. This cable is made up of P round

conductors, which are either solid or hollow as shown in Fig. 2.1. Outer radius of conductor p

is denoted by ap. If conductor p is hollow, we denote its inner radius with ap. The conductors

have permittivity ε, permeability µ, and conductivity σ1.

Our goal is to compute the frequency-dependent p.u.l. resistance matrix R(ω) and p.u.l.

inductance matrix L(ω), which appear in the Telegrapher’s equation

∂V

∂z= − [R(ω) + jωL(ω)] I , (2.1)

where vectors V =

[V1 V2 . . . VP

]Tand I =

[I1 I2 . . . IP

]T, are formed by the po-

tential Vp and the current Ip in each conductor. We make two assumptions: the conductors

are invariant along the z-direction, and the cross-section of each conductor is much smaller

than the wavelength at the frequencies of interest. These assumptions allow us to employ a 2D

1For the sake of simplicity, we assume in our discussion that all conductors have the same properties. Theproposed techniques can handle the most general case with trivial changes.

22

Chapter 2. Cables Surrounded by a Homogeneous Lossless Medium 23

εo, µo

ε, µ, σ

ap

(a)

εo, µoε, µ, σ

ap ap

(b)

Figure 2.1: Sample solid and hollow cross-sections

quasi-TM approach to calculate the impedance parameters. Both these assumptions hold for

practical cable geometries.

2.2 Surface Admittance Operator

In order to calculate the p.u.l. impedance matrix we employ the surface admittance formulation

introduced by De Zutter and Knockaert [38]. In this formulation, we replace all solid and hollow

conductors by the surrounding medium. In order to maintain the fields outside the conductors

unchanged, we introduce equivalent current sources on the conductors’ boundaries. We first

derive the surface admittance relationship for a solid conductor, following which we will consider

a hollow conductor.

2.2.1 Surface Admittance Operator for a Solid Conductor

We consider conductor p whose cross-section is solid, and its center position is given by (xp, yp).

The boundary cp of the conductor can be traced by the position vector rp(ap, θp) that is defined

by

rp(ρp, θp) = (xp + ρp cos θp) x + (yp + ρp sin θp) y , (2.2)

where (ρp, θp) constitute radial and azimuthal coordinates of a cylindrical coordinate system

centered about (xp, yp), as shown in Fig 2.2a. Unit vectors along x- and y-direction are denoted

by x and y, respectively.

Next, we expand the longitudinal electric field on the boundary of the conductor by a

truncated Fourier series

E(p)z (θp) =

Np∑n=−Np

E(p)n ejnθp , (2.3)


εo, µo

ε, µ, σ

θpρp

(a)

εo, µocp

J(p)s (θp)

(b)

Figure 2.2: Application of the equivalence theorem to a solid round conductor. The conductor

in (a) is replaced by the surrounding medium and an equivalent current J(p)s (θp) on its surface

as in (b).

where Np governs the accuracy of the field representation inside the conductor. By setting

Np = 0, the electric field is assumed to be circularly-symmetric. This assumption is valid for all

cases with no proximity effect. Otherwise, Np = 3 or Np = 4 is sufficient to capture proximity

effect in most practical cases, which will be demonstrated in Sec. 2.6. In Sec. 4.2, we will also

present an efficient algorithm to automatically estimate Np. Fourier coefficients E(p)n in (2.3)

are one of the unknown variables in this formulation.

The longitudinal electric field at any point inside the conductor, as described by the position

vector rp(ρp, θp), may be obtained by solving the homogeneous Helmholtz equation [50]

∇2E(p)z (ρp, θp) + k2E(p)z (ρp, θp) = 0 , (2.4)

subject to the boundary condition (2.3). In the equation above, k =√ωµ (ωε− jσ) is the

wavenumber inside the conductor. The solution of (2.4) is given by [50]

E(p)z (ρp, θp) =

Np∑n=−Np

E(p)n

J|n|(kap)J|n|(kρp) ejnθp , (2.5)

where J|n|(.) is the Bessel function of order |n| [51].

We now simplify the problem by replacing the conductor with the surrounding medium,

thereby, homogenizing the whole medium. In order to maintain the fields outside the con-

ductor unchanged, we introduce an equivalent surface current J(p)s (θp) on the boundary of the

conductor according to the equivalence theorem [50], as shown in Fig 2.2b. We expand this


equivalent current on the pth-conductor through a truncated Fourier series

J (p)s (θp) =

1

2πap

Np∑n=−Np

J (p)n ejnθp . (2.6)

In the above equation, the scaling factor 1/(2πap) has been introduced to make the coefficient

J(p)0 equal to the current flowing in the conductor, as shown in [38].

By equivalence principle [50], the value of the equivalent current is given by

J (p)s (θp) = H

(p)t (a−p , θp)−H

(p)t (a−p , θp) , (2.7)

where H(p)t is the tangential magnetic field at the boundary of the conductor in the original

configuration shown in Fig. 2.2a, and H(p)t is the tangential magnetic field at the boundary of

the conductor in the new configuration shown in Fig. 2.2b. Under the quasi-TM assumption,

the magnetic field quantities in (2.7) can be calculated from their corresponding electric field

quantities as

H(p)t (a−p , θp) =

1

jωµ

∂E(p)z

∂ρp

∣∣∣∣ρp=a

−p

(2.8)

H(p)t (a−p , θp) =

1

jωµ0

∂E(p)z∂ρp

∣∣∣∣ρp=a

−p

, (2.9)

where E(p)z is the electric field in the scenario after we replace the conductor by the surrounding

medium, as in Fig. 2.2b. This electric field, once again, can be found by solving the Helmholtz

equation

∇2E(p)z (ρp, θp) + k20 E(p)z (ρp, θp) = 0 (2.10)

subject to the boundary condition (2.3). In (2.10), k0 = ωµ0√ωε0 is the wavenumber of the

surrounding medium. Solving the homogeneous Helmholtz equation (2.10) gives

Ez(ρp, θp) =

Np∑n=−Np

E(p)n

J|n|(k0ap)J|n|(k0ρp) ejnθp . (2.11)

Now we can find the value of equivalent current density J(p)s (θp) in (2.7). To calculate


the equivalent current, we first relate the electric and magnetic fields on the boundary of the

conductor cp by substituting the Fourier expansion of electric fields E(p)z and E(p)z , which are

given in (2.5) and (2.11), into (2.8) and (2.9). These substitutions result in

H(p)t (a−p , θp) =

k

jωµ

Np∑n=−Np

E(p)n

J|n|(kap)J ′|n|(kap) ejnθp , (2.12)

H(p)t (a−p , θp) =

k0jωµ0

Np∑n=−Np

E(p)n

J|n|(k0ap)J ′|n|(k0ap) ejnθp , (2.13)

where J ′|n|(.) is the derivative of the Bessel function of order |n| [51]. Next, we substitute (2.12)

and (2.13) into (2.7) to obtain the expression for equivalent current density

J (p)s (θp) =

1

jω

Np∑n=−Np

E(p)n

[k

µ

J ′|n|(kap)J|n|(kap)

− k0µ0

J ′|n|(k0ap)J|n|(k0ap)

]ejnθp . (2.14)

We now substitute the Fourier expansion of the equivalent current (2.6) into the left-hand side

of (2.14) to get

1

2πap

Np∑n=−Np

J (p)n ejnθp =

1

jω

Np∑n=−Np

E(p)n

[k

µ


− k0µ0


]ejnθp . (2.15)

In order to relate coefficients J(p)n and E

(p)n we apply the method of moments. We project (2.15)

on to the Fourier basis functions ejn′θp by applying

ˆ 2π

0[.] e−jn

′θp dθp for n′ = −Np, . . . , Np (2.16)

to both sides of (2.15), and get

ˆ 2π

0

Np∑n=−Np

J (p)n ej(n−n

′)θp dθp =2πapjω

ˆ 2π

0

Np∑n=−Np

E(p)n

[k

µ


− k0µ0


]ej(n−n

′)θp dθp

(2.17)

for n′ = −Np, . . . , Np. The equation above may be simplified by interchanging the summation


and integration on both sides of the equation, and then evaluating the integrals. Since,

ˆ 2π

0ej(n−n

′)θp dθp =

2π n = n′

0 n 6= n′, (2.18)

we simplify (2.17) to

J (p)n = Y (p)

n E(p)n , (2.19)

where,

Y (p)n =

2π

jω

[kapJ ′|n|(kap)µJ|n|(kap)

−k0apJ ′|n|(k0ap)µ0J|n|(k0ap)

], (2.20)

for n = −Np, . . . , Np. Relation (2.19) defines the surface admittance relationship that relates

the electric field to the current density. In (2.19), Y(p)n is termed as the surface admittance

operator for a solid conductor [38].

The concept of surface admittance is analogous to the Norton theorem in circuit analysis.

Using the Norton theorem, any complex portion of a large circuit may be replaced by the Norton

current source and the Norton admittance. Similarly, using the surface admittance relationship

we can replace the conductor by an equivalent current source. In Norton theorem, the Norton

admittance is defined to be the ratio of Norton current and the open circuit voltage at the

input terminals of the network. Analogously, the surface admittance operator is the ratio of

equivalent current and the electric fields on the boundary of the conductor. In essence, the

surface admittance concept is a simpler representation of the conductor which will allow us to

simplify our original problem, just like Norton theorem helps simplify a complicated circuit.

To briefly summarize the process employed in this subsection, we first computed the electric

field inside the conductor subject to a Dirichlet boundary condition. This Dirichlet boundary

condition was expressed as a truncated Fourier series. Next, we replaced the conductor by

the surrounding medium. In order to preserve the fields outside the conductor, an equivalent

current was introduced on the boundary of pth-conductor. This equivalent current was also

expanded through a truncated Fourier series. Following this, we used the equivalence theorem

to relate the coefficients of equivalent current with the coefficients of electric field expansion to

obtain the surface admittance operator.


2.2.2 Surface Admittance Operator for a Hollow Conductor

We now apply the same procedure we employed for a solid conductor to a hollow conductor.

Let the cross-section of the pth-conductor be hollow, with inner boundary denoted by cp and

outer boundary denoted by cp. Outer boundary cp can once again be described by the same

position vector rp(ap, θp) defined in (2.2). Inner boundary cp can be traced by the position

vector rp(ap, θp).

The electric field on the inner and outer boundaries are denoted by E(p)z (θp) and E

(p)z (θp),

respectively, and are expanded through truncated Fourier series

E(p)z (θp) =

Np∑n=−Np

E(p)n ejnθp , (2.21a)

E(p)z (θp) =

Np∑n=−Np

E(p)n ejnθp . (2.21b)

The electric field at any arbitrary point inside the hollow conductor can be computed by solving

the homogeneous Helmholtz equation (2.4) subject to the two Dirichlet boundary conditions in

(2.21a)-(2.21b). By doing so, we get

E(p)z (ρp, θp) =

Np∑n=−Np

(Cn(k)H|n|(kρp) +Dn(k)K|n|(kρp)

)ejnθp , (2.22)

where Cn(k) and Dn(k) are computed by imposing the boundary conditions (2.21a)-(2.21b)

and are equal to

Cn(k) =E

(p)n K|n|(kap)− E

(p)n K|n|(kap)

H|n|(kap)K|n|(kap)−H|n|(kap)K|n|(kap), (2.23a)

Dn(k) =E

(p)n H|n|(kap)− E

(p)n H|n|(kap)

H|n|(kap)K|n|(kap)−H|n|(kap)K|n|(kap). (2.23b)

In these formulas, H|n|(.) and K|n|(.) denote Hankel functions of order |n| of, respectively, first

and second kind [51].

Analogous to the approach we took for a solid conductor, we next replace the hollow con-

ductor with the surrounding medium. We need to introduce two equivalent currents: J(p)s (θp)


εo, µoε, µ, σ

ap ap

(a)

εo, µo

J(p)s (θp) cp

cp

J(p)s (θp)

(b)

Figure 2.3: Application of the equivalence theorem to a hollow conductor. The actual conductor,

shown in (a), is replaced by the surrounding medium and equivalent currents J(p)s (θp) and

J(p)s (θp) are introduced on the inner and outer surface of the conductor as in (b) .

and J(p)s (θp), respectively, on the inner boundary and on the outer boundary to restore fields

outside the conductor, as shown in Fig 2.3b. Equivalent current J(p)s (θp) ensures that fields

outside the cavity (ρp > ap) are preserved, and current J(p)s (θp) ensures that fields inside the

cavity (ρp < ap) are preserved. These electric currents are again expanded into truncated

Fourier series

J (p)s (θp) =

1

2πap

Np∑n=−Np

J (p)n ejnθp , (2.24)

J (p)s (θp) =

1

2πap

Np∑n=−Np

J (p)n ejnθp . (2.25)

Scaling factors 1/(2πap) and 1/(2πap) ensure that the total current inside the pth-conductor

equals to J(p)0 + J

(p)0 . By applying the equivalence principle, we ensure that the fields outside

the conductor remain unchanged by setting currents to

J (p)s (θp) =

1

jω

[1

µ

∂E(p)z

∂ρp− 1

µ0

∂E(p)z∂ρp

]ρp=a

−p

, (2.26)

J (p)s (θp) =

1

jω

[1

µ0

∂E(p)z∂ρp

− 1

µ

∂E(p)z

∂ρp

]ρp=a

+p

, (2.27)

where E(p)z (ρp, θp) is the electric field inside the conductor after we replace the conductor with

the surrounding medium. This electric field may be found by imposing boundary conditions

(2.21a)-(2.21b) on the homogeneous Helmholtz equation (2.4) with wavenumber k replaced by

k0, which is the wavenumber of the surrounding medium. The solution of (2.4) will be the same


as (2.22), but with k replaced by k0.

Next, we find the equivalent currents J(p)s (θp) and Js(θp) by substituting the solutions of

electric fields E(p)z (ρp, θp) and E(p)z (ρp, θp) into (2.26) and (2.27)

J (p)s (θp) =

1

jω

Np∑n=−Np

[1

µ

(Cn(k)H′|n|(kap) +Dn(k)K′|n|(kap)

)− 1

µ0

(Cn(k0)H′|n|(k0ap) +Dn(k0)K′|n|(k0ap)

)]ejnθp , (2.28)

J (p)s (θp) =

1

jω

Np∑n=−Np

[1

µ0

(Cn(k0)H′|n|(kap) +Dn(k0)K′|n|(k0ap)

)− 1

µ

(Cn(k)H′|n|(kap) +Dn(k)K′|n|(kap)

)]ejnθp , (2.29)

where H′|n|(.) and K′|n|(.) are the derivatives of the Hankel functions H|n|(.) and K|n|(.), re-

spectively. Next, we substitute the Fourier expansion of currents, which are given in (2.24)

and (2.25), into the left-hand side of (2.28) and (2.29), and then apply the method of mo-

ments on the resulting equation by following the same procedure we used for solid conductors.

This method of moments procedure will allow us to relate the Fourier coefficients of equivalent

currents and electric fields on the boundary as

J (p)n

J(p)n

= Y(p)n

E(p)n

E(p)n

. (2.30)

Here, Y(p)n is a 2× 2 matrix

Y(p)n =

Y11,n Y12,n

Y21,n Y22,n

, (2.31)


where,

Y11,n =2π

jω

[χn(kap, kap)

mn(kap, kap)µ− χn(k0ap, k0ap)

mn(k0ap, k0ap)µ0

](2.32a)

Y12,n =2π

jω

[χn(k0ap, k0ap)

mn(k0ap, k0ap)µ0− χn(kap, kap)

mn(kap, kap)µ

](2.32b)

Y21,n =2π

jω

[χn(k0ap, k0ap)

mn(k0ap, k0ap)µ0− χn(kap, kap)

mn(kap, kap)µ

](2.32c)

Y22,n =2π

jω

[χn(kap, kap)

mn(kap, kap)µ− χn(k0ap, k0ap)

mn(k0ap, k0ap)µ0

](2.32d)

with χn(α, β) = β[H′|n|(β)K|n|(α)−H|n|(α)K′|n|(β)

], .

2.2.3 Surface Admittance Operator for Multiple Conductors

In sections 2.2.1 and 2.2.2, we derived the surface admittance operator for a solid and a hollow

conductor. We now extend this concept to all the conductors by replacing every conductor by

the surrounding medium and introducing an equivalent current on each conductor’s boundary.

To simplify the notation, we cast the coefficients of the electric field on the boundary of

each conductor into vector E(p). If pth-conductor is solid, we let

E(p) =

E(p)−Np...

E(p)0

...

E(p)Np

, (2.33)

otherwise, if pth-conductor is hollow then

E(p) =

E(p)−Np...

E(p)Np

E(p)−Np...

E(p)Np

. (2.34)


Next, we assemble the coefficients of the electric fields from all P conductors into a global vector

of unknowns

E =

E(1)

E(2)

...

E(P )

. (2.35)

The size of E is N × 1, where

N =

Ps∑p=1

(2Np + 1) +

Ph∑p=1

2(2Np + 1) , (2.36)

where Ps and Ph denote number of solid and hollow conductors, respectively. In a similar

manner, we collect all the Fourier coefficients of the equivalent currents into a global vector

J =

J(1)

J(2)

...

J(P )

, (2.37)

where J(p) is a vector containing the Fourier coefficients of the equivalent currents on p-th

conductor, and is analogous to (2.33) and (2.34).

At this point, we can relate vectors E and J by a global surface admittance operator as

J = YsE , (2.38)

where Ys is a block diagonal matrix, with entries of each block computed from (2.20) for solid

conductors or (2.31) for hollow conductors. Furthermore, we relate the equivalent currents J

with conduction currents I (2.1) through

I = UTJ , (2.39)

where U is a constant N × P matrix made by all zeros and one or two “1”s in each column.


If conductor p is solid, then “1” is in the pth column, and the same row as the coefficient J(p)0

in J. Otherwise, if conductor p is hollow, then “1” is in the pth column, and the same rows as

the coefficients J(p)0 and J

(p)0 in J .

In our formulation, we have a total of three unknown quantities: electric field coefficients

E, equivalent current coefficients J, and the p.u.l. impedance R(ω) + jωL(ω). So far we have

discussed one relationship between the equivalent currents and the electric fields, namely the

surface admittance operator (2.38). In the next section, we will discuss the electric field integral

equation which provides us with another relationship. A third relationship will stem from the

independence of the p.u.l. impedance from the excitation current and will be discussed in

Sec 2.4.

2.3 Electric Field Integral Equation

After the application of the equivalence theorem, we have homogenized the background medium

with equivalent current sources at the boundaries of the conductors. At this point, we relate

the electric field and the equivalent current sources through the electric field integral equation

(EFIE)

Ez(r) = −jωAz(r)− ∂V

∂z, (2.40)

where V is the scalar potential, and Az is the z-component of the magnetic vector potential.

Using the superposition principle, vector potential Az(r) can be expanded as

Az(r) =P∑q=1

Aq(r) , (2.41)

whereAq(r) is the contribution due to the equivalent currents on the contour cq of qth-conductor

(and cq if qth-conductor is hollow). If conductor q is solid, then

Aq(r) = −µ0ˆ 2π

0J (q)s (θ′q)G(r, rq(aq, θ

′q))aqdθ

′q , (2.42)


while, if conductor q is hollow

Aq(r) = −µ0ˆ 2π


′q))aqdθ

′q − µ0

ˆ 2π


′q))aqdθ

′q (2.43)

due to the superposition of the contributions from the inner and outer equivalent currents. In

(2.42) and (2.43), G(r, r′) is the Green’s function of a two-dimensional homogeneous space [50]

G(r, r′) =1

2πln∣∣r− r′

∣∣ . (2.44)

Here, we use the quasi-static Green’s function instead of the full-wave Green’s function. The

advantage of using the quasi-static Green’s function is that it is frequency independent, which

will translate into computational savings. The disadvantage is that the quasi-static Green’s

function is only accurate when a) the cross-section of the cable is much smaller than a wave-

length, so that the retardation effect may be neglected; and b) the surrounding medium is

lossless . The first assumption is always valid because the smallest wavelength of our interest is

30 m at 10 MHz, which is much smaller than the typical cross-section of a power cable. In this

chapter, we are considering lossless surrounding medium; hence the second assumption is also

valid. Both assumptions will be relaxed in Chapter 3 when we consider cables surrounded by

the lossy ground.

Next, we evaluate EFIE (2.40) on the boundary of each conductor. On the boundary of the

p-th conductor, EFIE reads

E(p)z (θp) = −jωAz(rp(ap, θp))−

∂Vp∂z

, (2.45)

where we substituted Ez(rp) with E(p)z (θp), which is the electric field on the boundary cp.

Furthermore, we relate the potential drop∂Vp∂z to the p.u.l. impedance parameters through the

Telegrapher’s equation (2.1) [52,53]

E(p)z (θp) = −jωAz(rp(ap, θp)) +

P∑q=1

[Rpq(ω) + jωLpq(ω] Iq , (2.46)

In (2.46), Rpq(ω) and Lpq(ω) represent the (p, q) entry of the p.u.l. resistance and inductance


matrices, respectively. If p-th conductor is hollow, then we also evaluate EFIE (2.40) on the

inner boundary cp, obtaining

E(p)z (θp) = −jωAz(rp(ap, θp)) +

P∑q=1

[Rpq(ω) + jωLpq(ω] Iq . (2.47)

Our goal is to solve the p.u.l. impedance parameters that appear in (2.46) and (2.47), which

will be done through the method of moments.

Solid conductors

For simplicity, we first consider the case where all conductors are solid, we will then generalize

the procedure to treat a mixture of solid and hollow conductors. We first substitute the Fourier

expansion of equivalent currents and fields in (2.6) and (2.3) into (2.47) and (2.42). Following

these substitutions, we obtain

Np∑n=−Np

E(p)n ejnθp =

jωµ02π

P∑q=1

Nq∑n=−Nq

J (q)n

ˆ 2π

0ejnθ

′q G(rp(ap, θp), rq(ap, θ

′q))dθ

′q

+P∑q=1

[Rpq(ω) + jωLpq(ω)] Iq . (2.48)

We discretize the equation above by the method of moments. We apply the Galerkin method

on the above equation by applying operator

ˆ 2π

0[.] e−jn

′θp dθp n′ = −Np, . . . , Np (2.49)

to both sides of (2.48), obtaining

E(p)n′ =jωµ0

P∑q=1

Nq∑n=−Nq

G(p,q)n′,n J

(q)n + δn′,0

P∑q=1

[Rpq(ω) + jωLpq(ω)] Iq , (2.50)

for p = 1, . . . , P , where

δn′,0 =

1 when n′ = 0

0 when n′ 6= 0

(2.51)


and where G(p,q)n′,n denotes the (n′, n) entry of the matrix G(p,q). Matrix G(p,q) describes the

contribution of the equivalent current on q-th conductor to the field on the boundary of the

p-th conductor. The entries of G(p,q) are given by the double integral

G(p,q)n′,n =

1

(2π)2

ˆ 2π

0

ˆ 2π

0G(rp(ap, θp), rq(aq, θ

′q)) ej(nθ

′−n′θ) dθpdθ′q , (2.52)

which may be evaluated analytically as shown in Appendix B.1.

Using the matrix notation established in this chapter, we rewrite (2.52) as

E = jωµ0GJ + U [R(ω) + jωL(ω)] I (2.53)

where Green’s matrix G is made up of several block matrices

G =

G(1,1) . . . G(1,P )

.... . .

...

G(P,1) . . . G(P,P )

. (2.54)

Equation (2.53) is the discretized EFIE that allows us to capture the proximity between the

conductors. Indeed, the non-diagonal blocks in (2.54) capture the mutual impedance between

the conductors.

Solid and hollow conductors

In presence of hollow conductors, the formulas presented in the previous section can be easily

generalized. For each hollow conductor, one must add the contribution of the equivalent current

present on the inner boundary. If the p-th conductor is solid and q-th conductor is hollow, then

we add to the right hand side of (2.48)

jωµ02π

Nq∑n=−Nq

J (q)n

ˆ 2π

0ejnθ

′q G(rp(ap, θp), rq(aq, θ

′q)dθ

′q . (2.55)


The inclusion of an additional term in (2.55) to the EFIE (2.48) modifies Green’s matrix G(p,q)

which may now be rewritten as

G(p,q) =

[G(cp,cq) G(cp,cq)

]. (2.56)

If both the p-th and q-th conductors are hollow, we define G(p,q) as

G(p,q) =

G(cp,cq) G(cp,cq)

G(cp,cq) G(cp,cq)

. (2.57)

This Green’s matrix describes the contribution of the current on the inner and outer boundaries,

cp and cp, to the field on the inner boundary and outer boundary. Entries of each of the four

block matrices are given by (2.52) for different values of rp and rq depending on which of

the four blocks is being evaluated. The analytic evaluation of double integral in (2.52) makes

the technique fast and readily scalable for large number of conductors. Otherwise, numerical

integration would be very slow. The analytic evaluation of (2.52) is a contribution of this thesis.

For the sake of readability, it is given in Appendix B.1, since it involves long mathematical

derivations.

To summarize this section, we evaluated the electric field on the contour of each conductor

by an application of the electric field integral equation. By applying the method of moments

we obtained (2.53), which is the second relation between E, J, and the p.u.l. impedance. In

Sec. 2.4, we will be able to solve for all three quantities using a third relation.

2.4 Computation of p.u.l. Impedance

The p.u.l. parameters can be computed by combining the surface admittance relation (2.38),

and the discretized EFIE (2.53). We first left multiply (2.53) by Ys, and use (2.38) to obtain

J = jωµ0YsGJ + YsU [R(ω) + jωL(ω)] I . (2.58)


In the equation above, equivalent current J may be expressed as

J = (1− jωµ0YsG)−1 YsU [R(ω) + jωL(ω)] I , (2.59)

where 1 is the identity matrix. Next, we left multiply (2.59) by UT to get

I =[(

UT (1− jωµ0YsG)−1 YsU)· (R(ω) + jωL(ω))

]I . (2.60)

Since all materials in the cable are linear, the p.u.l. impedance must be independent from

the current flowing in the conductors. Hence, the expression inside the square bracket on the

right-hand side of (2.60) must be equal to the identity matrix. Consequently, we have that

R(ω) + jωL(ω) =(UT (1− jωµ0YsG)−1 YsU

)−1. (2.61)

By taking the real and imaginary part of the equation above, we obtain the p.u.l. resistance

and inductance matrices

R(ω) = Re

(UT (1− jωµ0YsG)−1 YsU

)−1(2.62)

L(ω) = ω−1Im(

UT (1− jωµ0YsG)−1 YsU)−1

. (2.63)

Note that (2.62) and (2.63) give the partial p.u.l. resistance and inductance, since we have

referred all conductor potentials to infinity. We can obtain the p.u.l. impedance from the

partial p.u.l. impedance through the procedure outlined in Appendix A.

We clarify that even though we discretized the electric fields and current density, and stored

them into vectors E and J, we do not explicitly solve for these quantities. Solving for these

quantities add an additional overhead, as we have evaluate (2.53) and (2.59). However, we will

have to solve for E and J when we calculate the field distribution in Sec. 4.1.

2.5 Computational Cost

In this section, we briefly discuss the computational cost of our proposed approach–MoM-SO.


The computational cost of our technique varies with the number of conductors, and in

particular, the total number of electric field and equivalent current coefficients N , whose value

is given in (2.36). In order to compute the p.u.l. impedance, we first generate all the matrices

that appear on the right-hand side of (2.62) and (2.63), following which all the matrix operations

in these two equations are performed.

There exist three matrices in (2.62) and (2.63) that need to be generated, namely Ys, G,

and U. Among these three matrices, matrix U, which appears in (2.39), is a sparse matrix,

and so it is computationally inexpensive to generate. Surface admittance matrix Ys is a block

diagonal matrix of size N ×N , where each block is a diagonal matrix with two additional sub-

diagonals if the block corresponds to a hollow conductor. Hence, this matrix too is inexpensive

to generate. The Green’s matrix G is most expensive to generate among the three matrices.

Matrix G is of size N × N and is densely populated. However, analytical evaluation of the

entries of this matrix improves the time it takes to create G. Moreover, under the quasi-static

scenario, this matrix only needs to be generated once for all the frequency points.

Inversion of (1 − jωYsG) is the most expensive operation in our technique.2 However,

since MoM-SO is a surface method, the size of the matrix that needs to be inverted is much

smaller compared to a FEM technique. Therefore, the inversion cost is much less than the

FEM. Another advantage is that MoM-SO, unlike FEM, does not require finer meshing at high

frequencies to model skin effect because the surface admittance operator efficiently handles skin

effect. Consequently, the number of unknowns grows very mildly with increasing frequency.

2.6 Numerical Results

In order to demonstrate the accuracy and computational efficiency of the proposed approach,

we present four test cases: a two-wire line, a coaxial cable, a three-phase cable, and a pipe-type

cable.

2In practise, we do not invert this matrix, but apply LU factorize it and perform the operation UT (1 −jωYsG)−1Ys.


2.6.1 Example # 1: A Two-wire Line

We first validate the proposed approach by considering a system of two solid round conductors.

Each conductor has a radius of a = 10 mm, and is made up of copper with conductivity σ =

5.8 · 107 S/m. We consider two different center-to-center spacing between the two conductors:

D = 100 mm and D = 25 mm. In our configuration, current flows in one conductor and returns

through the other conductor.

Figures 2.4 and 2.5 show the resistance and inductance of the line. In these figures, we

compare the proposed approach against FEM (COMSOL Multiphysics [4]), and two sets of

analytic formulas. The first set of analytic formulas is a high frequency approximation with the

loop resistance and inductance given by [54]

R =Rsπa

D2a√(D2a

)2 − 1, (2.64)

Lext =µ0π

cosh−1(D

2a

). (2.65)

where Rs = (σδ)−1 is the surface resistance and

δ =1√

πfµ0σ(2.66)

is the skin depth inside the conductor. These formulas account for proximity effect only at high

frequencies. The second set of formulas are based on the internal impedance of the conductors

which is given by [54]

Zint =1√

2πaσδ

ber(ξ) + j bei(ξ)

bei′(ξ)− j ber′(ξ), (2.67)

where Kelvin functions ber(.) and bei(.) are the real and imaginary parts of J0(ξ ej3π/4), re-

spectively [51]. In this set of formulas, the total p.u.l. impedance is given by [54]

Z = 2Zint + jωLext . (2.68)

Equation (2.68) captures skin effect, but proximity effect is neglected.

Figures 2.4 and 2.5 show the p.u.l. inductance and resistance when the spacing between


100

102

104

106

10−4

10−3

10−2

Frequency [Hz]

Res

ista

nce

p.u

.l.

(Ω/m

)

Analytical (no proximity)

Analytical (high−freq)

MoM−SO

FEM

100

102

104

106

10−4

10−3

10−2

Frequency [Hz]

Res

ista

nce

p.u

.l.

(Ω/m

)



MoM−SO

FEM

Figure 2.4: P.u.l. resistance of the two-wire line of Sec. 2.6.1 for two different wire separations:100 mm (left panel) and 25 mm (right panel) [3]. Comparison between the high frequencyapproximation (2.64), the analytic formula without proximity (2.68), the proposed method(MoM-SO) and the finite elements method [4] is shown.

100

102

104

106

9

9.2

9.4

9.6

9.8

10

10.2

10.4

10.6

x 10−7

Frequency [Hz]

Induct

ance

p.u

.l.

[H/m

]



MoM−SO

FEM

100

102

104

106

2.5

3

3.5

4

4.5

5

5.5x 10

−7

Frequency [Hz]

Indu

ctan

ce p

.u.l

. [H

/m]



MoM−SO

FEM

Figure 2.5: P.u.l. inductance of the two-wire line of Sec. 2.6.1 for two different wire separations:100 mm (left panel) and 25 mm (right panel) [3]. Comparison between the high frequencyapproximation (2.65), the analytic formula without proximity (2.68), the proposed method(MoM-SO) and the finite elements method [4] is shown.

the conductors is 100 mm and 25 mm. In MoM-SO, electric fields and equivalent currents

are expanded with Np = 4 in (2.3) and (2.6) for both spacings.3 In these figures, the curves

labeled “Analytical (no proximity)”, “MoM-SO”, and “FEM” match very well when the spacing

between the conductors is 100 mm. These results suggest that there exist no proximity effect

between the two conductors when the spacing is 100 mm. On the contrary, there is significant

3Np = 0 is sufficient for spacing of 100 mm because there is little proximity effect.


proximity when the spacing is only 25 mm. However, this proximity effect is perfectly captured

by MoM-SO and FEM since the curves labeled by “Analytical (high-freq)”, “MoM-SO”, and

“FEM” match very well at high frequencies. Finally, we can see that the results from the

proposed method agree very well with the FEM. However, as we shall see in the next examples,

the CPU cost for the proposed method is significantly lower than the FEM [4].

2.6.2 Example # 2: Coaxial Cable

In the previous section, we validated MoM-SO against FEM and analytic formulas for a system

consisting of solid conductors. In this section, we validate the proposed approach by considering

a coaxial (SC) cable which has a hollow sheath. For our example, both core and sheath of the

coaxial cable are made up of copper with conductivity σ = 5.8 · 107 S/m. The radius of the

core is a = 22 mm, inner radius of the sheath is b = 39.5 mm, and outer radius of the sheath

is c = 44 mm.

The impedance of a coaxial cable can be computed analytically because of its symmetric

geometry. The analytic formula to compute the impedance of a coaxial cable can be derived

through its surface impedance and is given by [55]

Z = jωL′ + Za(ω) + Zb(ω) , (2.69)

where L′ is the external inductance for a lossless coaxial cable (σ = ∞), Za(ω) is the internal

impedance of the inner conductor, and Zb(ω) is the internal impedance of the sheath. These

quantities can be computed using the following formulas:

L′ =µ02π

log

(b

a

), (2.70)

Za(ω) =η

2πa

I0(γca)

I1(γca), (2.71)

Zb(ω) =η

2πb

[I0(γcb)K1(γcc) +K0(γcb)I1(γcc)I1(γcc)K1(γcb)− I1(γcb)K1(γcc)

], (2.72)


where

η =

√jωµ0σ

(2.73)

γc =√jωµ0σ , (2.74)

and In(.) and Kn(.) are the modified Bessel functions of first and second kind of order n.

100

102

104

106

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9x 10

−7

Frequency [Hz]

Ind

uct

ance

p.u

.l. [H

/m]

MoM−SO

Analytic

100

102

104

106

10−4

10−3

10−2

Frequency [Hz]

Res

ista

nce

p.u

.l.

[Ω/m

]

MoM−SO

Analytic

Figure 2.6: P.u.l. inductance(left panel) and resistance (right panel) of a coaxial cable consid-ered in Sec. 2.6.2. Comparison between the Analytic formulas (2.69) and the proposed approach(MoM-SO) is shown.

Figure 2.6 shows the inductance and resistance for the coaxial cable. It can be seen that

there is an excellent agreement between the proposed approach and the analytic formulas. In

other words, the surface admittance approach for hollow conductors accurately captures the

skin effect in a coaxial cable.

2.6.3 Example # 3: Three-Phase Cable

We now consider a three-phase cable made up of three SC cables in a symmetric arrangement.

Through this example we further validate the proposed approach, and also show its excellent

computational efficiency. The geometry of the three-phase cable is shown in Fig. 2.7. This is a

typical geometry of an underground power cable that was provided by our industrial partners,

and such cables are used to connect three-phase power systems. This cable consists of three core

conductors, three wire screens, and a steel armoring, for a total of 293 circular subconductors.


Figure 2.7: Three-phase cable considered in Sec. 2.6.3

Table 2.1: Characteristics of the cable in the example of Sec. 2.6.3.

Item Parameters

Core σ = 58 · 106 S/m, r = 10.0 mm

Insulation t = 4.0 mm, εr = 2.3

Wire screen 50 wires, r = 0.5 mm, σ = 58 · 106 S/m

Jacket t = 2 mm, εr = 2.3

Tables 2.1 and 2.2 list geometrical and material parameters of the cable.

Before discussing our example any further, we define the positive-, zero-, and negative-

sequence impedances. In a three-phase line, the unbalanced voltages and currents may be

decomposed into the zero-, positive-, and negative-sequence voltages and currents. The ra-

tios between sequence voltages and sequence currents are defined as the zero-, positive-, and

negative-sequence impedances. The sequence impedances are obtained from the line impedance

matrix Z as follows [10]

Z0

Z+

Z−

=1

3

1 1 1

1 h h2

1 h2 h

Z

1 1 1

1 h2 h

1 h h2

, (2.75)


Table 2.2: Armor characteristics for the structure considered in Sec. 2.6.3.

Item Parameters

Armor outer diameter 88.26 mm

Wire diameter 3 mm

Conductivity 107 S/m

µr 100

No. wires per layer 70

100

102

104

106

10−2

10−1

100

101

Frequency [Hz]

Res

ista

nce

p.u

.l.

[Oh

m/k

m]

Positive sequence

Zero sequence

FEM

MoM−SO, Np=0.

MoM−SO, Np=3.

MoM−SO, Np=7.

100

102

104

106

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Frequency [Hz]

Ind

uct

ance

p.u

.l.

[mH

/km

]

Positive sequence

Zero sequence

FEM

MoM−SO, Np=0.

MoM−SO, Np=3.

MoM−SO, Np=7.

Figure 2.8: P.u.l. resistance (left panel) and p.u.l. inductance (right panel) of the three-phasecable of Sec. 2.6.3, obtained with MoM-SO and FEM ©[2013] IEEE [3].

where h = ej2π/3, Z0 is the zero sequence impedance, Z+ is the positive sequence impedance,

and Z− is the negative sequence impedance. Throughout this thesis, we will plot the sequence

impedances to validate our technique.

Using MoM-SO, we compute the 3× 3 series impedance matrix of the three phase conduc-

tors taking the screens as reference conductor. We also assume that screens are continuously

grounded, i.e. the potential of each screen is set to zero. Table 2.3 shows the calculated positive-

and zero-sequence resistance and reactance per kilometer at 50 Hz. The computation has been

performed with three different discretizations for electric fields and current density in (2.3) and

(2.6): Np = 0, Np = 3 and Np = 7. As a validation we used a MATLAB implementation of the

FEM [1] with a very fine mesh (177,456 triangles). Results obtained with Np = 3 and Np = 7

deviate from the FEM result by less than 1%. These results further confirm the accuracy of

the proposed technique.

Figure 2.8 shows the positive- and zero-sequence resistance and inductance as a function


Table 2.3: Positive- and zero-sequence impedance of the three-phase cable of Sec. 2.6.3 at 50 Hz.MoM-SO is compared against a finite element approach [1].

MoM-SO (proposed)

Np = 0 Np = 3 Np = 7 FEM

R+[Ω/km] 0.06905 0.07259 0.07261 0.07218

X+[Ω/km] 0.08703 0.09042 0.09048 0.09041

R0[Ω/km] 0.2386 0.2437 0.2438 0.2459

X0[Ω/km] 0.08033 0.08943 0.08958 0.08975

Table 2.4: Computation time for MoM-SO and FEM for Three-phase cable example of Sec. 2.6.3

MoM-SO (proposed)

Np = 0 Np = 3 Np = 7 FEM

Number of unknowns 586 4102 8790

Green’s function discretization 11.6 s 13.7 s 16.5 s

Impedance computation (per fre-quency sample)

0.085 s 2.01 s 15.5 s 440 s*

All computations were performed on a systemwith a 2.5 GHz CPU and 16 GB of memory.

*Mesh size: 177,456 triangles.

of frequency, from 1 Hz to 1 MHz. It is observed that with Np = 0, significant errors result

as proximity effects are ignored. Indeed, by setting Np = 0 in the Fourier expansions (2.3)

and (2.6) for the fields and currents one assumes a circularly-symmetric current distribution on

the conductors. With Np = 3 and Np = 7, almost identical results are achieved which agree

very well with the FEM result [3]. The FEM results deviate from MoM-SO results at high

frequencies because the FEM mesh is not sufficiently fine to properly account for skin effect.

In the proposed technique, instead, skin effect is implicitly and fully described by the surface

admittance operator, and does not affect the discretization of the problem, which depends only

on the proximity of the conductors. As a result, the level of discretization, controlled by Np,

does not have to be increased significantly as frequency grows, making MoM-SO much more

efficient than FEM.

Timing results, reported in Table 2.4, demonstrate the excellent performance of the devel-

oped algorithm. Table 2.4 breaks down the computational time of MoM-SO into time required

to generate the Green’s matrix G, and time to assemble and solve the system (2.61). Since


-60 -40 -20 0 20 40 60

-30

0

30

60

0

0

1

16

5

4

x[mm]

y[m

m]

Insulation

Jacket

Screen

Core

Figure 2.9: Three single-core cables inside a conducting pipe (seven conductors).

Np = 3 is sufficient for obtaining accurate results, the total computational cost for computing

the 31 impedance samples in this example is T = 11.6 + 31× 2.01 = 73.9 s. The computation

time in FEM is 440 s per frequency sample, which is 220 times slower than the per-sample

computation time of 2.01 s in MoM-SO with Np = 3. We can therefore safely state that MoM-

SO is at least 100 times faster than the FEM approach when several frequency samples are

needed. MoM-SO requires less unknowns to solve the problem thanks to the surface-approach.

Less unknowns lead to less time to assemble and solve the linear system (2.61), as a result

MoM-SO is much faster than FEM. For all validation scenarios, we use FEM tools [1, 4] that

are used in power industry. There may exist other volumetric techniques such as [56]4 which

may be more optimized than [1,4]. However, these tools are not commonly used. Furthermore,

the fundamental reason for large speed up in our technique is due to surface approach. Hence,

MoM-SO should be faster than the accelerated volumetric techniques as well.

2.6.4 Example # 4: Pipe-Type Cable

Finally, we consider a pipe-type cable containing three single-core cables that are placed asym-

metrically inside a conducting pipe. Each single-core cable features a metallic screen inside an

insulating jacket, giving a system of seven insulated conductors as shown in Fig 2.9. Table 2.5

shows material and geometrical parameters of this cable.

4This particular technique is for 3D geometries.


Table 2.5: Characteristics of the cable system shown in Fig. 2.9.

Item Parameters

Core σ = 58 · 106 S/m, radius = 10.0 mm

Insulation Thickness = 4.0 mm, εr = 2.3

Screen Thickness = 0.2 mm, σ = 58 · 106 S/m

Jacket Thickness = 2 mm, εr = 2.3

Steel pipe Outer diameter = 100 mm, thickness = 5 mm, σ = 107 S/m , µr = 100

Series Impedance Computation

Using MoM-SO, we computed the 6 × 6 series impedance matrix of the six conductors (three

phase conductors plus three screens), between 1 Hz and 1 MHz using 120 logarithmically spaced

samples. Next, the screens were eliminated by assuming them to be continuously grounded,

giving a 3× 3 impedance matrix. Figure 2.10 shows the calculated positive-sequence resistance

and inductance per km, for different number of basis functions Np. It is observed once again that

orders Np = 3 and Np = 7 practically give the same result, and so Np = 3 is deemed sufficient.

For validation, we compared MoM-SO against a FEM simulation [1] performed with a fine

mesh (108,418 triangles). The MoM-SO result agrees very closely with the FEM result, thereby

validating the proposed algorithm. Figure 2.10 also shows the cable parameters obtained with

Np = 0. With this setting, we assume a circularly symmetric current distribution in the

conductors, neglecting proximity effects. This assumption is made by the analytic formulas

commonly employed by most electromagnetic transient programs used by power engineers.

From Fig. 2.10, we can observe that this approximation is adequate for the resistive part of the

impedance. However, it leads to an underestimation of the p.u.l. inductance at low frequency.

Analytic formulas can deviate significantly from the actual impedance whenever there are

significant proximity effects. By continuously grounding the screens of the cable, we can mini-

mize the proximity effect. However, in practise screens are not always continuously grounded,

but are often left open. Figure 2.11 shows the p.u.l. resistance and inductance obtained when

the screens are left open at both ends, i.e. with zero net current flowing in each screen. It

is observed that we now get large errors also at high frequencies when ignoring the proximity

effect (Np = 0), while the result with MoM-SO (Np = 3, Np = 7) agrees closely with the FEM

solution. As before, Np = 3 is a sufficient order for MoM-SO since increasing the order to


100

102

104

106

10−2

10−1

100

101

102

Frequency [Hz]

Res

ista

nce

p.u

.l.

[ Ω

/km

]

FEM

MoM−SO, Np=0.

MoM−SO, Np=3.

MoM−SO, Np=7.

100

102

104

106

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Frequency [Hz]

Ind

uct

ance

p.u

.l.

[mH

/km

]

FEM

MoM−SO, Np=0.

MoM−SO, Np=3.

MoM−SO, Np=7.

Figure 2.10: P.u.l. positive sequence resistance and inductance of the cable system, obtainedwith MoM-SO and FEM. Screens are continuously grounded [5].

100

102

104

106

10−2

10−1

100

101

102

Frequency [Hz]

Res

ista

nce

p.u

.l. [

Ω/k

m]

FEM

MoM−SO, Np=0.

MoM−SO, Np=3.

MoM−SO, Np=7.

100

102

104

106

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Frequency [Hz]

Induct

ance

p.u

.l. [m

H/k

m]

FEM

MoM−SO, Np=0.

MoM−SO, Np=3.

MoM−SO, Np=7.

Figure 2.11: P.u.l. positive sequence resistance and inductance of the cable system, obtainedwith MoM-SO and FEM. Screens are open [5].

Np = 7 does not change the result. Note, the assumption of open or grounded screens are

only made to reduce our 6 × 6 matrices to the 3 × 3 matrices for the purpose of comparison.

Practically, the cable models are derived from the 6× 6, without any reduction.

Timing Results

Table 2.6 reports the computational cost for the alternative approaches. FEM takes 3.5 hours

to compute the cable impedance, while MoM-SO with Np = 3 takes only 5.57 s. The proposed

method is therefore faster by a factor of about 2200. These results confirm the outstanding

efficiency of the proposed method.


Table 2.6: Computation time for the proposed method (MoM-SO) and finite elements (FEM)applied to the cable system of Sec. 2.6.4.

Method Computation time Speed up

FEM 12,600 s = 3.5 hours -

MoM-SO, Np = 0 3.57 s 3529 X

MoM-SO, Np = 3 5.57 s 2262 X

MoM-SO, Np = 7 7.93 s 1589 X

All computations were performed on a workstationwith a 2.5 GHz CPU and 16 GB of memory.

V1 V7

V8

1 km

Figure 2.12: First configuration considered in Sec. 2.6.4. A unit step voltage is applied to thetop phase conductor of the cable of Fig. 2.9. All other phases and sheaths are grounded at oneend [5].

Transient Analysis

We now present two transient simulation scenarios to show the effects of proximity on the

transient waveform. In the first scenario, we applied a unit step voltage to the phase conductor

of the top cable in Fig. 2.9, with all other phases and sheaths grounded at one end. Such a

scenario represents a switching condition. All conductors are open at the far end, as shown in

Fig. 2.12. In this configuration, we measured the core-sheath voltage (V7 − V8), and sheath-

ground voltage (V8). In the second scenario shown in Fig. 2.15, we applied a unit step voltage

between the screens of the two SC cables in the bottom in Fig. 2.9, with all other cables

grounded. This scenario may arise, for example, when there is an insulation breakdown and

the core and the screens are shorted, which will drive current inside the sheaths of the cables

exciting an intersheath mode. In this case, we measured the differential intersheath voltage

(V10 − V12).

In order to perform the transient simulation, we obtained the capacitance matrix using

analytic formulas [26]. Using the time domain implementation of the Universal Line Model [47]

available in PSCAD v4.2 [21], we simulated the transient scenario in Fig. 2.12. Simulations were


0 20 40 60 80 100−0.5

0

0.5

1

1.5

2

Time [µs]

Vo

ltag

e [V

]

FEM

MoM−SO, Np=0.

MoM−SO, Np=3.

Figure 2.13: Phase-screen voltage at the receiving end (V7− V8) for the configuration shown inFig. 2.12 [5].

0 20 40 60 80 100−0.06

−0.04

−0.02

0

0.02

0.04

Time [µs]

Volt

age

[V]

FEM

MoM−SO, Np=0.

MoM−SO, Np=3.

Figure 2.14: Sheath voltage at the receiving end (V8) for the configuration shown in Fig. 2.12 [5].

performed three times using the series impedance obtained with different methods: FEM, MoM-

SO with Np = 0 (only skin effect taken into account), and MoM-SO with Np = 3 (both skin and

proximity effect taken into account). By setting Np = 0, we obtain the same results as those

obtained using the analytic formulas commonly-used in electromagnetic transient programs.

Figures 2.13 and 2.14 show the resulting voltage waveform between the phase and screen, and

between the phase and ground, respectively. The voltage waveform in Fig. 2.13 shows excellent

agreement between all three curves. From this waveform, we notice that the voltage wave

bounces off back-and-forth from voltage source to the open-end, until it gradually reaches the

steady state of 1V. In this scenario, there is little proximity effect. However, the influence

of proximity effect is visible in Fig. 2.14 because there is a significant difference between the


Np = 0 curve, and the Np = 3 and FEM curves. This proximity effect is due to the presence

of proximity currents and asymetrical geometry of the cable. It is observed that the result

by MoM-SO (Np = 3) and FEM agree closely. The deviation from the FEM result is further

reduced when increasing the MoM-SO order.

The influence of proximity effect is even more pronounced for waves that travel between the

sheaths, i.e. the intersheath modes. To see this, we applied a step voltage between the sheaths

of the two lower cables in Fig. 2.9, as shown in Fig. 2.15. The simulated results in Fig. 2.16

show that ignoring proximity effect leads to very large errors in the transient response.

V12

V10

1 km

Figure 2.15: Second configuration considered in Sec. 2.6.4. A differential voltage excitation isapplied between two screens [5].

0 20 40 60 80 100−0.5

0

0.5

1

1.5

2

Time [µs]

Vo

ltag

e [V

]

FEM

MoM−SO, Np=0.

MoM−SO, Np=3.

Figure 2.16: Differential voltage at the receiving end (V10 − V12) for the configuration ofFig. 2.15 [5].

The two scenarios that we considered are coaxial and intersheath modes. Any transient

voltage excitation may be decomposed into coaxial, intersheath, and ground modes.5 Although

in our setup we normalized the excitation voltages to one, in reality the voltages may be

hundreds of kV. In such situations, the difference in magnitude of curves obtained with and

5We only considered two of the six total modes.


without proximity will be magnified, giving results that are incorrect.

This study highlights why we need to calculate the accurate impedance parameters in order

to predict transients accurately.

2.7 Conclusions

In this chapter, we developed MoM-SO, an elegant technique to compute the p.u.l. impedance

of a cable system made up of solid and hollow round conductors. Our approach relied on two

concepts: the surface admittance operator, and the electric field integral equation. Through

the surface admittance operator, we homogenized the medium by replacing all conductors with

the surrounding medium and equivalent currents on conductors’ boundaries. Thereafter, by

applying the method of moments to the electric field integral equation, we related the equivalent

currents with fields and the p.u.l. impedance. Examples discussed in Sec. 2.6 validated that

skin and proximity effects, two phenomena which significantly alter transient waveform inside

the cable, were successfully captured by MoM-SO. Hence, MoM-SO is as accurate as other

computational tools based on FEM and conductor partitioning which are used in the industry.

The main advantage of the proposed technique when compared to FEM is that it is very

fast. Speedups of upto 2000 times were observed in the examples presented. These speedups

in computation were due to two reasons: employment of a surface approach and analytic

evaluation of all the matrices. Due to surface approach, we had to discretize fields and currents

only on the boundaries of the conductors, as opposed to the whole volume. Fewer unknowns

led to significantly less time to assemble and solve the linear system. Furthermore, analytic

evaluations of the method of moments integrals reduced the time required to assemble Green’s

matrix G. Finally, unlike FEM, MoM-SO requires no volumetric meshing that makes the

tool simple to use. Instead, discretization in MoM-SO is controlled by a single parameter Np,

which can be fully automated as we will see in Sec.4.2. Furthermore, the absence of volumetric

meshing in MoM-SO also leads to another advantage of not having remesh the geometry at high

frequencies to account for skin effect. Instead, skin effect in our technique is easily accounted

through the surface admittance operator. This advantage also greatly simplifies MoM-SO so

it can be used by everyone, even those without any working knowledge of electromagnetic


simulation methods. In the next chapter, we will further develop the framework of MoM-SO to

include the effect of multilayer ground on the p.u.l. impedance parameters.

Chapter 3

Cables Surrounded by a Lossy and

Multilayered Medium

The MoM-SO technique that we derived in Chapter 2 allows us to compute the p.u.l. impedance

of a cable surrounded by a lossless homogeneous medium, such as a two-wire system shown in

Fig 3.1a. The focus of this chapter is on the inclusion of the ground return effect, which is

important whenever current flows through the earth, which is common during fault scenarios.

The technique derived in this chapter will allow us to compute the p.u.l. impedance of system

where earth is the return path for current, as in the example of Fig. 3.1b.

A straightforward approach to extend the theory presented in the previous chapter to include

the ground return effect involves replacing the quasi-static Green’s function (2.44) by a full-

wave Green’s function. Hence, by simply changing the entries of Green’s matrix G (2.52) we

can accurately include the ground return effect. However, this approach has two disadvantages:

• It does not scale well with the number of conductors. The Green’s function of a lossy

multilayered medium requires numerical evaluation of an infinite integral, and since there

are a lot of entries in G (2.52) that depend on the Green’s function, this approach ends

up being computationally expensive.

• The approach of Chapter 2 cannot be readily extended to include the effects of a tunnel or

a hole. However, we want to include these effects because there does not exist any accurate

technique that can account for tunnel effects, despite of their practical relevance.

55

Chapter 3. Cables Surrounded by a Lossy and Multilayered Medium 56

(a) Two-wire system (b) Wire-ground system

Figure 3.1: Sample circuit configurations

In order to circumvent these two problems, we will introduce the concept of surface admittance

operator of a cable-hole system. This surface admittance operator will allow us to efficiently

model the tunnel or hole, and all the cables inside it, by a single equivalent current on the

boundary of the hole. The effects of holes will be included via the surface admittance operator

of the cable-hole system. Moreover, since we will simplify the whole system by a single equiv-

alent current source, our new Green’s matrix will be very small. Hence, we will require fewer

numerical integrations, which will make our technique faster.

In Sec. 3.1, we formulate our new problem by setting the notations for conductors and holes.

In the following section, we will derive surface admittance operator of the cable-hole system.

In Sec. 3.3, we will couple the equivalent current source with the Green’s function of a layered

medium to include the ground return effect. Next, we will calculate the p.u.l. impedance.

Numerical results are presented in Sec. 3.8 to demonstrate the accuracy and computational

speed of the proposed technique.

3.1 Problem Formulation

Our goal is to compute the p.u.l. impedance of cables made up of round solid and hollow

conductors buried in one or more holes dug inside conductive soil. For the sake of clarity, we

present the theory considering only solid conductors and a single hole. We will also assume

that the background medium has two layers. Later, in Secs. 3.5 and 3.6, we will discuss how

the proposed method can be easily generalized to include multilayered ground medium, hollow

conductors, and multiple holes.

A sample configuration that we consider is shown in Fig. 3.2. Let the cable contain P


y

x

(x, y)a

c

ρθ

apcp

θp(xp, yp)

air (ε0, µ0)

ground (ε0, µ0, σg)

conductors (ε, µ, σ)

hole (ε, µ)

Figure 3.2: Cross-section of a simple cable-hole system with two conductors used to illustratethe proposed method. The coordinate system used in this chapter is also presented [6].

round solid conductors which are buried inside a hole. We retain the notation established for

geometrical and material parameters of the conductors in Chapter 2. Therefore, as shown in

Fig. 3.2, the p-th conductor is centered at (xp, yp) and has radius ap. Each conductor has

conductivity σ, magnetic permeability µ, and relative permittivity ε. Conductors are placed

inside a hole which is centered at (x, y) and has a radius a. The material inside the hole has

permeability µ, and permittivity ε. The background medium is air for y > 0 and is lossy soil

with conductivity σg for y < 0, as shown in Fig. 3.2. Both air and ground have permittivity ε0

and permeability µ0.

3.2 Surface Admittance Representation for the Cable-Hole Sys-

tem

In this section, we extend the concept of surface admittance operator to the cable-hole system.

The surface admittance representation of the cable-hole system simplifies the original problem

shown in Fig. 3.2 to that shown in the right panel of Fig. 3.3. In other words, we will be able

replace the hole (and conductors inside it) with the surrounding ground and a single equivalent

current source on the boundary of the hole.

3.2.1 Surface Admittance Operator for the Conductors

The first step of this formulation is to replace all the conductors inside the cable by the sur-

rounding hole medium as in Chapter 2. The fields outside the conductors are restored by

introducing equivalent current J(p)s (θp), which is expanded using the truncated Fourier series


µ0, ε0

σg, µ0, ε0

µ, ε µ, ε

J(p)s (θp)

y

x

µ0, ε0

σg, µ0, ε0

σg, µ0, ε0

Js(θ)

y

x

c

Figure 3.3: Left panel: cross-section of the cable in Fig. 3.2 after all conductors have been

replaced by the surrounding hole medium. Equivalent currents J(p)s (θp) are introduced on their

contours. Right panel: cross-section of the cable after application of the equivalence theoremto the boundary of the hole. An equivalent current Js(θ) is introduced on the hole boundary c.

in (2.6), on the boundary of each conductor, as shown in the left panel of Fig. 3.3. Equivalent

currents are related to the electric fields, which are expanded using the truncated Fourier series

in (2.3), through the surface admittance operator

J = YsE , (3.1)

where vectors J and E, defined by (2.37) and (2.35) respectively, and contain the equivalent

current and the electric field coefficients on the surface of each conductor. The surface admit-

tance operator Ys was derived in Chapter 2, and its diagonal entries are given by (2.20) with

k0 replaced by k = ω√µε, which is the wavenumber inside the hole.

3.2.2 Surface Admittance Operator for the Cable-Hole System

Next, we simplify the problem even further by representing the entire cable-hole system with

a unique equivalent current density Js(θ) placed on the boundary of the hole, as shown in the

right panel of Fig. 3.3. The boundary of the hole is denoted by c and can be traced by the

position vector r(a, θ) where

r(ρ, θ) =(x+ ρ cos θ

)x +

(y + ρ sin θ

)y , (3.2)

for ρ ∈ [0, a], and θ ∈ [0, 2π). In (3.2), (ρ, θ) form a cylindrical coordinate system centered at

(x, y), as shown in Fig. 3.2.


Similarly to the our approach for conductors, we represent the magnetic vector potential on

the boundary of the hole through the truncated Fourier series expansion

Az(θ) =

N∑n=−N

An ejnθ , (3.3)

where N is analogous to N , and represents how accurately we want to represent the vector

potential on the boundary of the hole. The value of N depends on the distribution of the

scattered field inside and outside the hole. Results in Sec. 3.8 will show that N = 4 is sufficient

in all applicative scenarios. The Fourier coefficients of expansion in (3.3) are cast into a column

vector A =

[A−N . . . A

N

]T. Next, we replace the hole and all the equivalent currents inside

it with the material of the surrounding (ground) medium, and introduce an equivalent current

source on the boundary of the hole in order to maintain the fields outside the hole unchanged.

This equivalent current density is also expanded through the truncated Fourier series

Js(θ) =1

2πa

N∑n=−N

Jn ejnθ . (3.4)

Fourier coefficients in (3.4) are cast into vector J =

[J−N . . . J

N

]T. The value of equivalent

current Js(θ) follows from the equivalence theorem [50]

Js(θ) =

[1

µ0

∂Az(ρ, θ)∂ρ

− 1

µ

∂Az(ρ, θ)∂ρ

]ρ=a−

. (3.5)

In (3.5), Az is the magnetic vector potential inside the hole in the configuration shown in the

left panel of Fig. 3.3, i.e. before the equivalence theorem has been applied. Instead, Az is

the magnetic vector potential inside the hole for the configuration shown in the right panel of

Fig. 3.3, i.e. after the equivalence theorem has been applied. To evaluate the value of equivalent

hole current we must first compute the two magnetic vector potentials Az and Az.


Vector Potential Az

By definition, the magnetic vector potential satisfies the inhomogeneous Helmholtz equation

inside the hole [50]

∇2Az + k2Az = −µP∑q=1

J (p)s (θp) (3.6)

subject to the Dirichlet boundary condition (3.3) on c. The forcing term on the right-hand side

of (3.6) is the sum of conductor equivalent currents inside the hole. The solution of (3.6) may

be decomposed into the sum of particular and general solution

Az(ρ, θ) = A′z(ρ, θ) + A′′z(ρ, θ) , (3.7)

where particular solution A′′z(ρ, θ) carries the effect of excitation terms on the right-hand side

of (3.6), and general solution A′z(ρ, θ) satisfies the homogeneous Helmholtz equation with right-

hand side of (3.6) set to zero.

Particular Solution A′′z

The particular solution of (3.6) at an arbitrary point inside the hole is given by [50]

A′′z(ρ, θ) = −µP∑q=1

ˆ 2π

0J (q)s (θ′q)G

(r(ρ, θ), rq(θ

′q))aq dθ

′q , (3.8)

where integral kernel

G(r, r′

)=j

4K0

(k∣∣r− r′

∣∣) (3.9)

is the Green’s function of a homogeneous medium with K0(.) being the Hankel function of

second kind of order 0 [51]. Equation (3.8) is the superposition of fields due to current source

on each conductor’s boundary. In order to evaluate (3.5), we only require vector potential inside

the hole. In this region the medium is homogeneous, therefore we use the homogeneous medium

Green’s function to compute A′′z , regardless of inhomogeneity in the surrounding medium.


General Solution A′z

The general solution of (3.6) is given by [50]

A′z(ρ, θ) =

N∑n=−N

CnJ|n|(kρ)

ejnθ , (3.10)

where coefficients Cn are determined by enforcing the boundary condition (3.3). These co-

efficients are cast into the column vector C =

[C−N . . . C

N

]T. To enforce the boundary

condition, we substitute (3.8) and (3.10) evaluated on c into (3.7) and equate it to the vector

potential on c as given in (3.3)

Az(θ) = A′z(a, θ) + A′′z(a, θ)

N∑n=−N

An ejnθ =N∑

n=−N

CnJ|n|(ka) ejnθ −µP∑q=1

ˆ 2π

0J (q)s (θ′q)G

(r(a, θ), rq(θ

′q))aq dθ

′q . (3.11)

This equation is similar to the EFIE (2.48) given in Chapter 2 and can be solved with a similar

method of moments procedure. In particular, we apply operator

ˆ 2π

0[.] e−jn

′θ dθ (3.12)

on both sides of (3.11) for n′ = −N , . . . , N . After integrating over θ, this step results in

An′ = Cn′J|n′|(ka)− µNq∑

n=−Nq

J (q)n

1

2π2

ˆ 2π

0

ˆ 2π

0G(r(a, θ), rq(θ

′q))

ej(nθ′q−n′θ) dθ′qdθ , (3.13)

which in vector notation is given by

C = D1

(A + µG0J

), (3.14)

where D1 is a diagonal matrix with the (n, n) entry given by

[D1][n,n] =(J|n|

(ka))−1

, (3.15)


and G0 is a Green’s matrix which relates the magnetic vector potential on the boundary of

the hole to the conductor equivalent currents inside the hole. Matrix G0 is of dimension

(2N + 1)×N , and may be expressed as

G0 =

[G

(1)0 . . . G

(p)0 . . . G

(P )0

](3.16)

where block G(p)0 is of dimension (2N + 1)× (2Np + 1), and it describes the contribution of the

equivalent current on the surface of p-th conductor to the particular solution of vector potential

on the boundary c. The (n′, n) entry of this matrix is found by evaluating

[G

(p)0

](n′,n)

=1

(2π)2j

4

ˆ 2π

0

ˆ 2π

0K0

(k∣∣∣rp(ap, θ′p)− r(a, θ)

∣∣∣) ej(nθ′p−n′θ) dθ′pdθ , (3.17)

which may be solved analytically as shown in Appendix B.3.

We now have both the particular solution (3.8) and the general solution (3.10) of vector

potential inside the hole Az.

Vector potential Az

Next, we findAz which is the vector potential inside the hole for the configuration shown in the

right panel of Fig. 3.3, i.e. after we have replaced the hole and equivalent conductor currents

inside it by the ground medium. In this configuration, there exists no source term because there

are no currents inside the hole. Hence, the solution of Az satisfies the homogeneous Helmholtz

equation

∇2Az + k2gAz = 0 (3.18)

subject to the Dirichlet boundary condition (3.3). The solution of (3.18) is given by

Az(ρ, θ) =

N∑n=−N

AnJ|n|(kgρ)

J|n|(kga)ejnθ , (3.19)

where An are the Fourier coefficients in (3.3).


Equivalent Hole Current

We now find the value of equivalent current Js(θ) on the boundary of the hole. We calculate

the vector potential

Az(ρ, θ) =N∑

n=−N

CnJ|n|(kρ)

ejnθ −µP∑q=1

ˆ 2π

0J (q)s (θ′q)G

(r(ρ, θ), rq(θ

′q))aqdθ

′q , (3.20)

which is found by substituting the general and particular solutions in (3.10) and (3.8) into (3.7).

Next we substitute Az andAz in (3.20) and (3.19) into the expression of equivalent hole current

in (3.5) to obtain

1

2πa

N∑n=−N

Jn ejnθ =

1

µ0

N∑n=−N

AnkgJ ′|n|(kga)

J|n|(kga)ejnθ − 1

µ

N∑n=−N

kCnJ ′|n|(ka)

ejnθ (3.21)

+

P∑q=1

ˆ 2π

0J (q)s (θ′q)

∂G

∂ρaqdθ

′q

,where we also substituted the Fourier expansion of equivalent hole current in (3.4). We compute

coefficients Jn in the above equation with the method of moments by applying the Galerkin

operator (3.12) to both sides of (3.21). The result of this step, expressed through vectors and

matrices, is given by

J = YsA + TJ . (3.22)

The above equation shows that the equivalent current Js(θ) has two contributions. First com-

ponent is analogous to the surface admittance matrix of solid conductors, and involves diagonal

matrix Ys whose entries

[Ys

][n,n]

= 2πa

[kgµ0

J ′|n|(kga)

J|n|(kga)− k

µ

J ′|n|(ka)

J|n|(ka)

], (3.23)

for n = −N , . . . , N . The first term in (3.22) exists whenever the surrounding ground and the

hole have different material parameters. Physically, it accounts for reflections and scattering

of fields that occur due to the differences in material properties of the hole and the ground.

Second term in (3.22) contains a transformation matrix T which maps the equivalent currents


J(p)s (θp) on conductor boundaries to the equivalent current Js(θ) on the hole boundary. This

matrix is given by

T = 2πa[G0 −D2G0

], (3.24)

where D2 is a diagonal matrix with entries

[D2][n,n] = kJ ′|n|(ka)

J|n|(ka), (3.25)

and G0 is a Green’s matrix of size (2N + 1)×N , which is generated similarly to the matrix G0

which appears in (3.16), this is shown in Appendix B.4. The second term in (3.22) is present

whenever there are equivalent currents inside the hole. In other words, the second term exists

whenever the original problem has one or more conductors inside the hole.

Expression (3.22) is one of the main contribution of this thesis, since it allows us to replace

one or more complicated cable (with hundreds of conductors) and its surrounding hole by a

single equivalent current. In Sec. 2.2.1, we discussed how the surface admittance relation of a

solid conductor is analogous to the Norton’s theorem of circuit analysis. In the same way, we

can draw an analogy between (3.22) and the Norton equivalence theorem. In this analogy, the

cable-hole system is analogous to a large circuit, and the conductors are analogous to smaller

circuit blocks that make the large circuit. In this section, we first took the smaller circuits

blocks (conductors) and applied Norton’s theorem (equivalence theorem) on them. Through

Norton’s theorem (equivalence theorem), we replaced each of the smaller circuit blocks (conduc-

tors) by their Norton equivalent admittances (surface admittance operator of the conductor)

and Norton currents (equivalent conductor current). In the second stage, we applied Norton’s

theorem (equivalence theorem) again. This time, we replaced all the previously calculated Nor-

ton admittances and currents (equivalent conductor current) by a new set of Norton admittance

(surface admittance operator of the hole) and Norton current (equivalent hole currents). Notice

that by two successive application of Norton’s theorem (equivalence theorem) we were able to

simplify a large circuit (cable-hole system) to a single admittance (surface admittance operator

of the cable-hole system) and Norton source (equivalent hole current).

By simplifying our problem, we can now include the ground return effects very efficiently


because we will only have to couple the effects of layered ground medium with a single equivalent

current source, as opposed to many conductors in our original problem. This simplification leads

to an enormous speedup because we will require fewer numerical integrations to include ground

return effect.

3.3 Inclusion of Ground Return Effect

At this point, we have simplified our problem considerably, and now have a single equivalent

current source inside the ground as shown in the right panel of Fig. 3.3. Next, we include

the ground return effect by invoking an integral equation which couples our cable-hole system

with the surrounding ground-air medium. This integral equation will allow us to compute the

magnetic vector potential Az on the boundary of the hole, which in turn will help us evaluate

fields inside the hole and thereby the p.u.l. transmission line parameters.

By definition of the magnetic vector potential, we can relate currents and vector potential

on the boundary of the hole through the integral equation [50]

Az(a, θ) = −µ0ˆ 2π

0Js(θ

′)Gg

(r(a, θ), r(a, θ′)

)adθ′ , (3.26)

where Gg is the Green’s function of the background medium. In our case, shown in Fig. 3.3, the

medium consists of one layer of air and one of conductive material. For this case, the Green’s

function reads [57]

Gg(x, y, x′, y′) =

1

4π

ˆ ∞−∞

exp (−jβx (x− x′))√β2x − k2g

[exp

(−∣∣y − y′∣∣√β2x − k2g) (3.27)

+RTM exp(

(y + y′)√β2x − k2g

)]dβx ,

where

RTM =

√β2x − k2g −

√β2x − k20√

β2x − k2g +√β2x − k20

(3.28)

and k0 = ω√µ0ε0 is the wavenumber of air. In (3.27), we used x, y, x′, and y′ to express the

x-component and y-component of the position vectors r(a, θ) and r(a, θ′).


Next, we substitute the Fourier expansion of vector potential Az(θ) and current Js(θ) as

given in (3.3) and (3.4) into the integral equation (3.26), following which we solve the integral

equation through the method of moments. The method of moments procedure is the same as

that used for solving integral equation (2.48), and will not be repeated here. The solution of

this equation is

A = −µ0GgJ , (3.29)

where Gg is another Green’s matrix whose entry (n′, n) can be computed by solving the following

triple integral 1

[Gg](n′,n) =1

(2π)2

ˆ 2π

0

ˆ 2π

0Gg

(x+ a cos θ, y + a sin θ, x+ a cos θ′, y + a sin θ′

)ej(nθ

′−n′θ) dθdθ′ , (3.30)

for n, n′ = −N , . . . , N . Two of the three integrals in the above expression can be evaluated

analytically, and the last one must be solved numerically. Discussion on the evaluation of (3.30)

is postponed to Sec. 3.5.2. By substituting the relationship (3.22) into (3.29) we obtain the

expression for magnetic vector potential

A = −µ0(1 + µ0GgYs

)−1GgTJ , (3.31)

where 1 is the identity matrix.

In order to compute the p.u.l. parameters we need to solve for five unknowns: J, E,

A, J, and Z(ω). By now we have discussed three relationships: i) two surface admittance

relationships – one is (3.1), which relates conductor equivalent currents J and fields on the

conductors’ boundaries E, and another is (3.22), which relates the hole equivalent current J

and vector potential on the hole’s boundary A; and ii) radiation relationship in (3.29), which

relates the vector potential on the boundary of the hole to the hole equivalent current. In the

next section, we invoke two additional relations and solve for the p.u.l. parameters.

1This is a triple integral because there is one integral inside Gg.


3.4 Computation of Per-Unit Length Parameters

In Sec. 3.3, we computed the vector potential Az(a, θ) on the boundary of the hole. We now

relate this vector potential with the electric field on the boundaries of the conductors by applying

the EFIE. Thereafter, we will be able to compute the p.u.l. impedance of the cable.

In a manner similar to Chapter 2, we evaluate the EFIE on the boundary of each conductor

inside the hole. On the boundary cp of p-th conductor, the EFIE reads

E(p)z (θp) = −jωAz(rp)−

∂Vp∂z

. (3.32)

The scalar potential on cp is related to the p.u.l. impedance parameters by the Telegrapher’s

equation (2.1). Furthermore, the magnetic vector potential on contour cp is given by the sum

of general (3.10) and particular solution (3.8)

A′′z(rp(ap, θp) =N∑

n=−N

CnJ|n|(kρ′)

ejnθ′ −µ

P∑q=1

ˆ 2π

0J (q)s (θ′q)G(rp(ap, θp), rq(aq, θ

′q))aqdθ

′q ,

(3.33)

where the auxiliary variables ρ′, and θ′ are defined through the following equality

ρ′(cos θ′x + sin θ′y

)= rp(ap, θp)− (xx + yy) . (3.34)

Next, we substitute the Telegrapher’s equation (2.1), vector potential (3.33), and the Fourier

expansion of the electric field into (3.32) to obtain

E(p)z (θp) = jωµ

P∑q=1

ˆ 2π

0J (q)s (θ′q)G(rp(ap, θp), rq(ap, θ

′q))aqdθ

′q (3.35)

− jωN∑

n=−N

CnJ|n|(kρ′)

ejnθ′+

P∑q=1

[Rpq(ω) + jωLpq(ω] Iq .

Finally, we substitute the Fourier expansions of the electric field and the equivalent current

density in (2.3) and (2.6) into the equation above. Following the rearranging of the summation


and integration we obtain

Np∑n=−Np

E(p)n ejnθp =

jωµ

2π

P∑q=1

Nq∑n=−Nq

J (q)n

ˆ 2π

0G(rp(ap, θp), rq(aq, θ

′q)) ejnθq dθ′q

− jωN∑

n=−N

CnJ|n|(kρ′)

ejnθ′+

P∑q=1

[Rpq(ω) + jωLpq(ω] Iq . (3.36)

We evaluate (3.36) on the boundary of each conductor (p = 1, . . . , P ). We solve the integral

equation (3.36) using the method of moments procedure, and obtain

E = −jωHC + jωµGcJ + U [R(ω) + jωL(ω)] UTJ . (3.37)

In (3.37), Gc is a Green’s matrix of size N × N which computes the vector potential on each

conductor’s boundary due to the equivalent currents on the surface of all the other conductors.

Appendix B.2 shows how to generate matrix Gc. Matrix H in (3.37) is of size N × (2N + 1)

and is given by

H =

[H(1) . . . H(p) . . . H(P )

]T, (3.38)

where matrix H(p) is of dimension (2N +1)× (2Np+1), and is used to find the general solution

on the boundary of p-th conductor. The (n, n′) entry of H(p) is given by

[H(p)

](n,n′)

=1

2π

ˆ 2π

0J|n|

(kρ′)

ej(nθ′−n′θp) dθpdθ

′ , (3.39)

which can be evaluated analytically as shown in Appendix B.5.

By substituting (3.14) and (3.31) into (3.37) we get

E = jωµ0ΨJ + U [R(ω) + jωL(ω)] UTJ , (3.40)

where

Ψ =1

µ0

[HD1

[µ0

(1 + µ0GgYs

)−1GgT− µG0

]+ µGc

]. (3.41)

Equation (3.40) is the discretized form of the EFIE. Matrix Ψ has a very meaningful interpre-

tation which we can see by comparing (3.40) with the EFIE (2.58) in Chapter 2. We notice that


both (3.40) and (2.58) are exactly the same, except G in (2.58) is replaced by Ψ. Therefore,

Ψ is simply the Green’s matrix of a lossy layered medium with holes inside it.

We now manipulate (3.40) in order to calculate the p.u.l. impedance matrix. We left-

multiply (3.40) by Ys and use (3.1) to simplify (3.40) to

J = jωµ0YsΨJ + YsU [R(ω) + jωL(ω)] I . (3.42)

The equation above is of the same form as (2.58), and the final resistance and inductance

matrices are give by

R(ω) = Re

(UT (1− jωµ0YsΨ)−1 YsU

)−1, (3.43)

L(ω) = ω−1Im(

UT (1− jωµ0YsΨ)−1 YsU)−1

. (3.44)

With these formulas, one can readily compute the pul impedance of the buried cable from the

surface admittance matrix of the conductors, the matrix Ψ from (3.41) and U from (2.39).

3.5 Extension to a Multilayered Surrounding Medium

We now extend the theory discussed so far in this chapter to include multilayer ground medium.

As mentioned earlier, the ground return effect is fully captured by Gg, hence by simply modi-

fying this matrix we can account for a variety of background media.

Here we consider the most general case of a background medium made up of L layers, with

layer 1 being the top-most layer and layer L being the bottom-most layer. Each layer is assumed

to be invariant along the x-direction. Upper and lower bounds of the l-th layer are denoted by

yl−1 and yl, respectively, as shown in Fig. 3.4. We assume that the cable and hole are inside

layer s. Layer l has electrical conductivity σl, permittivity εl, and permeability µl.

In order to compute entries of matrix Gg, we require the Green’s function of a multilayer

medium. In the next section we calculate the Green’s function of a multilayer medium. Then,

in Sec. 3.5.2, we use this Green’s function to compute entries of matrix Gg.


Layer 1

Layer s

Layer L

(x′, y′)

y = y1

y = y0 =∞

y = ys−1

y = ys

y = yL−1

y = yL = −∞

(x, y)

Figure 3.4: Geometry of a multilayer background medium

3.5.1 Green’s Function of a Multilayer Media

By definition, the Green’s function Gg(x, y, x′, y′) relates the vector potential at point (x, y)

due to a point source at (x′, y′). The point source is oriented along the z-axis because all the

equivalent currents are in this direction. Both (x, y) and (x′, y′) are assumed to be in layer s,

as shown in Fig. 3.4.

To find the vector potential at (x, y), we solve the Helmholtz equation subject to an excita-

tion at (x′, y′). This means that in layer l 6= s, where there are no sources, the vector potential

must satisfy the Helmholtz equation2

∇2Aδz(x, y) + k2lAδz(x, y) = 0 , (3.45)

where kl = ωµl√ωεl − jσl is the wavenumber inside layer l, and we use Aδz to denote the vector

potential in our current problem. Instead, in layer s the magnetic vector potential satisfies the

non-homogeneous Helmholtz equation

∇2Aδz(x, y) + k2sAδz(x, y) = −µlδ(x− x′, y − y′) , (3.46)

2In this thesis, the Laplacian operator is equal to ∂2

∂x2+ ∂2

∂y2


due to the presence of a point source δ(x− x′, y − y′) at (x′, y′). We employ a spectral domain

approach [57] to solve the Helmholtz equations (3.45) and (3.46) because of the homogeneity in

the media along the x-direction. To transform the Helmholtz equation (3.45) into the spectral

domain, we expand the vector potential Aδz(x, y) using the Fourier transform

Aδz(x, y) =1

2π

ˆ ∞−∞Aδz(βx, y) e−jβxx dβx . (3.47)

Expansion of Helmholtz equation results in

ˆ ∞−∞

(∂2

∂2x+

∂2

∂2y+ k2l

)Aδz(βx, y) e−jβxx dβx = 0

ˆ ∞−∞

(∂2

∂2y− (β2x − k2l )

)Aδz(βx, y) e−jβxx dβx = 0 , (3.48)

which implies3

∂2

∂2yAδz(βx, y)− (β2x − k2l )Aδz(βx, y) = 0 . (3.49)

The solution of (3.49) is made by travelling waves in ±y directions

Aδz(βx, y) = C+l exp

(−y√β2x − k2l

)+ C−l exp

(y√β2x − k2l

), (3.50)

where C+l and C−l are the constants that are found from the boundary conditions at the interface

of each layer. It should be noted that C+L is zero because a non-zero value implies that there

is a travelling-wave originating from y = −∞, which is physically impossible. Similarly, C−1 is

set to zero as there cannot exist a wave originating at y =∞ and travelling in −y direction.

In layer s, which contains the point source, the magnetic vector potential may be decom-

posed into general and particular solution. The general solution in spectral domain is given by

(3.50). The particular solution [50]

Aδz ′′(βx, y) =µs2

1√β2x − k2s

exp(jβxx

′) exp(−∣∣y − y′∣∣√β2x − k2s) , (3.51)

3the left-hand side of (3.48) represents a Fourier transform, and only the Fourier transform of zero can resultin zero


Layer sσs , µs , εs

Layer s+ 1σs+1 , µs+1 , εs+1

Layer s− 1σs−1 , µs−1 , εs−1

Layer 1σ1 , µ1 , ε1

y = ys−1y = y′y = ys y = ys−2 y = y0 =∞y = yL−1y = yL = −∞

Layer LσL , µL , εL

ZL, γL Zs+1, γs+1

Zs, γs Zs, γs

Zs−1, γs−1 Z1, γ1

Zeq,LZeq,s+2 Zeq,s+1 Zeq,s−1 Zeq,s−2

Zeq,1

Isrc

Figure 3.5: Multilayer media transmission line model

is obtained through the spectral transformation of the Green’s function of a 2D homogeneous

medium (3.9).

To find the vector potential at point (x, y), we must solve for coefficients C+l and C−l for

l = 1, . . . , L, for a total of 2(L − 1) unknowns. We may solve these coefficients by forcing

continuity of the vector potential and its derivative across each interface, which translates

into the continuity of electric and magnetic field. Indeed, this is how we obtained the Green’s

function of a two-layered media (3.27). For a large number of total layers L, it is more convenient

to solve the problem using a transmission line model. We now present how to formulate and solve

this transmission line model in order to derive the Green’s function of an L-layered medium.

We model the problem of finding the spectral domain vector potential Aδz(βx, y) in a

multilayer media through the transmission line shown in Fig. 3.5. In this transmission line

model, each layer of ground is represented by a segment of transmission line with characteristic

impedance [58]

Zl =1√

β2x − k2l, (3.52)

and propagation constant

γl =√β2x − k2l . (3.53)


Zeq,s+1 Zeq,s−1

y = ys−1y = y′y = ys

Zs, γs Zs, γs

Isrc

Figure 3.6: Simplified transmission line circuit to model layer s

Furthermore, the current source

Isrc = µs exp(jβxx

′) (3.54)

represents the point source inside the multilayer medium. Our goal is to compute the voltage

V (y) in the transmission line which is equal to Aδz(βx, y).

Since we just want to find the vector potential inside layer s, we can further simplify the

transmission line model in Fig. 3.5 to that in Fig. 3.6. In Fig. 3.6, Zeq,s+1 is the equivalent

impedance looking into layer s+1 from layer s at y = ys, as shown in Fig. 3.5. Likewise, Zeq,s−1

is the equivalent impedance looking into layer s − 1 from layer s at y = ys−1. The value of

Zeq,s+1 may be found as follows:

1. Replace segment L by an equivalent impedance Zeq,L = ZL at y = yL−1. This is done

because layer L is semi-infinite, and therefore its input impedance is the same as its

characteristic impedance.

2. Find the equivalent impedance of layers L and L − 1 seen by looking into L − 1 from

y = yL−2. This can be found with the well-known formula to compute an equivalent

impedance of a transmission line of length (yL−2 − yL−1) and characteristic impedance

ZL−1, which is terminated with the load of Zeq,L [25]

Zeq,L−1 = ZL−1Zeq,L + jZL−1 tanh (γl−1(yL−2 − yL−1))ZL−1 + jZeq,L tanh (γl−1 (yL−2 − yL−1))

. (3.55)

3. Recursively apply step 2. to find equivalent impedances Zeq,L−2, Zeq,L−3, . . ., Zeq,s+1.

The impedance Zeq,s−1 may be computed similarly. Finally, the solution of circuit in Fig. 3.5


is given by [59]

V (y) =

(ZsIs

2

)exp

(−∣∣y − y′∣∣ γs)+

(ZsIs

2

)[1

1− ΓRΓL exp (−2(ys−1 − ys)γs)

]·[

ΓL exp((2ys − y′ − y)γs

)+ ΓR exp((−2ys−1 + y + y′)γs)

+ ΓRΓL exp((

2ys − 2ys−1 + y′ − y)γs)

+ ΓRΓL exp((

2ys − 2ys−1 + y − y′)γs) ]

, (3.56)

where

ΓL =Zeq,s+1 − ZsZeq,s+1 + Zs

, (3.57)

ΓR =Zeq,s−1 − ZsZeq,s−1 + Zs

, (3.58)

are the reflection coefficients of both loads in Fig. 3.6. Voltage V (y) found in (3.56) is also

equal to the spectral domain vector potential Aδz(βx, y).

The Green’s function Gg(x, y, x′, y′) of the layered medium is then found by inverse Fourier

transformation of Aδz(βx, y) found in (3.56)

Gg(x, y, x′, y′) =

−1

2πµs

ˆ ∞−∞Aδz(βx, y) exp (−jβxx) dβx . (3.59)

Note that the equation above has been scaled by −1µs in order to be consistent with the definition

of the Green’s function in rest of the thesis.4

3.5.2 Entries of Green’s Matrix Gg

We now use the Green’s function of a multilayer medium in (3.59) to derive entries of Green’s

matrix Gg. This matrix represent discretized Green’s function which relates vector potential

to equivalent currents on the boundary of the hole, as we derived in the field integral equation

in (3.29).

4In this section, we only computed the vector potential due to a point-source. However, vector potential andthe Green’s function are related by scalar −µs, which is offsetted here.


We substitute (3.59) into (3.30)

[Gg](n′,n) =−1

(2π)3

ˆ 2π

0

ˆ 2π

0

ˆ ∞−∞

(ZsIs2µs

)e−|y−y

′|γs e−jβxx ej(nθ′−n′θ) dβx dθ dθ

′︸︷︷︸Term 1

(3.60)

+−1

(2π)3

ˆ 2π

0

ˆ 2π

0

ˆ ∞−∞

(ZsIs2µs

)[ΓL e(2ys−y

′−y)γs

1− ΓRΓL e−2(ys−1−ys)γs

]e−jβxx ej(nθ

′−n′θ) dβx dθ dθ′

︸︷︷︸Term 2

+−1

(2π)3

ˆ 2π

0

ˆ 2π

0

ˆ ∞−∞

(ZsIs2µs

)[ΓRe

(−2ys−1+y+y′)γs

1− ΓRΓLe−2(ys−1−ys)γs

]e−jβxxej(nθ

′−n′θ)dβx dθ dθ′

︸︷︷︸Term 3

+−1

(2π)3

ˆ 2π

0

ˆ 2π

0

ˆ ∞−∞

(ZsIs

2

)[ΓRΓLe

(2ys−2ys−1+y′−y)γs


]e−jβxxejnθ

′−n′θdβx dθ dθ′

︸︷︷︸Term 4

+−1

(2π)3

ˆ 2π

0

ˆ 2π

0

ˆ ∞−∞

(ZsIs

2

)[ΓRΓLe

(2ys−2ys−1+y−y′)γs


]e−jβxxej(nθ

′−n′θ)dβx dθ dθ′

︸︷︷︸Term 5

,

where x, y, x′, and y′ are defined as follows

x = x+ a cos θ (3.61a)

y = y + a sin θ (3.61b)

x′ = x+ a cos θ′ (3.61c)

y′ = y + a sin θ′ . (3.61d)

Solving (3.60) numerically is computationally expensive. However, we can solve some of the

integrals in (3.60) analytically. We now discuss how to evaluate each of the five terms in (3.60).

Term 1

Term 1 can be evaluated analytically. Term 1 is the most dominant of the five terms in (3.60)

because it represents the field radiated by the current source in presence of a homogeneous

medium. This can be seen by evaluating Term 1 in (3.60) over βx to obtain

1

(2π)2j

4

ˆ 2π

0

ˆ 2π

0K0

(ks

∣∣∣r(θ)− r(θ′)∣∣∣) e(j(nθ′−n′θ)dθ dθ′ , (3.62)


which is similar to entries of Green’s matrix Gc, which appears in (3.37). The solution of Gc,

and therefore Term 1 in (3.60), is given in Appendix B.2.

Term 2

Terms 2-5 in (3.60) only account for reflections due to top and bottom interfaces of layer s.

We can solve the two finite integrals in Terms 2-5 analytically. By doing so, we significantly

reduce the time it takes to evaluate these terms. We begin from Term 2 in (3.60), and rearrange

it to

−1

16π3

ˆ ∞−∞

(1√

β2x − k2s

)[ΓLe

2ysγs


]ˆ 2π

0e−jβx(x+a cos θ)−γs(y+a sin θ)−jn

′θdθ

ˆ 2π

0ejβx(x+a cos θ

′)−γs(y+a sin θ′)+jnθ′dθ′dβx ,

(3.63)

where x, y, x′, and y′, are as defined in (3.61a)-(3.61d). Using the following identities from [60]5

ˆ 2π

0e−jβx(x+a cos θ)−γs(y+a sin θ)−jn

′θdθ = (sgn(−βx))n′2πe−jβxx−γsy

(βx − γsβx + γs

)n′/2In′ (jaks)

(3.64)

ˆ 2π

0ejβx(x+a cos θ

′)−γs(y+a sin θ′)+jnθ′dθ′ = (sgn(βx))n2πejβxx−γsy(βx − γsβx + γs

)n/2I−n (jaks) ,

(3.65)

with ,

sgn(τ) =

1 τ > 0

−1 τ ≤ 0 ,

(3.66)

we can simplify the triple integral (3.63) to

−ˆ ∞−∞

((sgn(βx))n(sgn(−βx))n

′

4π√β2x − k2s

)[ΓLe

2ysγse−2γsyIn′ (jaks) I−n (jaks)

1− ΓRΓLe(−2(ys−1−ys)γs)

](βx − γsβx + γs

)(n+n′)/2

dβx .

(3.67)

5Note that in [60], sgn(βx) terms are missing. But these terms are included by comparing numerical integrationresults.


The infinite integral in (3.67) cannot be evaluated analytically. It will be integrated numerically

as discussed later in this Section.

Term 3

Term 3 may be simplified to

−ˆ ∞−∞

(sgn(βx))n(sgn(−βx))n′

4π√β2x − k2s

(ΓRe

−2ys−1γse2γsyI−n(jksa)In′(jksa)


)(βx + γsβx − γs

)(n+n′)/2

dβx

(3.68)

by following the same procedure described for Term 2.

Term 4

Term 4 can be simplified to

ˆ ∞−∞


4π√β2x − k2s

(ΓRΓLe

(2ys−2ys−1)γsI−n(jksa)In′(jksa)



)(n−n′)/2dβx

(3.69)

by following the same procedure described for Term 2.

Term 5

Term 5 can be simplified to

ˆ ∞−∞


4π√β2x − k2s

(ΓRΓLe

(2ys−2ys−1)γsI−n(jksa)In′(jksa)



)−(n−n′)/2dβx

(3.70)

by following the same procedure as in Term 2.


Entries of Green matrix

By combining (3.62), (3.67), (3.68), (3.69), and (3.70), we obtain the (n′, n) entry of matrix Gg

[Gg](n′,n) =j

(16π2

ˆ 2π

0

ˆ 2π

0K0

(ks

∣∣∣r(θ)− r(θ′)∣∣∣) e(j(nθ′−n′θ)dθ dθ′− (3.71)

ˆ ∞−∞


4π√β2x − k2s

(I−n(jksa)In′(jksa)


)[

ΓRe−2ys−1γs

(βx + γsβx − γs

)(n+n′)/2

e2γsy + ΓLΓRe2γs(ys−ys−1)


)(n−n′)/2

+ΓLΓRe2γs(ys−ys−1)


)−(n−n′)/2+ ΓLe

2ysγs

(βx − γsβx + γs

)(n+n′)/2

e−2γsy

]dβx ,

where the first term may be evaluated analytically as discussed earlier, and the second term

which combines the infinite integrals in Terms 2-5 must be evaluated numerically.

Both terms in (3.71) have a physical meaning, as the first term corresponds to radiation in

a homogeneous unbounded medium, and the second term is the scattering component of the

vector potential due to the presence of different layers of materials. Since the second term on

the right-hand side of (3.71) corresponds to the scattering component of the field, it is usually

smaller than the first term of (3.71) due to two reasons:

1. The reflection coefficient between any two interface is small for practical ground con-

ductivities between 0 and 5 S/m. Therefore, most of the wave originating from layer s

transmits to the adjacent layers, and only a small portion of it is reflected.

2. In most underground scenarios, layer s is a lossy medium. Hence, the incident field which

originates from the equivalent current on the hole boundary decays as it propagates from

the boundary of the hole, to the interface, back to the boundary of the hole. This decay

ensures that the second term on the right-hand side of (3.71) is smaller compared to

Term 1.

Even though the second term on the right-hand side of (3.71) is small relative to the first

term, we cannot completely neglect it. However, from a practical point-of-view, we only need

to calculate the second term in (3.71) for n = 0 and n′ = 0. This simplification can be made

because the infinite integral in (3.71) for n 6= 0 or n′ 6= 0 represents higher order interactions


between the scattered field and hole equivalent currents, and this interaction is extremely low

when compared to the overall result. This assumption has been numerically verified for many

different test scenarios.

We have now discussed how to derive the Green’s matrix Gg for a multilayered medium. The

final expression that we derived in (3.71) is very efficient because we only need to evaluate one

infinite integral per hole. In the introduction of this chapter, we rejected the idea of including

ground effect by replacing the quasi-static Green’s function, which is used to compute Green’s

matrix G (2.52), by the Green’s function of multilayered medium given by (3.59) because of

its poor efficiency. Had we employed that approach, we would have had to evaluate N2 infinite

integrals, making the technique extremely slow6. Hence, the approach discussed in this chapter

is extremely efficient.

In our current formulation, we introduced several new matrices: Gc, G0, G0, D1, D2 and

H. However, all these matrices can be generated analytically. The advantage that we gained

by introducing these matrices is that we were able to derive the surface admittance operator of

the hole, which reduced the number of entries in matrix Gg, and hence the number times we

have to numerically evaluate an infinite integral.

In summary, to generate the Green’s matrix Gg of a multilayer medium, we first found

the Green’s function of a multilayer medium by computing the vector potential due to a point

source in layer s. We solved the vector potential in spectral domain through a transmission line

model. Thereafter, we found the spatial domain solution through the inverse spectral transfor-

mation. Next, the Green’s function was decomposed into two components: homogeneous and

inhomogeneous. The homogeneous component (Term 1 in (3.60)) was evaluated analytically.

The inhomogeneous component (Terms 2-5 in (3.60)) was partially solved analytically, and

the rest was evaluated through numerical integration. By generating Gg, we can now include

ground return effect of any multilayer background medium.

6N is the total number of equivalent current or field coefficients and is given in (2.36)


3.6 Extension to Hollow Conductors and Multiple Holes

For the sake of clarity, we described the proposed method considering only solid round conduc-

tors buried into a single hole. The proposed technique can handle, however, the most general

case, where we have multiple holes each one containing an arbitrary arrangement of solid and

hollow conductors. In this section, we briefly discuss how hollow conductors and multiple holes

can be introduced to the theoretical framework discussed so far.

When hollow conductors are introduced, we have equivalent currents on inner and outer

contours of hollow conductors. The inclusion of hollow conductors will change the surface

admittance operator Ys as discussed in Sec. 2.2.3. Additionally, all Green’s matrices that

relate equivalent conductor currents to the fields need to be modified in order to include the

fields or equivalent currents on the inner contour of any hollow conductors. In the presence of

multiple holes, the process of Sec. 3.2 is first applied to each hole independently. An equivalent

current (3.4) is introduced on the boundary of each hole, and related to the equivalent currents

inside that specific hole through (3.22). Then, one integral per hole is added to the right-hand

side of (3.26) to account for the superposition effect on the vector potential due to a current on

the boundary of each hole. These changes, would require appropriate algebraic modifications

to the expression to compute p.u.l. resistance and inductance in (3.43) and (3.44) .

3.7 An Approximate Technique to Include the Ground Return

Effect

In this chapter, we developed a versatile technique to include accurately the ground return effect

of a cable placed inside a tunnel dug in multilayered ground. There exists a simpler method

to include ground return effect directly to the MoM-SO technique developed in Chapter 2.

The approximate technique was published in [7], and was developed prior to the accurate

ground return formulation that we derived in this chapter. This method relies on MoM-SO to

capture proximity effect, and the analytic formulas to capture skin and ground return effect.

Approximate ground return technique is faster than the technique developed in this Chapter

because it relies on analytic formulas. However, all the disadvantages of analytic formulas for


ground return impedance that we discussed in Sec. 1.3.4 are also present in this technique.

Hence, this technique cannot be used a) for cables buried inside a multilayered ground, b) when

there exists proximity effect inside ground, and c) when cables are placed inside a tunnel. In

addition, this technique can only be used for cable geometries whose ground return impedance

can be calculated using the analytic formulas, such as three-phase and SC cables [18].

We now show how this technique works. The series impedance Z of any cable can be

decomposed as

Z = (Zc + Zg) + ∆Zprox , (3.72)

where Zc denotes the cable impedance inclusive of skin effect, Zg denotes the cable impedance

due to ground return effect, and ∆Zprox denote the cable impedance due to proximity effect.

Commercial EMTP tools [21, 61] compute Zc + Zg, but neglect ∆Zprox. Instead, MoM-SO

presented in Chapter 2 can calculate Zc + ∆Zprox by setting Np > 0 in the Fourier expansions

of fields and equivalent currents in (2.3) and (2.6). MoM-SO can also calculate Zc by setting

Np = 0. In essence, we run two simulations in MoM-SO, one by setting Np = 0 and another by

setting Np > 0, and combine the results of the two simulations to calculate

∆Zprox = ZMoM-SO(Np > 0)− ZMoM-SO(Np = 0) . (3.73)

By adding ∆Zprox (3.73) obtained using MoM-SO to Zc+Zg obtained using EMTP tools we can

calculate the total impedance Z (3.72) inclusive of skin, proximity, and ground return effects.

This approximate technique was published in [7], and will be used to compare the new

MoM-SO technique developed in this chapter.

3.8 Numerical Results

In this section, we validate the proposed technique against FEM (COMSOL Multiphysics [4])

and analytic formulas for underground cables. We present a total of four test cases. The

first test case will show that MoM-SO accurately captures skin, proximity and ground return

effects. The second test case will show that MoM-SO also accounts for proximity effect in the

ground which is disregarded by analytic formulas commonly-used in existing simulation tools.


spacing

depth


air (ε0, µ0)

Figure 3.7: System of three single core cables used for validation in Sec. 3.8.1. Conductivemedia are shown in gray while insulating media are shown in white.

The third test case will consider a three-phase cable inside a tunnel dug in earth. The final

example will demonstrate the influence of multilayer ground medium on the cable impedance

parameters.

3.8.1 Example # 1: Three Single Core Cables Buried in Earth

In this example, we validate the impedance computed through the proposed MoM-SO method

against a commercial FEM solver (COMSOL Multiphysics [4]), the “cable constant” (analytic)

formulas [62], and the hybrid technique presented in Sec. 3.7. We consider a system made up

of three SC cables buried in ground at a depth of 1 m, as shown in Fig. 3.7. This example also

demonstrates that MoM-SO can handle multiple holes and hollow conductors because we use

a hollow conductor to model each screen of the SC cables. Also, as shown in Fig. 3.7, each SC

cable is placed into a separate tunnel (hole).

Geometrical and Material Properties

The geometrical and material parameters of the three SC cables are presented in Table 3.1. We

consider two different values for cable spacing: s = 2 m and s = 85 mm. With the spacing

of s = 2 m, there is low proximity effect between the cables. Conversely, there is a significant

proximity effect between the cables with the spacing of s = 85 mm. The background medium

is two-layered with air in the top-half (y > 0) and lossy ground in the bottom-half (y < 0).

Ground conductivity is set to 0.01 S/m.


Table 3.1: Single core cables of Sec. 3.8.1: geometrical and material parameters

Core Outer diameter = 39 mm, ρ = 3.365 · 10−8 Ω ·mInsulation t = 18.25 mm, εr = 2.85

Sheath t = 0.22 mm, ρ = 1.718 · 10−8 Ω ·mJacket t = 4.53 mm, εr = 2.51

Simulation Setup

We extract a 6 × 6 impedance matrix of the system of six conductors (three core conductors

and three hollow screens) with FEM and MoM-SO. Ground is indirectly taken as a reference

conductor for this system by setting the return path to be at infinity. Impedance is evaluated

at 31 frequency points logarithmically spaced between 1 Hz and 1 MHz.

In MoM-SO, we set Np and N to 4 to accurately represent the fields in (2.3), and (3.3), and

equivalent currents in (2.6) and (3.4) on the conductors’ and holes’ boundaries. In the FEM

solver, the mesh has to be carefully set up to achieve good accuracy. Ground has to be meshed

up to a distance of three times the skin depth at all frequencies in order to properly incorporate

ground return effect. For the first 25 frequency points, we used a mesh with 725,020 triangles

for the s = 85 mm case, and 837,618 triangles for the s = 2 m case. These many triangles are

required because the dimension of the ground is very large, especially at low frequencies where

the skin depth is equal to 1500 m. Additionally, very fine triangles are required to mesh the

screens which are very thin. At the last six frequency points, which are spread between 100

kHz and 1 MHz, skin depth becomes extremely small, and the mesh has to be refined inside

the conductors. This required the use of the so-called boundary layer elements, which increased

mesh size to 1,053,638 for the s = 2 m case. The mesh distribution in FEM [4] for the case of

s = 85 mm is shown Fig. 3.8.

Continuously-grounded Screens

We first consider the case where cable screens are grounded. In this configuration, we calculate

the 3×3 impedance matrix of the cable from the 6×6 impedance matrix by setting the potential

of the screens to zero. By continuously grounding the screens, we can minimize proximity

effect between the SC cables. The positive-sequence resistance and inductance obtained with


(a)

(b)

Figure 3.8: Mesh used for the three single-core cable case in Sec. 3.8.1. Top panel shows themesh distribution inside the SC cables, and the bottom panel shows the mesh distribution inthe surrounding ground.


100

102

104

106

10−1

100

s = 85mm

s = 2m

Res

ista

nce

p.u

.l. [Ω

/km

]

Frequency [Hz]

100

102

104

106

0.2

0.4

0.6

0.8

1

s = 85mm

s = 2m

Induct

ance

p.u

.l. [m

H/k

m]

Frequency [Hz]

Figure 3.9: Cable system of Sec. 3.8.1: positive-sequence resistance (top panel) and inductance(bottom panel) computed using FEM (), MoM-SO (·), and cable constant (- -). Screens arecontinuously grounded [6].

MoM-SO, FEM and cable constant formulas are presented in Fig. 3.9. The zero-sequence

resistance and inductance are shown in Fig. 3.10. The zero- and positive-sequence resistance

and inductance are calculated using the post-processing technique shown in Sec. 2.6.3. The

excellent agreement observed between FEM and MoM-SO validates the proposed technique.

Cable constant formulas also provide accurate results because there is very little proximity

effect.

Open Screens

We now consider another scenario where the screens are left opens at both ends. When screens

are left open, there is a significant proximity effect between the three SC cables. Figure 3.11

shows the positive-sequence resistance and inductance for the case where cables are close to-

gether (s = 85 mm). Both MoM-SO and FEM accurately capture skin and proximity effects in


100

102

104

106

10−1

100

101

s = 85mm

s = 2m

Res

ista

nce

p.u

.l.

[ Ω

/km

]

Frequency [Hz]

100

102

104

106

0

2

4

6

8

s = 85mm

s = 2m

Induct

ance

p.u

.l.

[mH

/km

]

Frequency [Hz]

Figure 3.10: Cable system of Sec. 3.8.1: zero-sequence resistance (top panel) and inductance(bottom panel) computed using FEM (), MoM-SO (·), and cable constant (- -). Screens arecontinuously grounded [6].

conductors and ground. However, cable constant formulas return accurate results only at low

frequencies, and become inaccurate beyond 100 Hz where proximity effect is more pronounced.

Figure 3.11 also shows the resistance and inductance obtained with an approximate method

to include ground effect presented in Sec. 3.7. We can see that the approximate technique of

Sec. 3.7 can accurately compute skin, proximity, and ground return effects up to 100 kHz. How-

ever, it fails after 100 kHz because it relies on analytic formulas which are inaccurate beyond

this frequency. The proposed method of this chapter, instead, remains accurate over the entire

frequency range.

Timing Results

Table 3.2 shows the CPU time taken by MoM-SO and FEM to analyze the cable system. FEM

requires more than 6 minutes per frequency point, while MoM-SO only 0.8 s. This dramatic

speed up, beyond 400X, comes from the fact that, with the MoM-SO method, one has to mesh


100

102

104

106

10−2

10−1

100

101

Res

ista

nce

p.u

.l. [

Ω/k

m]

Frequency [Hz]

100

102

104

106

0.25

0.3

0.35

0.4

Induct

ance

p.u

.l.

[mH

/km

]

Frequency [Hz]

Figure 3.11: Cable system of Sec. 3.8.1: positive-sequence resistance (top panel) and inductance(bottom panel) computed using FEM (), MoM-SO (·), cable constant ( ), and MoM-SO withapproximate ground return effects [7] ( ). The screens of the cables are open [6].

Table 3.2: Example of Sec. 3.8.1: CPU time required to compute the impedance at one frequency

Case MoM-SO (Proposed) Hybrid (Approximate) FEM Speed-up

s = 85mm 0.80 s 0.08 s 371.21 s 464 X

s = 2m 0.80 s 0.08 s 452.77 s 566 X

All computations were performed on a workstationwith a 3.40 GHz CPU and 16 GB of memory.

neither the cross-section of the conductors nor the surrounding ground where return current

may flow. On the other hand, the complex mesh needed to capture ground return effects and

skin effect at high frequency makes FEM very time consuming. Moreover, with FEM, the user

must spend extra time to properly set up the mesh generator, since the default settings may

not lead to accurate results. MoM-SO, instead, being meshless, is much easier to use, and can

be fully automated, as shown in Sec. 4.2 [63].


3.8.2 Example # 2: Proximity Effect in Surrounding Ground

100

102

104

106

100

102

104

positive sequence

zero sequence

Res

ista

nce

p.u

.l. [Ω

/km

]

Frequency [Hz]

100

102

104

106

−0.1

0

0.1

0.2

0.3

positive sequence

Induct

ance

p.u

.l. [m

H/k

m]

Frequency [Hz]

Figure 3.12: Cable system considered in Sec. 3.8.2: resistance (top panel) and inductance(bottom panel) computed using FEM (), MoM-SO (·), cable constant ( ), and MoM-SOwith approximate ground return effects [7] ( ). Phase conductors are open, and current isinjected in the sheaths [6].

In this example, we consider the same system of three SC cables with spacing s = 85 mm and

ground conductivity σg = 100 S/m. Such a high ground conductivity is only to demonstrate the

proximity effects inside ground. At this ground conductivity value, proximity effect develops

inside the ground, and this proximity effect influences cable impedance. In this example, we

leave the phase conductors open, and inject currents in the sheaths. In other words, the three-

phase excitation is applied to the sheaths. Figure 3.12 shows the resistance and inductance

obtained in this scenario with MoM-SO, FEM and the approximate ground return method

discussed in Sec. 3.7. The excellent agreement between MoM-SO and FEM shows that the

proposed method correctly captures proximity effect in both conductors and ground. Proximity

effect in ground becomes visible above 100 Hz, and beyond this frequencies analytic formulas


give incorrect results because they neglect the proximity effect. Even the technique in Sec. 3.7

is inaccurate beyond 100 Hz because it also relies on analytic formulas to calculate the ground

return impedance.

3.8.3 Example # 3: Three Single-Core Cables Inside a Tunnel

2 m1 m


air (ε0, µ0)

tunnel (ε0, µ0)

1.5 m

Figure 3.13: System of three single-core cables in a tunnel considered in Sec. 3.8.3. Conductivemedia are shown in gray while insulating media are shown in white.

We investigate the effect of tunnel on the p.u.l. impedance parameters. We consider the

same system of three SC cables placed inside a tunnel. The cross-section of the system is

depicted in Fig. 3.13. Cables are spaced by s = 85 mm, and their characteristics are reported

in Table 3.1. Sheaths are left open at both ends, so only proximity current flows inside them.

We perform two simulations. In the first simulation, FEM and MoM-SO are used to compute

the positive- and zero-sequence impedance of the cable in presence of the tunnel. We then repeat

the computation with the tunnel removed, and cables buried directly in ground. The resistance

and inductance values obtained for both cases are shown in Fig. 3.14. From Fig. 3.14, we can

conclude that the influence of the tunnel on the cable impedance is visible above 3 MHz, because

below this frequency impedance obtained using MoM-SO with and without the tunnel match

very well. Some transients, such as those induced by disconnector switching in gas insulated

switchgear, contain components even beyond 3 MHz. These transients cannot be accurately

predicted without accounting for the tunnel effect. Since there is an excellent agreement between

MoM-SO and FEM in Fig. 3.14, we can conclude that MoM-SO can accurately predict tunnel

effects.


104

105

106

107

100

105

zero sequence

positive sequence

Frequency [Hz]

Res

ista

nce

p.u

.l. [Ω

/km

]

104

105

106

107

1

2

3

4

5

Induct

ance

p.u

.l. [m

H/k

m]

Frequency [Hz]

zero sequence

Figure 3.14: System of three SC cables in a tunnel considered in Sec. 3.8.3: resistance (toppanel), and inductance (bottom panel) computed with FEM () and MoM-SO (·). In orderto show the effect of the tunnel, the resistance and inductance of the cables buried directly inground are also shown (×) [6].

MoM-SO took only 0.29 s per frequency point against the 498.3 s taken by FEM, for a

speed up of 1,734 times7. Note, the time taken by MoM-SO for this example is lower than the

example in Sec. 3.8.1 because all cables in this example are placed inside the same hole. By

doing so, the Green’s matrix Gg in (3.29) is made smaller and requires less time to perform the

numerical integration in (3.71).

3.8.4 Example # 4: Two Conductors in a Four-layer Medium

Finally, we consider a two conductor system in a four-layer medium, as shown in Fig. 3.15.

Each conductor is of radius 22 mm and conductivity σ = 5.8 · 106S/m. Center-to-center dis-

tance between these conductors is 100 mm. The depth of the cable is 7.90 m from the earth-air

interface. The surrounding medium, as shown in Fig. 3.15, consists of four-layers with con-

7All computations were performed on a workstation with 3.40 GHz CPU and 16 GB memory.


Air (ε0, µ0)

Sea (ε0, µ0, σ1 = 3 S/m)

Seabed (ε0, µ0, σ2 = 0.2 S/m)

Soil (ε0, µ0, σ3 = 0.01 S/m)

y = −12 m

y = −8 m

y = 0 m

7.9 m

0.1 m

Figure 3.15: A two-conductor system inside a four-layer medium considered in Sec. 3.8.4

ductivities (from top layer to bottom layer) of 0 S/m, 3 S/m, 0.2 S/m, and 0.01 S/m. Such a

surrounding medium is representative of a shallow river or sea. The seabed, due to its wetness,

has higher conductivity (0.2 S/m) than typical conductivity of earth (0.01 S/m). The cable

that we consider is impractical for power system applications because a typical cable would

have an insulation around the conductors and screens. Nevertheless, we can use this system to

validate our proposed multilayer approach.

Firstly, we compute the 2 × 2 p.u.l. impedance matrix with MoM-SO setting the meshing

parameters N = 4, and Np = 4. Next, we compute the same 2× 2 impedance matrix in COM-

SOL Multiphysics, which requires a mesh size of 303,010 domain elements and 18,120 boundary

elements. Figures 3.16 and 3.17 show loop- and common-mode inductance and resistance com-

puted with the two techniques. Excellent agreement between the two techniques validates the

proposed MoM-SO approach for a multilayer medium.

Analytic formulas only support a single layer ground. In order to demonstrate the influence

of multiple layers, we calculate the impedance of our system in MoM-SO using a single layer

ground model. The conductivity of the ground is set to σ = 3 S/m. Note that the ground

conductivity of σ = 3 S/m is also impractical because it implies that the conductors are floating

in the sea. Nevertheless, here we are only trying to show the influence of a multilayered ground

model. The impedance obtained using this single layer ground model is shown Figs. 3.16 and


3.17 and is denoted by “MoM-SO: 2 layers”. As seen in Figs. 3.16, there is a significant variation

in common-mode inductance obtained using two different models. This result highlights why

we require a multilayer ground model to simulate subsea cables.

To simulate this example, MoM-SO took only 0.20 s per frequency point, which was 255

times faster than the 51.10 s taken by COMSOL Multiphysics. The computations were per-

formed on a workstation with 3.40 GHz CPU and 16 GB memory.

100

102

104

106

5.6

5.8

6

6.2

6.4

6.6

6.8

7

x 10−7

Frequency [Hz]

Lo

op

−M

od

e In

du

ctan

ce p

.u.l

. [H

/m]

COMSOL

MoM−SO: 4 layers

MoM−SO: 2 layers

100

102

104

106

0

0.2

0.4

0.6

0.8

1x 10

−5

Frequency [Hz]

Co

mm

on

−M

od

e In

du

ctan

ce p

.u.l

. [H

/m]

COMSOL

MoM−SO: 4 layers

MoM−SO: 2 layers

Figure 3.16: Loop-mode and common-mode inductance of the two-conductor system consideredin Sec. 3.8.4.

100

102

104

106

10−4

10−3

10−2

10−1

100

Frequency [Hz]

Lo

op

−M

od

e R

esis

tan

ce p

.u.l

. [Ω

/m]

COMSOL

MoM−SO: 4 layers

MoM−SO: 2 layers

100

102

104

106

10−4

10−3

10−2

10−1

100

101

Frequency [Hz]

Co

mm

on

−M

od

e R

esis

tan

ce p

.u.l

. [Ω

/m]

COMSOL

MoM−SO: 4 layers

MoM−SO: 2 layers

Figure 3.17: Loop-mode and common-mode resistance of a two-conductor system considered inSec. 3.8.4


3.9 Conclusions

In this chapter, we extended MoM-SO to include ground return effect. In the process, we made

three main scientific contributions: introduce the surface admittance operator of a cable-hole

system, introduce the effects of multilayered background medium, and introduce the effects

of tunnels. We derived the surface admittance operator of a cable-hole system to make our

calculations efficient and to introduce the effects of tunnels. Through this operator, we greatly

reduced the complexity of the problem by replacing the hole, and the cables inside, with a single

equivalent current source. The reduced system was then efficiently coupled with multilayered

surrounding medium to account for ground effects.

The new technique that we developed is better than both the analytic formulas widely used

by the industry, and the FEM. MoM-SO is better than the analytic formulas because it is more

accurate. MoM-SO can i) handle any cable made up of solid and hollow round conductors,

ii) account for proximity effect both inside the conductor and inside the surrounding ground,

iii) account for tunnel effects, and iv) account for ground effects of a multilayer ground medium.

None of the above mentioned features are handled by analytic formulas. Even though FEM

can handle all the features mentioned above, MoM-SO remains to be overall a better cable

modelling tool. Compared to FEM, MoM-SO is i) faster, and ii) simple to use. Results showed

that MoM-SO was up to 1700 times faster than FEM for the case of three-phase cable buried

inside a tunnel dug in ground. Such speedups are observed because MoM-SO is a surface

method, hence, very few unknowns are required to solve the problem. Fewer unknowns lead to

faster solution time. Furthermore, owing to the surface approach, MoM-SO is also simpler to

use than FEM. In particular, user does not have to mesh the geometry of the cable, which may

be annoying to do at very high and very low frequencies in FEM tools. Instead, with MoM-SO,

the user has to specify two discretization parameters (N , N), based on which the impedance

parameters are calculated. Even these two discretization parameters, will be fully automated

in Sec. 4.2. In conclusion, computational speed and ease-of-use make MoM-SO a very practical

tool for cable modelling.

Chapter 4

Fields Computation and Adaptive

Discretization

In Chapters 2 and 3, we developed MoM-SO which computes the p.u.l. impedance of cables

surrounded by a homogeneous or a lossy multilayered media. This chapter presents two addi-

tional developments of MoM-SO: fields computation and automatic prediction of the number

of basis functions. Information on the field and current distribution can help engineers decide

appropriate materials that are best-suited for their application and voltage-levels. Field distri-

bution outside the cable is also useful to predict electromagnetic compatibility related issues,

like signal interference, associated with the cables. Analytic formulas that are widely-used can-

not give field information. FEM tools can plot field distributions, but they are slow and require

additional setup time.

Choice of mesh in a FEM tool is often a “guessing game”, and its influence on the impedance

calculation should not be underestimated. Even though the discretization in MoM-SO is very

straightforward, and is governed by parameters Np and N . It is convenient to have an algorithm

that automatically sets these parameters. Advantages of such an algorithm are: a) any power

engineer, even without advanced training in electromagnetics, can use the tool; b) MoM-SO

can be easily integrated into existing transient tools; and c) simulation time can be reduced

because if the algorithm is efficient then the number of basis functions are never overestimated.

This chapter is divided into two main sections. In the first section, we will explore how

94

Chapter 4. Fields Computation and Adaptive Discretization 95

we can extend MoM-SO to plot fields distributions inside and outside the cable. In the second

half of this chapter, we present an adaptive discretization scheme which can automatically and

efficiently predict the number of basis functions for each conductor.

4.1 Fields and Current Distribution

From the equations presented in Chapters 2 and 3, we can compute the electric field and the

equivalent current density on the boundaries of the conductors and the holes. Then, from that

information, we show how one can obtain fields everywhere.

4.1.1 Fields and Equivalent Currents on the Boundaries

In order to calculate fields everywhere, we must first calculate a) the p.u.l. impedance pa-

rameters given in (3.43) and (3.44); b) the field coefficients on the holes’ and the conductors’

boundaries from (3.31) and (3.40); and c) the equivalent currents on the holes’ and conductors’

boundaries from (3.22) and (3.42). In order to calculate the quantities mentioned above, we

must set the total current I in each conductor. Once the Fourier coefficients of fields and cur-

rents on the boundaries are known, we can easily compute the fields everywhere. We calculate

the field distribution inside a conductor in Sec. 4.1.2, inside a hole in Sec. 4.1.3, and inside

ground in Sec. 4.1.4.

4.1.2 Inside a Conductor

While deriving the surface admittance operator in Chapter 2, we expanded the total electric

field E(p)z (ρp, θp) inside a solid and a hollow conductor as in (2.5) and (2.22), respectively. The

total electric field may be decomposed into two quantities: impressed electric field Ep,impz and

induced electric field Ep,indz . In order to derive an expression for the electric field, we ideally

apply a voltage sources which excite the conductors. These voltage sources produce impressed

fields Ep,impz , which can be directly related to the scalar potential of conductors [53]. For p-th

conductor, this relationship is given as

Ep,impz = −∂Vp

∂z. (4.1)


The electric field impressed on conductor p drives current in the same conductor, which in turn

induces electric field and currents in all conductors and in the surrounding medium. We choose

to plot the induced electric field inside the conductor, which can be obtained by

Ep,indz (ρp, θp) = Epz (ρp, θp)− Ep,impz

= Epz (ρp, θp)−P∑q=1

[Rpq(ω) + jωLpq(ω)] Iq . (4.2)

If the pth conductor is solid, then we substitute (2.5) into (4.2)

Ep,indz (ρp, θp) =

Np∑n=−Np

E(p)n

J|n|(kap)J|n|(kρp) ejnθp −

P∑q=1

[Rpq(ω) + jωLpq(ω)] Iq , (4.3)

and if the pth conductor is hollow, then we substitute (2.22) into (4.2)

Ep,indz (ρp, θp) =

Np∑n=−Np


)ejnθp (4.4)

−P∑q=1

[Rpq(ω) + jωLpq(ω)] Iq ,

where Cn(k) and Dn(k) are given in (2.23a)-(2.23b). Equation (4.3) and (4.4) provide expres-

sions to calculate induced electric field anywhere inside a solid or a hollow conductor.

Now we calculate current density inside a conductor. Since current is driven by both im-

pressed and induced electric fields, the current density inside a solid conductor is given by

J (p)z (ρp, θp) = σ

(Ep,impz + Ep,indz

)︸︷︷︸

Ez(ρp,θp)

(4.5)

= σ

Np∑n=−Np

E(p)n

J|n|(kap)J|n|(kρp)ejnθp . (4.6)

Similarly, the current density inside a hollow conductor is obtained by multiplying (2.22) by σ

J (p)z (ρp, θp) = σ

Np∑n=−Np


)ejnθp . (4.7)


We now discuss how to calculate magnetic field inside the p-th conductor. The magnetic

field inside conductor p is a vector with two components (under quasi-TM)

H(p)(ρp, θp) = H(p)ρ (ρp, θp)ρ +H

(p)θ (ρp, θp)θ , (4.8)

where ρ and θ are unit vectors in radial and azimuthal directions, respectively, of a cylindrical

coordinate system centered at (xp, yp), and from Maxwell’s equations under the quasi-TM

assumption

H(p)ρ (ρp, θp) = − 1

jωµ

1

ρp

∂E(p)z (ρp, θp)

∂θp, (4.9a)

H(p)θ (ρp, θp) =

1

jωµ

∂E(p)z (ρp, θp)

∂ρp. (4.9b)

We substitute (2.5) into (4.9a)-(4.9b) to calculate the magnetic field inside the solid conductor

H(p)ρ (ρp, θp) =

−1

ωµρp

Np∑n=−Np

nE(p)n

J|n|(kap)J|n|(kρp)ejnθp (4.10a)

H(p)θ (ρp, θp) =

k

jωµ

Np∑n=−Np

E(p)n

J|n|(kap)J ′|n|(kρp)e

jnθp . (4.10b)

Similarly, by substituting (2.22) into (4.9a)-(4.9b) we can find the magnetic field inside a hollow

conductor

H(p)ρ (ρp, θp) = − 1

ωµ

1

ρ

Np∑n=−Np

n[Cn(k)H|n|(kρp) +Dn(k)K|n|(kρp)

]ejnθp (4.11a)

H(p)θ (ρp, θp) =

k

jωµ

Np∑n=−Np

[Cn(k)H′|n|(kρp) +Dn(k)K′|n|(kρp)

]ejnθp , (4.11b)

where Cn(k) and Dn(k) are constants that are given in (2.23a)-(2.23b).


4.1.3 Inside a Hole

We can find the electric and magnetic fields inside the hole by first calculating the vector

potential Az. To find the vector potential at each point inside the hole, we need to compute

the general solution (3.10)

A′z(ρ ′, θ ′) =N∑

n=−N

CnJ|n|(kρ ′)

ejnθ′, (4.12)

and the particular solution (3.8)

A′′z(ρ ′, θ′) = −µP∑q=1

ˆ 2π

0J (q)s (θ)G(r(ρ ′, θ ′), rq(aq, θ

′q))aqdθq ,

= −µ 1

2π

j

4

P∑q=1

Nq∑n=−Nq

J (q)n

ˆ 2π

0K0(kd

′′)ejnθqdθq , (4.13)

where (ρ ′, θ ′) describes, in the cylindrical coordinate system centered at (x, y), point where

we want to calculate the fields. In (4.13), d′′ represents the distance between r(ρ ′, θ ′) and

rq(aq, θ′q). We can express (4.13) also as

A′′z(ρ ′, θ ′) = −µGJ , (4.14)

where

G =

[G(1) G(2) . . . G(P )

], (4.15)

with G(p) of dimension 1× (2Np+ 1) if p-th conductor is solid, and of dimension 1×2(2Np+ 1)

if p-th conductor is hollow. Entry (1, n) of G(p) is given by

[G(q)

]1,n

=1

2π

j

4

ˆ 2π

0K0(kd

′′)ejnθqdθq , (4.16)

which can be evaluated similarly to the entries of Gc as shown in Appendix B.2.


The magnetic vector potential can be related to the electric field through1

Ez(ρ ′, θ ′) = −jωAz(ρ ′, θ ′) , (4.17)

and to the magnetic field through

H(ρ ′, θ ′) =

[1

µ∇× (Az(ρ, θ)z)

]ρ=ρ ′ ,θ=θ ′

,

=

[1

ρ

1

µ

∂Az(ρ, θ)∂θ

ρ− 1

µ

∂Az(ρ, θ)

∂ρθ

]ρ=ρ ′ ,θ=θ ′

, (4.18)

where z is the unit vector along the longitudinal direction. By substituting the general solu-

tion (4.12) and the particular solution (4.13) into (4.17) and (4.18), we can find the electric and

magnetic field at any point inside the hole. Current density inside the hole can be obtained by

multiplying the electric field by the conductivity of the material within the hole.

4.1.4 Inside Ground

We find the electric field, the magnetic field, and the current density by first calculating the

magnetic vector potential in ground.

To calculate the vector potential in the surrounding ground we make use of the theory on

multilayer Green’s function we discussed in Sec. 3.5.1. We first consider the case where the

observation point is inside layer s. In this case, the solution of vector potential is given by

Az(x, y) = −µsGgJ (4.19)

where Gg is a matrix of size 1× (2N + 1)2. The entry (1, n) of Gg is given by

[Gg]1,n =−1

2πµs

ˆ 2π

0

ˆ ∞−∞Aδz(βx, y) e−jβxx ejnθ dβxdθ , (4.20)

where Aδz(βx, y) is the vector potential due to a point source. The expression for Aδz(βx, y) is

provided in (3.56). In (3.56), we set x′ = x + a cos θ and y′ = y + a sin θ. Matrix Gg describes

1Note there is no impressed field in this domain2assuming a single hole


Zeq,s+2+–

y = ys+1 y = ys

Az(βx, ys)Zs+1, γs+1

(a)

Zeq,s−2+–

y = ys−2y = ys−1

Az(βx, ys−1) Zs−1, γs−1

(b)

Figure 4.1: Equivalent Transmission line models

the vector potential at a point (x, y) due to the equivalent current on c. This matrix is similar

to Gg which relates the vector potential to the equivalent current on the boundary of the hole

as shown in (3.29). Integral on the right-hand side of (4.20) can be evaluated through the

procedure discussed in Sec. 3.5.2.

Outside layer s, the vector potential can be solved recursively. We start out by evaluating

the vector potential in spectral domain at the two interfaces of layer s, i.e. at y = ys and

y = ys−1 (refer to Fig. 3.5). The vector potential at y = ys is given by

Az(βx, ys) = −µsˆ 2π

0Js(θ)Gg(x, y, x

′, y′) dθ , (4.21)

where the Gg is Green’s function was found in (3.59) . We can compute Az(βx, ys−1) similarly.

By knowing Az(βx, ys), we can find the vector potential anywhere on the left-hand side of layer

s in the transmission line model shown in Fig. 3.5. Similarly, solution of Az(βx, ys−1) allows us

to find the vector potential to the right-hand side of layer s.

To solve for vector potential in layer s − 1, we simplify the transmission line model in

Fig. 3.5 to that in Fig. 4.1a. Here we replaced all layers to the left of layer s + 1 by an

equivalent impedance Zeq,s+2. The solution of the transmission line problem in Fig. 4.1a is

Az(βx, y) =Az(βx, ys)

1 + ΓL e2γs+1(−ys+ys+1)

[eγs+1(y−ys) +ΓL eγs+1(2ys+1−y−ys)

], (4.22)

where

ΓL =Zeq,s+2 − Zs+1

Zeq,s+2 + Zs+1.


Similarly, we can find the vector potential in layer s − 1 through the simplified model in

Fig. 4.1b. The vector potential in layer s− 1 is given by

Az(βx, y) =Az(βx, ys−1)

1 + ΓL e2γs−1(ys−1−ys−2)

[eγs−1(ys−1−y) +ΓR eγs−1(−2ys−2+y+ys−1)

], (4.23)

where

ΓR =Zeq,s−2 − Zs−1Zeq,s−2 + Zs−1

.

The vector potential in the other layers can be solved with the same procedure applied in a

recursive fashion.

Finally, we apply the inverse Fourier transform

Az(x, y) =1

2π

ˆ ∞−∞Az(βx, y) e−jβxx dβx (4.24)

to find the vector potential in the spatial domain. The integral above is evaluated using a

procedure similar to that outlined in Sec. 3.5.2.

4.1.5 Validation Plots

In this section, we present two examples to validate the formulas to calculate the electric field,

the magnetic field, and the current density.

Two conductors

Our first example is a system of two solid conductors with radii a1 = 1 mm and a2 = 2 mm,

and center-to-center spacing of D = 3.5 mm. Both conductors have conductivity of σ =

5.8 · 107 S/m. We excited both conductors with two currents of magnitude 1 A and opposite

phase. Figures 4.2, 4.3, and 4.4 show the magnitudes of the electric field, magnetic field,

and current density at three different frequencies: low (1Hz), medium (6kHz), and high (200

kHz). We compare the results obtained using our technique against the results obtained from

FEM(COMSOL Multiphysics [4]). In FEM plots, the domain outside the black solid circle

represents infinite element domain, and so the fields outside this line is fictitious. All the plots


obtained with our formulas have an excellent agreement against the plots obtained with FEM

(COMSOL Multiphysics), which validates the formulas presented in this Chapter.


(a) Electric field intensity at 1 Hz

(b) Magnetic field intensity at 1 Hz

(c) Current density at 1 Hz

Figure 4.2: The magnitudes of the electric field, magnetic field, and current density obtainedwith MoM-SO (left panel) and COMSOL Multiphysics (right panel) for the two-conductor casein Sec. 4.1.5 at 1 Hz. Conductors are excited with 1 A of currents flowing in opposite directions.


(a) Electric field intensity at 6 kHz

(b) Magnetic field intensity at 6 kHz

(c) Current density at 6 kHz

Figure 4.3: The magnitudes of the electric field, magnetic field, and current density obtainedwith MoM-SO (left panel) and COMSOL Multiphysics (right panel) for the two-conductor casein Sec. 4.1.5 at 6 kHz. Conductors are excited with 1 A of currents flowing in opposite directions.


(a) Electric field intensity at 200 kHz

(b) Magnetic field intensity at 200 kHz

(c) Current density at 200 kHz

Figure 4.4: The magnitudes of the electric field, magnetic field, and current density obtainedwith MoM-SO (left panel) and COMSOL Multiphysics (right panel) for the two-conductor casein Sec. 4.1.5 at 200 kHz. Conductors are excited with 1 A of currents flowing in oppositedirections.


One Conductor Inside a Multilayer Ground

In our second example, we validate the field in the surrounding multilayer medium due to

the presence of a current carrying wire. The wire has a radius of 22 mm and conductivity of

5.8 · 107 S/m. Surrounding medium has three layers with conductivities of 0 S/m, 3.33 S/m,

and 0.01 S/m from top to bottom. The second layer stretches from y = −4 to y = 0, as

shown in Fig. 4.5. Fig. 4.5 shows the magnitudes of the magnetic field and current density at

5 kHz obtained using MoM-SO, and using COMSOL Multiphysics [4]. The excellent agreement

between the set of plots validates the fields computation formulas in MoM-SO.

(a) MoM-SO

(b) COMSOL Multiphysics

Figure 4.5: The Magnitudes of magnetic field and current distribution inside the ground dueto a wire carrying a current of 1 A, as considered in Sec. 4.1.5. Fields are compared usingMoM-SO and COMSOL at 5 kHz.


4.2 Automatic Basis Function Estimation

In this section we discuss methods to choose truncation order Np of the Fourier expansion of

fields (2.3), (2.21a), (2.21b) and equivalent currents (2.6), (2.24), (2.25). Choice of the number

of basis function is an important aspect of MoM-SO because if Np is underestimated then

proximity effect will not be accurately predicted, where as, an overestimation of Np leads to

high computational time. In this thesis, we employed Fourier basis functions which are already

very efficient for round geometries that we consider. None of the other basis functions, for

example pulse or roof-tops, would lead to fewer basis functions to represent the conductors.

4.2.1 Trial & Error Estimation

A classical way to determine the setting for meshing parameters is through trial and error. In

our case, we can let Np be the same in (2.3), (2.21a), (2.21b), (2.6), (2.24), and (2.25) for all

conductors, and iteratively increase Np by one until a certain convergence criteria is satisfied.

Convergence criteria may be, for example, the difference in p.u.l. impedance between two

successive iterations ∣∣∣Z[i](ω)− Z[i−1](ω)∣∣∣ < α

∣∣∣Z[i](ω)∣∣∣ , (4.25)

where Z[i](ω) is the impedance computed in the i-th iteration.

The trial and error estimation, while simple, has two shortcomings. First, it assumes that

Np is the same for all conductors, thereby overestimating Np for some conductors. Additionally,

it requires multiple solutions of the problem at hand.

4.2.2 Adaptive Estimation

A better strategy can be devised based on energy in the Fourier series of equivalent current

density J(p)s (θp). The truncated Fourier series expansion of J

(p)s (θp) was given in (2.6) as

J (p)s =

1

2πap

Np∑n=−Np

J (p)n ejnθp . (4.26)

Our goal is to choose the truncation order Np in (4.26) such that the energy in (4.26) is very

close to energy in (4.26) with Np = ∞. i.e. We want to ensure that energy in the truncated


Fourier expansion is almost equal to the energy in complete Fourier expansion. Furthermore,

note that for n 6= 0, coefficient J(p)n is non-zero if and only if the equivalent current distribution

is radially asymmetric. An asymmetric current distribution can only arise if there are proximity

effect in the system. Hence, the problem at hand is to determine Np in (4.26) that captures

proximity effect sufficiently well.

Our strategy stems from the following insights:

1. We can check whether there is sufficient number of basis functions by comparing the

energy in the highest Fourier coefficients (n = ±Np) against energy of other coefficients

(|n| < Np). To do this we use the following convergence criteria:

∣∣∣J (p)Mp

∣∣∣2 +∣∣∣J (p)−Mp

∣∣∣2 < β

Mp−1∑n=−Mp+1

∣∣∣J (p)n

∣∣∣2 , (4.27)

which tests whether a given order Mp sufficiently represents fields and current. In (4.27),

β is a threshold constant which was experimentally determined to be 0.05 for error3 of

approximately 1% [63]. Lower values of β give smaller relative error.

2. We observed that the values of Np required to achieve a given accuracy are larger at high

frequency. Typically, at low frequencies an Np of the order of 1-2 is sufficient due to low

proximity effect, while at higher frequencies an order of 5-6 may be required.

Based on these two insights, we propose the following scheme to adaptively determine Np. We

denote the required frequency points as ωl with l = 1, . . . ,W , and assume that they are sorted

in ascending order.

Phase 1: At the highest frequency, determine the maximum Np which will be used for each

conductor with the following steps [63]:

1. Set l = W , and set Np = 1 for all conductors;

2. Compute G (2.54), Ys (2.38), R(ω) (2.62), and L(ω) (2.63) for ω = ωl;

3Error here is defined as the difference between p.u.l. parameter obtained with order Mp and exact p.u.l.parameter, normalized by exact p.u.l. parameter. Exact p.u.l. parameter are obtained by setting Np to a veryhigh value.


3. Compute J assuming a current of 1 A in each line. A line can be made by a single round

conductor or by a group of conductors, as in the USB cable of Fig. 4.6;

4. Set Mp = Np and check (4.27) for each conductor. If all conductors pass this test, go

to step 5. Otherwise, for the conductors that fail the test increase Np by one, and go to

step 2.

Phase 2: Compute the p.u.l. parameters at the other frequency points, reducing Np when

appropriate [63]:

5. Move to the previous frequency (l = l − 1);

6. Compute Ys (2.38), R(ω) (2.62), and L(ω) (2.63) for ω = ωl;

7. Compute J assuming a current of 1 A in each line;

8. Set Mp = Np − 1 and check (4.27) for each conductor. For the conductors that satisfy

this test, if Np > 1 decrease Np by one (we keep Np ≥ 1 at all times);

9. If discretization has changed, update G. Go to step 5.

The above procedure is applicable to homogeneous lossless surrounding medium. If background

is lossy, then instead of computing G we need to update additional matrices (i.e. Gg, G0, G0,

and Gc). The advantage of this procedure is that it automatically chooses an appropriate

number of basis functions for each conductor. Additionally, the reduction phase (Phase 2),

reduces the number of unknowns at low frequencies making the procedure faster.

4.2.3 Numerical Results

USB Cable

We consider the example of USB 2.0 cable shown in Fig. 4.6 [63]. This cable is made up of 115

round copper conductors. The cable includes lines for high-speed signaling, a line for electric

power, the reference (ground) line and a coaxial shield. The drain line is electrically connected

to the shield.

Fig. 4.7 shows an excellent agreement in resistance and inductance curves obtained using

MoM-SO with adaptive discretization and a MATLAB implementation of the FEM [1]. FEM


-1.5 -1.0 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

mm

mm

Shield

Signal 2

Ground

Power

Drain

Signal 1

Figure 4.6: Typical cross section of a USB 2.0 cable [8].

simulation mesh size was 40, 000 triangles, and it took 23.40 seconds per frequency point to

compute the impedance matrix. On the other hand, MoM-SO with adaptive meshing algorithm

took only took 0.68 seconds per frequency point.

With the Trial-and-Error approach, the simulation time was found to be 75% higher than

adaptive meshing. Adaptive meshing algorithm ensures that the fields and currents are neither

under-estimated nor over-estimated, which leads to faster calculations compared to the trial-

and-error approach. The algorithm adjusts Np for each conductor based on the amount of

proximity effect experienced by the conductor. Fig. 4.8 shows Np used for various frequencies.

These plots demonstrate that the prediction of proximity effect and Np is non-trivial.

In order to perform transient simulations, one also requires capacitance and conductance

parameters of the USB cable. There exist analytic formulas to compute capacitance, but these

formulas neglect proximity effects, which could be significant in our case because the signals

are not shielded. Proximity-aware capacitance can be obtained using numerical methods based

on FEM. In this thesis, we will not consider transient simulation of this cable.

Trial-and-Error approach vs. Adaptive Discretization for Other Geometries

Now we show a numerical comparison between the trial-and-error approach in Sec. 4.2.1, and

the adaptive discretization scheme in Sec. 4.2.2. We compare four different systems: a two-wire

line, a coaxial line, a micro-coaxial assembly, and a USB 2.0. The coaxial line we consider is


103

104

105

106

107

108

109

1010

1

2

3

4

5

6

7

8x 10

−7

Frequency [Hz]

Inducta

nce p

.u.l [H

/m]

← L(s2,s2)

L(s1,s1)

← L(p,p)

L(p,s1) & L(p,s1) →

← L(s1,s2)

MoM−SO

FEM

103

104

105

106

107

108

109

1010

10−1

100

101

Frequency [Hz]

Resis

tance p

.u.l. [ Ω

/m

]

R(s1,s1)

R(p,p)

R(p,s1)

R(s1, s2)MoM−SO

FEM

Figure 4.7: Inductance and resistance of the USB 2.0 cable of Fig. 4.6 computed with MoM-SOand finite elements (FEM). The plots show the self and mutual impedance between the signal1 (s1), signal 2 (s2), and power (p) lines with reference to the ground line. Mutual parametersare shown in magnitude.

made up of a core conductor and a wire screen (50 round solid conductors). The micro-coaxial

assembly (MCX) cable is made up of 8 coaxial lines. MCX is used for high-speed communication

in electronic products with turnable or foldable parts, like laptops [63]. The MCX cable we

consider has a total of 232 solid round conductors.

Table 4.1 shows the comparison between timing, accuracy, and complexity of both estimation

schemes for MoM-SO. In adaptive scheme, all the test cases were performed with the β value


−1 0 1

x 10−3

−1.5

−1

−0.5

0

0.5

1

1.5x 10

−3

(m)

(m)

Freq = 10000 MHz.

1

2

3

4

5

6

−1 0 1

x 10−3

−1.5

−1

−0.5

0

0.5

1

1.5x 10

−3

(m)

(m)

Freq = 3.856620e+01 MHz.

1

2

3

4

5

6

−1 0 1

x 10−3

−1.5

−1

−0.5

0

0.5

1

1.5x 10

−3

(m)

(m)

Freq = 7.278954e+00 MHz.

1

2

3

4

5

6

−1 0 1

x 10−3

−1.5

−1

−0.5

0

0.5

1

1.5x 10

−3

(m)

(m)

Freq = 1.373824e+00 MHz.

1

2

3

4

5

6

−1 0 1

x 10−3

−1.5

−1

−0.5

0

0.5

1

1.5x 10

−3

(m)

(m)

Freq = 1.487352e−01 MHz.

1

2

3

4

5

6

−1 0 1

x 10−3

−1.5

−1

−0.5

0

0.5

1

1.5x 10

−3

(m)

(m)

Freq = 1.000000e−03 MHz.

1

2

3

4

5

6

Figure 4.8: The number of basis functions (Np) used to expand fields and currents inside eachconductor of a USB cable in Sec. 4.2.3 at various frequency points. Each color corresponds toa specific value of Np as indicated in the colorbar.

in (4.27) of 0.05. The accuracy of the two methods were compared against “actual” impedance

results which were obtained using MoM-SO with a very large number of basis functions (Np =

12). From Table 4.1 it is evident that the adaptive estimation scheme is superior than the


trial-and-error approach.

Table 4.1: Comparison of the trial and error (TE) and adaptive (AD) estimation schemes forMoM-SO.

CPU Time (s) Max Error on R(ω)and L(ω) (%)

Problem size N(min-max)

TE AD TE AD TE AD

Two-wires 0.693 0.362 0.922 1.70 14 6-14

Coaxial 8.84 3.38 0.0942 0.0942 357 153-353

MCX 146 57.3 0.376 0.376 1592 696-1592

USB 2.0 39.0 22.2 0.356 0.328 805 345-831

Computations were performed on a workstationwith a 3.5 GHz CPU and 16 GB memory.

4.3 Conclusions

In this chapter, we developed two features for MoM-SO: fields computation and the adaptive

algorithm to predict the number of basis functions. Fields computation can help power engineers

with the design of power cables, and in general with the design of power systems. After running

MoM-SO once, we have access to the fields and equivalent currents on the boundaries of the

conductors and holes. This information is used to calculate the field distribution. The second

feature of automatic basis function is especially useful to make MoM-SO a simple-to-use tool

in the power community. The adaptive discretization algorithm has another advantage as it

prevents overestimation of the number of basis functions, which improves the performance of

MoM-SO as shown in Sec. 4.2.3.

Chapter 5

Conclusions

5.1 Summary

Submarine and underground power cables are in growing demand due to their application in

various new underground and underwater power transmission projects. These cables are highly

complex with hundreds of tightly packed conductors, and their modelling poses an exciting

challenge. Cable models are created from the p.u.l. impedance and admittance parameters,

and they are used to predict transients induced by phenomena such as fault, switching, and

breaker operations in power systems. Existing cable modelling techniques based on analytic

formulas and FEM may be inaccurate or very slow in computing the p.u.l. impedance under

several scenarios of applicative relevance. Hence, an improved modelling technique was needed

at the beginning of this research work.

In this thesis, we addressed that need for a better technique to characterize next-generation

submarine and underground power cables. We improved the state of the art by focusing on

two tangible criterias: accuracy and speed. Our technique is accurate because it includes skin,

proximity, tunnel, and ground return effects. In particular, our technique is better because it

models the following effects that are neglected by analytic formulas

• Proximity effect, which are especially important in submarine and underground cables,

are accurately accounted.

• Ground return effect of a cable buried in a multilayer ground is accurately accounted.

114

Chapter 5. Conclusions 115

• Effect of a tunnel on the p.u.l. impedance is accurately accounted.

Our technique is as accurate as FEM, however it is also much faster than FEM, which makes

MoM-SO a lot more practical to use in power industry.

The proposed approach is faster than FEM because it solves the problem by reducing its

complexity (number of unknowns), and by reducing the number of expensive operations. We

reduced the complexity of the system by employing a surface approach. In order to turn

a volumetric problem to a surface problem we introduced two key concepts that were not

previously found in the literature. These concepts are:

1. The surface admittance operator for a hollow round conductor, which combined with the

surface admittance operator for a solid round conductor, allows us to represent any solid or

hollow round conductor using a single equivalent current distribution on the conductor’s

boundary.

2. The surface admittance operator for a cable-hole system which allows us to represent the

entire cable, including all the conductors inside, and the surrounding hole or tunnel with

a single equivalent current distribution.

As shown throughout this thesis, these two ideas are powerful because we no longer have to

discretize fields or currents inside the conductors. Instead, the discretization of fields and

equivalent currents on the boundaries of the conductors and holes is sufficient. Discretization

of fields and currents only on the boundary means we have to solve for fewer unknowns, which

leads to less time to assemble and solve the system. We further ensured that the number of

basis functions (and unknowns) are neither underestimated nor overestimated by devising an

adaptive algorithm which was presented in Sec. 4.2.2 of this thesis. In order to further speed

up the technique, we reduced the number of computationally expensive operations. One of the

most expensive operations in a typical method of moments technique is the evaluations of the

method of moments integrals. This operation was optimized in our technique by analytically

evaluating all, but one, method of moments integrals. These evaluations are one of the key

contributions of this thesis, and all of them are deferred to the Appendix. Even the one method

of moments integral which was evaluated numerically was partly optimized through analytic


integration techniques, as shown in Sec. 3.5.2. The combined effect of all the fundamental and

mathematical optimization was evident from the timing result of various test cases presented

in this thesis, to list a few

• A speedup of 220 times was observed when simulating the three-phase cable made up of

293 conductors in Sec. 2.6.3.

• A speedup of 2200 times was observed when simulating the pipe-type cable with screens

and armor made up of solid conductor in Sec. 2.6.4.

• A speedup of 1700 times was observed when simulating the system of three single-core

cables placed inside a tunnel in a two-layer ground, as shown in Sec. 3.8.3.

All the speedups given above are relative to the FEM. These results are conclusive enough to

suggest that MoM-SO is significantly faster than FEM, which makes MoM-SO a practical tool

for power cable modelling.

An additional advantage of the proposed technique is that it is simple to use. Analytic

formulas to compute p.u.l. impedance are persistently being used in the commercial power

transient simulation tools because they are simple to use. The adaptive algorithm devised in

Sec. 4.2.2 makes MoM-SO too, a simple to use tool to compute the p.u.l. impedance parameters.

MoM-SO simply needs the user to specify the geometry, and the p.u.l. impedance can be found

right away.

In conclusion, MoM-SO is a very fast, versatile, and simple to use cable modelling tool

that calculates accurately the p.u.l. impedance parameters of any arbitrarily complex cable

structure made up of solid and hollow round conductors. MoM-SO is already being used by

the members of the European consortium on EM Transients that supported this project. This

consortium includes many European transmission system operators and other large industry

partners, such as SINTEF Energi, RTE (Reseau de transport d’electricite), EdF (Electrite de

France), EirGrid PLC, Siemens AG, DONG Energy Power, and Nexans Norway AS to name

a few. Furthermore, in the near future, MoM-SO will be adopted in an EMTP tool used by

SINTEF Energi.


5.2 Contributions

The research work presented in this thesis has resulted in several scientific publications. These

publications are listed below, and the main contributions of each paper are highlighted.

1. U. R. Patel, B. Gustavsen, and P. Triverio, “An Equivalent Surface Current Approach

for the Computation of the Series Impedance of Power Cables with Inclusion of Skin and

Proximity Effects,” IEEE Trans. Power Del., vol. 28, pp. 2474-2482, 2013.

• Application of the surface admittance operator of a solid conductor to calculate the

p.u.l. impedance parameters of cables made up of solid round conductors.

• Analytic formulas to evaluate the Green’s matrix

2. U. R. Patel, B. Gustavsen, and P. Triverio, “MoM-SO: a Fast and Fully-Automated

Method for Resistance and Inductance Computation in High-Speed Cable,” in 17th IEEE

Workshop on Signal and Power Integrity, Paris, France, May 12-15, 2013.

• Application of MoM-SO to electronic cables.

• Automatic basis function estimation.

3. U. R. Patel, B. Gustavsen, and P. Triverio, “Application of the MoM-SO Method for

Accurate Impedance Calculation of Single-Core Cables Enclosed by a Conducting Pipe,”

in 10th International Conference on Power Systems Transient (IPST 2013), Vancouver,

Canada, July 18-20, 2013.

• Application of MoM-SO to pipe-type cables with hollow conductors

• Demonstration of the influence of proximity effect on transient waveforms.

4. U. R. Patel, B. Gustavsen, and P. Triverio, “Proximity-Aware Calculation of Cable Series

Impedance for Systems of Solid and Hollow Conductors,” IEEE Trans. Power Del., in

press.

• Derivation of the surface admittance operator for a hollow conductor.

• Derivation of p.u.l. impedance of systems made up of solid and hollow round con-

ductors.


• Approximate approach to include ground return effect combining MoM-SO with

analytic formulas.

5. U. R. Patel, and P. Triverio, “MoM-SO: a Complete Method for Computing the Impedance

of Cable Systems Including Skin, Proximity, and Ground Return Effects,” IEEE Trans.

Power Del., submitted.

• Derivation of the surface admittance operator for a cable-hole system.

• Inclusion of tunnels or holes.

• Demonstration of proximity effect inside ground.

An additional contribution of this thesis is the inclusion of a multilayered ground model

in the MoM-SO framework. This material, presented in Sec. 3.5, will be soon submitted for

publication.

5.3 Future Directions

Several directions may be explored along the lines drawn by this thesis. Some of these include:

calculation of shunt admittance matrix, derivation of the surface admittance operator for wedge-

shaped conductors, inclusion of finite length effect, and inclusion of twisting of the cables.

Calculation of Shunt Admittance Matrix

In order to solve for voltages and currents in a transmission line model, both the series

impedance and the shunt admittance are required. In this thesis, we only developed a technique

to compute the series impedance. A logical extension of this research work is the computation

of shunt admittance matrix of a cable system using a surface approach.

Surface Operator for Other Conductor Geometries

The technique developed in this thesis was limited to solid and hollow round conductors. Most

power cables are made up of solid and hollow round conductors, but not all. For example,

there exist cables that are made up of wedge-shaped conductors. Currently, wedge-shaped


conductors can be handled by decomposing the conductor into many smaller rectangular and

triangular partitions, whose surface admittance operators are found in the literature [38], but

this approach will be inefficient. Hence, it is better to derive the surface admittance operator

which is specific to a wedge-shaped geometry. This investigation, however, will require the use

of more complex basis functions, which may not be as efficient as the Fourier basis functions

that we employed.

Finite Length Effect

One of the assumptions that we made in our derivation was that all conductors are of infinite

length. In practise, this assumption is not true, and may lead to incorrect impedance calcu-

lations, especially for short-length cables. A finite cable length destroys the symmetry of the

fields across the longitudinal direction, which makes it very challenging to include finite-length

effects of the cable in a 2D solver. An interesting extension to this research work will be to

compensate for the finite length of a cable without resorting to a 3D approach.

Twisted Cables

Many cables contain conductors which are periodically twisted in order to reduce mutual cou-

pling. Currently, we cannot compute impedance for such systems, since we assumed that the

conductors are uniform across the longitudinal directions. Computing the p.u.l. impedance of

a cable inclusive of the twisting effects of the conductors in a 2D approach is also an interesting

extension of the research work presented in this thesis.

Appendix A

Post-Processing of the p.u.l.

Parameters

The algorithms proposed in Chapters 2 and 3 compute the partial p.u.l. impedance, which

relates the scalar potential Vp and current Ip in each conductor of a cable. In this section, we

discuss how we can a) find the p.u.l. impedance when multiple conductors are lumped into a

single line; and b) find the total p.u.l. impedance parameters from the partial p.u.l. parameters

of a cable.

There may be scenarios, eg. in presence of wire screens, where multiple conductors are set

to the same potential and carry the same signal or phase current. In such scenarios, we have

no information regarding the current in each conductor. Instead, we have information on the

total current in many lumped conductors. For example, we may not know what the current in

each conductor of a wire screen is, but we may know the value of total current in the entire

screen. In order to use this information, we must perform reduction on the p.u.l. impedance

parameters that we calculated in Chapters 2 and 3.

For practical applications, we are often interested in computing the total p.u.l. impedance,

a quantity which relates currents and voltages. We need to apply a reduction procedure to turn

the partial p.u.l. impedance parameters to the total p.u.l. impedance parameters by setting

any conductor as reference for the voltages and as return path for the currents [54].

We define a line to be a group of one or more conductors which are kept at the same potential

120

Appendix A. Post-Processing of the p.u.l. Parameters 121

12

3

4

56

7 8

9

Figure A.1: Example to demonstrate the post-processing procedure outlined in Appendix A.Each conductor with the same color belongs to the same line. Two different shades of grayshow two different return lines

and carry the same signal. We classify lines into two types: active and return. The return line

is used as reference for the voltage of the corresponding line, and carries the return current. If

multiple active lines have the same return line, then the total current in that return line is the

sum of the currents in the corresponding active lines. Line voltages are defined as the difference

between the potential of an active line and its corresponding return line. To illustrate this

concept we consider the example in Fig. A.1. This example features nine different conductors,

and five distinct lines, out of which, three are active (in green, blue, and brown) and two are

return lines (in gray). Conductors in each line and their return path are shown in the table

below

Color Type of Line Line # Conductors in the line Return line #

Green Active 1 1, 2, 3 3

Blue Active 2 4, 5 3

Dark gray Return 3 6 N/A

Light gray Return 4 7,8 N/A

Brown Active 5 9 4

A.1 Procedure to Calculate Line Impedances

For a configuration in which multiple conductors are lumped into the same line, we would like to

relate line potential Vl with line current Il. We relate these quantities through the Telegrapher’s


equation

∂Vl

∂z= −ZlIl , (A.1)

where Zl is the impedance matrix of dimension equal to number of lines (active and reference).

In (A.1), vectors Vl and Il contain lines voltages and line currents, respectively, of all lines. For

the example in Fig. A.1, Zl is of dimension 5×5. We next relate line currents Il with conductor

current I through

Il = QI , (A.2)

where Q is made up of ‘1’s and ‘0’s. On the i-th row of Q, ‘1’s are present in the columns

corresponding to the conductors which are part of the i-th line, and all other columns are zero.

For our example, Q reads

Q =

1 1 1 0 0 0 0 0 0

0 0 0 1 1 0 0 0 0

0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 1 1 0

0 0 0 0 0 0 0 0 1

. (A.3)

Through matrix Q we show that the line current is equal to the sum of currents in conductors

of that particular line. Incidence matrix Q can also relate line potentials Vl and conductor

potentials V as

V = QTVl , (A.4)

which enforces that the scalar potentials of all conductors in the same line are identical. The

Telegrapher’s equation which relates conductor currents and conductor potentials is given by

∂V

∂z= −ZI , (A.5)


where Z is the partial p.u.l. impedance that is computed by the theory presented in this thesis.

We substitute (A.2) and (A.4) into (A.5) to get

∂Vl

∂z= −

(QZ−1QT

)−1Il , (A.6)

which implies

Zl =(QZ−1QT

)−1, (A.7)

when we compare (A.6) to (A.1). Hence, we can reduce any conductor impedance matrix Z to

line impedance matrix Zl using (A.7).

A.2 Procedure to Calculate the Total P.u.l. Impedance

Now we compute the total p.u.l. impedance from the partial p.u.l. impedance. The total p.u.l.

impedance relates voltages with currents Ig of all active lines. We store line voltages of each

active line with respect to the reference line in vector Vg. Likewise, the line currents of all

active lines are stored in vector Ig. The Telegrapher’s equation

∂Vg

∂z= −ZgIg , (A.8)

relates the line voltages and currents by the total p.u.l. impedance matrix Zg.

We relate voltages Vg and potentials Vl, and currents Ig and Il through a simple incidence

matrix S

Vg = STVl , (A.9)

Il = SIg . (A.10)

In the above equation, S is made up of ‘1’s, ‘0’s, and ‘-1’s. In the i-th row, we have a “1” in the

column corresponding to the active line’s line number, and “-1” in the column corresponding


to the line number of its return line. So for the example in Fig. A.1,

ST =

1 0 −1 0 0

0 1 −1 0 0

0 0 0 −1 1

. (A.11)

By substituting (A.9), and (A.10) into (A.6), we find

Zg = ST(QZ−1QT

)S . (A.12)

In conclusion, through the procedure outlined in this appendix we can calculate the impedance

of systems where conductors are grouped in a line and have a particular reference line that car-

ries the return current.

Appendix B

Analytic Evaluation of Matrices

B.1 Green’s Matrix G

In this section, we show how to analytically evaluate (2.52). The final results are summarized

at the end in Table B.1. We first substitute (2.44) into (2.52) to get the following integral

G(p,q)n′,n =

1

8π2

ˆ 2π

0

ˆ 2π

0ln∣∣rp(ap, θp)− rq(aq, θ

′q)∣∣ ej(nθ′q−n′θp) dθ′qdθp . (B.1)

In order to solve the above integral we introduce an auxiliary vector

r′′(ap, θp) = rp(ap, θp)− (xqx + yqy) (B.2)

= r′′(cos θ′′x + sin θ′′y

). (B.3)

Additionally, we introduce xq,p = (xq − xp), yp,q = (yq − yp), dpq =√x2p,q + y2p,q, and θp,q =

tan−1 (yq,p/xq,p) .

125

Appendix B. Analytic Evaluation of Matrices 126

Using auxiliary vector r′′(ap, θp), we can rewrite (B.1) as

G(p,q)n′,n =

1

8π2

ˆ 2π

0

ˆ 2π

0ln∣∣r′′(ap, θp)− aq (cos θ′qx + sin θ′qy

)∣∣ ej(nθ′q−n′θp) dθ′qdθp , (B.4)

=1

4π2

ˆ 2π

0e−jn

′θp

ˆ 2π

0ln∣∣r′′ (cos θ′′x + sin θ′′y

)− aq

(cos θ′qx + sin θ′qy

)∣∣ ejnθ′q dθ′q︸︷︷︸fn(ap,θp)

dθp ,

=1

4π2

ˆ 2π

0e−jn

′θp

ˆ 2π

0ln[(r′′)2 + a2q − 2aqr

′′ cos(θ′′ − θ′q)] ejnθ′q

4πdθ′q︸︷︷︸

fn(ap,θp)

dθp ,

The integral marked as fn(ap, θp) has analytic solution [60]. This solution depends on an

auxiliary variable α = |aq/(ap − dpq)|. If we have α < 1 then [60]

fn(ap, θp) =

ln(r′′) if n = 0

− a|n|q ejnθ

′′

2|n|(r′′)|n| if n 6= 0 ,

(B.5)

and if α > 1 then [60]

fn(ap, θp) =

ln(aq) if n = 0

− (r′′)|n|ejnθ′′

2|n|a|n|qif n 6= 0 .

(B.6)

We consider both cases for α (with two subcases) one after another.

First, we consider the case where α > 1, and n = 0. In this case, if we substitute (B.5) into

the last line of (B.4)

G(p,q)n′,0 =

1

4π2

ˆ 2π

0ln∣∣r′′∣∣ e−jn′θp dθ′p ,

=1

4π2

ˆ 2π

0ln |dpq (cos θpqx + sin θpqy) + rp(ap, θp)| e−jn

′θp dθ′p ,

=1

8π2

ˆ 2π

0ln(a2p + d2q,p + 2dpqap cos (θ − θqp)

)e−jn

′θp dθp . (B.7)

The integrand of the equation above is similar to fn(ap, θp), and its solution depends on another


auxiliary variable α = dq,p/ap. If α < 1 then we get

G(p,q)n′,0 =

12π ln ap n′ = 0

− e−jn′θpq

4π|n′|

(− |d|ap

)|n′|n′ 6= 0 ,

(B.8)

and if α > 1 then

G(p,q)n′,0 =

12π ln d n′ = 0

− e−jn′θpq

4π|n′|(−ap

d

)|n′|n′ = 0 .

(B.9)

Now we discuss the second subcase of α < 1 for n 6= 0. We will only consider n > 0, and

obtain, from symmetry, n < 0 later. We substitute (B.5) into (B.4) to obtain

G(p,q)n′,n = − 1

4π2

ˆ 2π

0

ejnθ′′

2|n|

(aqr′′

)|n|e−jn

′θp dθp . (B.10)

The integral above can be rewrite as a contour integral

−janq8π2nan′p

˛zn′−1

(xp,q − jyp,q + z)ndz , (B.11)

where z = ap e−jθp . The integral in (B.1) can be solved using Cauchy-Residue theorem [64].

The following subcases arise in this situation.

• If |dpq| > ap, then we get

G(p,q)n′,n =

0 n′ > 1

− π(2π)2n

anqan′p

(−1)n′ (n−n′−1

−n′)

(xp,q − jyp,q)−n+n′n′ ≤ 0 ,

(B.12)

with(nm

)being the binomial coefficients.

• If |d| = 0, , then we get

G(p,q)n′,n =

0 n′ 6= n

−anq4πnan′p

n′ = n .

(B.13)

• If |dpq| < ap


and if n′ < 1, then we get

G(p,q)n′,n = −

anq4πnan′p

[(−1)−n

′(n− n′ − 1

−n′

)(dx − jdy)−n+n

′

+ (−1)n−1(n− n′ − 1

n− 1

)(−xp,q + jyp,q)

n′−n ] (B.14)

and if n′ ≥ 1, then we get

G(p,q)n′,n =

− anq

4πnan′p

(n′−1n−1

)(−xp,q + jyp,q)

n′−n n′ ≥ n and n′ ≥ n

0 n′ ≥ 1 and n′ < n .

(B.15)

Now we consider the second case of α > 1. In this case, we will first consider n = 0. We

substitute (B.6) into (B.4) to obtain

G(p,q)n′,0 =

1

(2π)2

ˆ 2π

0ln aq e−jn

′θp dθp , (B.16)

which can be evaluated analytically. The final result is

G(p,q)n′,0 =

12π ln aq n′ = 0

0 n′ 6= 0 .

(B.17)

If we have α > 1 and n > 0, then the resulting integral is

G(p,q)n′,n =

1

(2π)2

ˆ 2π

0e−jn

′θ

(−e

jnθ′′

2 |n|

(r′′

aq

)n)dθp . (B.18)

The right-hand side of the above equation can be formulated as

G(p,q)n′,n =

jan′p

8π2nanq

˛z−n

′−1 (xp,q + jyp,q + z)n dz , (B.19)

where z = ap ejθp . The integral in the equation above can be solved by Cauchy-Residue theorem


[64]. The final solution is

G(p,q)n′,n =

−(ap)n

′

4π(aq)nn

(nn′

)(xp,q + jyp,q)

n−n′ n ≥ n′ and n′ ≥ 0

0 otherwise .

(B.20)

Finally, we note that the entries for n < 0 can be obtained by symmetry

G(p,q)n′,n = (G

(p,q)−n′,−n)∗ , (B.21)

where ∗ denotes the complex conjugate.

Summary

We summarize all the solution of all the integrals below

Table B.1: Solution of the integral in (B.1)

Conditions Solution

∣∣∣ aqap−dpq

∣∣∣ < 1 ,dpqap

< 1 , n = 0 (B.8)


∣∣∣ < 1 ,dpqap

> 1 , n = 0 (B.9)


∣∣∣ < 1 ,dpqap

> 1 , n > 0 (B.12)


∣∣∣ < 1 ,dpqap

= 0 , n > 0 (B.13)


∣∣∣ < 1 ,dpqap

< 1 , n > 0 , n′ < 1 (B.14)


∣∣∣ < 1 ,dpqap

< 1 , n > 0 , n′ ≥ 1 (B.15)


∣∣∣ > 1 , n = 0 (B.17)


∣∣∣ > 1 , n > 0 (B.20)


n < 0 (B.21)

B.2 Green’s Matrix Gc

In this section, we discuss how to generate matrix Gc, which appears in (3.37). This matrix

may be expressed as

Gc =

G

(1,1)c . . . G

(1,P )c

.... . .

...

G(P,1)c . . . G

(P,P )c

, (B.22)

where G(p,q)c describes the field on boundary cp of the p-th conductor due to the equivalent

current on the boundary cq of the q-th conductor. In general, G(p,q)c is made up of four blocks

similarly to G in (2.57). Each of the four blocks are generated similarly, hence we only present

here how to generate a single block.

Entry (n′, n) of G(p,q)c is given by

[G(p,q)c

]n′,n

=1

(2π)2

ˆ 2π

0

ˆ 2π

0G (rp(ap, θp), rq(aq, θq)) e

j(nθq−n′θp) dθpdθq

1

(2π)21

4j

ˆ 2π

0

ˆ 2π

0K0(k |rp(ap, θp)− rq(aq, θq)|)ej(nθq−nθp) dθpdθq . (B.23)

In order to evaluate this integral we apply the Bessel function addition theorem (also known as

Graf’s theorem) [51,65]. This theorem states that

exp (jvθR)Kv(R) =∞∑

m=−∞Kv+m(Z)Jm(z) ejmθ , (B.24)

where z and Z are magnitudes of vectors z and Z, and R = |Z− z| is the distance between two

vectors |z| < |Z|. Angles θ and θR are defined according to Figure B.1.


θθR

z

Z R

Figure B.1: Lengths and angles used in (B.24).

We expand the Hankel function in the integrand of (B.23) through the addition theorem

K0

(k |rp(θp)− rq(θq)|

)=

∞∑m=−∞

Km(kr′′)Jm(kaq) ejm(θ′′−θq) for |aq| <∣∣r′′∣∣

∞∑m=−∞

Jm(kr′′)Km(kaq) ejm(θq−θ′′) for |aq| >∣∣r′′∣∣ , (B.25)

where the vector r′′ is defined by the following equality

r′′(θ′′) = (ap cos(θp) + xp − xq)x + (ap sin(θp) + yp − yq)y

= r′′( cos θ′′ x + sin θ′′ y ) . (B.26)

In general, r′′ is a function of θp. However, inequality aq < |r′′| will never depend on θp due

to physical reasons. If the inequality depends on θp then it would imply that two different

contours cp and cq are intersecting which is never true.

Case 1 (|aq| < |r′′|)

We divided our discussion into two cases, depending on whether |aq| < |r′′| or |a|q > |r′′|. We

first consider the case where |aq| < |r′′|. In this situation, we substitute (B.25) into (B.23) to


get:

[G(p,q)c

]n′,n

=1

(2π)21

4j

ˆ 2π

0

ˆ 2π

0

∞∑m=−∞

Km(kr′′)Jm(kaq) ejm(θ′′−θq) ej(nθq−n′θp) dθpdθq

=1

(2π)21

4j

ˆ 2π

0

∞∑m=−∞

Km(kr′′)Jm(kaq) ejmθ′′

e−jn′θp

ˆ 2π

0ejθq(n−m) dθqdθp

=1

(2π)

1

4j

ˆ 2π

0

∞∑m=−∞

Km(kr′′)Jm(kaq) ejmθ′′

e−jn′θp δn,mdθp

=1

(2π)

1

4jJn(kaq)

ˆ 2π

0Kn(kr′′) ejnθ

′′e−jn

′θp dθp . (B.27)

Once again, we apply the addition theorem to the Hankel function in the last line of (B.27) to

get

Kn(kr′′) ejnθ′′

=

for |dpq| > |ap| :

ejn(π+θqp)∞∑

m=−∞Kn+m(kdqp)Jm(kap) ejm(θqp−θp)

for |dpq| < |ap| :

ejnθp∞∑

m=−∞Km+n(kap)Jm(kdqp) ejm(θp−θqp)

, (B.28)

where dqp and θqp are defined using the following equality

dqp(cos θqpx + sin θqpy) = (xq − xp) x + (yq − yp) y . (B.29)

We consider the first case in (B.28). If |dpq| > |ap|, we substitute (B.28) into (B.27) and


solve the integral as follows

[G(p,q)c

]n′,n

=1

(2π)

1

4jJn(kaq)

ˆ 2π

0Kn(kr′′) ejnθ

′′e−jn

′θp dθp

=1

(2π)

1

4jJn(kaq)

ˆ 2π

0ejn(π+θqp)

∞∑m=−∞

Kn+m(kdqp)Jm(kap) ejm(θqp−θp) e−jn′θp dθp

=1

(2π)

1

4jJn(kaq)

∞∑m=−∞

Kn+m(kdqp)Jm(kap) ejn(π+θqp) ejmθqpˆ 2π

0e−jmθp e−jn

′θp dθp

=1

4jJn(kaq)

∞∑m=−∞

Kn+m(kdqp)Jm(kap) ejn(π+θqp) ejmθqp δm,−n′

=1

4jJn(kaq)Kn−n′(kdqp)J−n′(kap) ejn(π+θqp) e−jn

′θqp

=1

4jJn(kaq)Kn−n′(kdqp)J−n′(kap) ejn(π+θqp) e−jn

′θqp . (B.30)

Similarly, when |dpq| < |ap|, substitution of (B.28) into (B.27) allows us to simplify the integral

as

[G(p,q)c

]n′,n

=1

(2π)

1

4jJn(kaq)

ˆ 2π

0Kn(kr′′) ejnθ

′′e−jn

′θp dθp

=1

(2π)

1

4jJn(kaq)

ˆ 2π

0ejn(θpq+π+θp)

∞∑m=−∞

Jm(kdpq)Kn+m(kap) ejm(θp−θpq−π) e−jn′θp dθp

=1

(2π)

1

4jJn(kaq)

∞∑m=−∞

Jm(kdpq)Kn+m(kap) ejn(θpq+π) e−jm(θpq+π)

ˆ 2π

0ejθp(n+m−n

′) dθp

=1

4jJn(kaq)

∞∑m=−∞

Jm(kdpq)Kn+m(kap) ejn(θpq+π) e−jm(θpq+π) δm,n′−n

=1

4jJn(kaq)Jn′−n(kdpq)Kn′(kap) ejn(θpq+π) e−j(n

′−n)(θpq+π)

=1

4jJn(kaq)Jn′−n(kdpq)Kn′(kap) ej(2n−n

′)(θpq+π)

=1

4jJn(kaq)Kn′(kap)Jn′−n(kdpq) e−j(n

′−n)θqp . (B.31)


Case 2 (|aq| > |r′′|)

Now we substitute the second case in (B.25) into (B.23)

[G(p,q)c

]n′,n

=1

(2π)21

4j

ˆ 2π

0

ˆ 2π

0K0

(k |rp(θp)− rq(θq)|

)ej(nθq−n

′θp) dθpdθq

=1

(2π)21

4j

ˆ 2π

0

ˆ 2π

0

∞∑m=−∞

Jm(kr′′)Km(kaq) ejm(θq−θ′′) ej(nθq−n′θp) dθpdθq

=1

(2π)21

4j

ˆ 2π

0

∞∑m=−∞

Jm(kr′′)Km(kaq) e−jmθ′′

e−jn′θp

ˆ 2π

0ej(n+m)θ′ dθqdθp

=1

2π

1

4j

ˆ 2π

0

∞∑m=−∞

Jm(kr′′)Km(kaq) e−jmθ′′

e−jn′θp δm,−ndθp

=1

2π

1

4jK−n(kaq)

ˆ 2π

0J−n(kr′′) ejnθ

′′e−jn

′θp dθp . (B.32)

Next, we expand the Bessel function in the last expression of (B.32) using the following addition

theorem

J−n(kr′′) exp(jnθ′′) =

for |dpq| > |ap| :

ejn(π+θqp)m=∞∑m=−∞

J−n+m(kdqp)Jm(kap) e−jm(θqp−θp)

for |dpq| < |ap| :

ejnθp∞∑

m=−∞J−n+m(kap)Jm(kdqp) e−jm(θp−θqp)

(B.33)

which is analogous of (B.24) for Bessel functions.

When |dpq| > |ap|, we substitute case 1 of (B.33) into (B.32) and evaluate the integral by

doing manipulations similar to (B.31). The result of this evaluation is

[G(p,q)c

]n′,n

=1

4jK−n(kaq)J−n+n′(kdqp)Jn′(kap) e−jn

′θqp ejn(π+θqp) . (B.34)

When |ap| > |dpq|, we substitute case 2 of (B.33) into (B.32) and evaluate the resulting integral

by doing manipulations similar to (B.31). The result of this evaluation is

[G(p,q)c

]n′,n

=1

4jH(2)−n(kaq)J−n′(kap)Jn−n′(kdqp) ej(n−n

′)θqp . (B.35)


Summary

We have now fully solved the double integral (B.23). The table below summarizes the final

analytic formulas for[G

(p,q)c

]n′,n

that we derived.

Case Condition Solution

Case 1 – subcase 1 aq < |r′′| & dpq > ap (B.30)

Case 1 – subcase 2 aq < |r′′| & dpq < ap (B.31)

Case 2 – subcase 1 aq > |r′′| & dpq > ap (B.34)

Case 2 – subcase 2 aq > |r′′| & dpq < ap (B.35)

(B.36)

The given expressions have been verified via numerical integration.

B.3 Green’s Matrix G0

Green’s matrix G0 used in (3.16) requires the evaluation of

[G

(q)0

](n′,n)

=1

(2π)2j

4

ˆ 2π

0

ˆ 2π

0K0

(k∣∣∣rq(aq, θ′q)− r(a, θ)

∣∣∣) ej(nθ′q−n′θ)dθ′qdθ . (B.37)

The equation above is very similar to (B.23) with rp(ap, θp) replaced by r(a, θ). Since both rp

and r trace circles, we can reuse the solution presented in Appendix B.2. Only “Case 1 - subcase

2” from (B.36) is required here because rest of the conditions are never satisfied. Therefore,

entry (n, n′) of G(p)0 is given by

[G

(q)0

]n,n′

=1

4jJn(kaq)Kn′(ka)Jn′−n(kdpq) e−j(n

′−n)θqp , (B.38)

where dqp and θqp are define by the following equality

dqp (cos θqp x + sin θqp y) = (xq − x)x + (yq − y)y . (B.39)


In this section, dqp is defined differently than (B.29). In both cases the physical meaning of the

quantity is the same, that is dqp is the difference between centers of two circular contours. In

Appendix B.2, the two contours where cp and cq, while in this section the circular contours are

c and cq.

B.4 Green’s Matrix G0

Matrix G0, which is used in (3.24), may be decomposed into block matrices just like G0 in

(3.16). Entry (n, n′) of block G(q)0 , is given by

[G

(q)0

](n′,n)

=∂

∂ρ

[1

(2π)2j

4

ˆ 2π

0

ˆ 2π

0K0

(k∣∣∣rq(aq, θ′q)− r(ρ, θ)

∣∣∣) ej(nθ′q−n′θ)dθ′qdθ]ρ=a

. (B.40)

Note that the integral inside the square brackets in (B.40) is equal to the right-hand side of

(B.37) with a replaced by ρ. Therefore, we simplify (B.40)

[G

(q)0

]n,n′

=∂

∂ρ

[1

4jJn(kaq)Kn′(kρ)Jn′−n(kdpq) e−j(n

′−n)θqp

]ρ=a

, (B.41)

which can be evaluated to be

[G

(q)0

]n,n′

=k

4jJn(kaq)K′n′(ka)Jn′−n(kdpq) e−j(n

′−n)θqp , (B.42)

with the same expression for dqp as (B.39).

B.5 Matrix H

Matrix H, which is used in (3.37), can be decomposed into P block matrices as shown in (3.38).

In block H(p), entry (n, n′) is given by

[H(p)

](n,n′)

=1

2π

ˆ 2π

0J|n|

(kρ′)ej(nθ

′−n′θp) dθpdθ′ , (B.43)


for n = −N , . . . , N , and n′ = −Np, . . . , Np. In order to evaluate the right-hand side of the

equation above, we expand the Bessel function through the addition theorem (B.24)

Jn(kρ′) ejnθ′

=∞∑

m=−∞Jm(kd)Jn+m(kap) ej(m+n)θp e−jmθd , (B.44)

where d and θd are defined by the following equality

d (cos θd x + sin θd y) = (x− xp)x + (y − yp)y . (B.45)

Additionally, we use the following identity [51]

Jβ(α) = (−1)βJ−β(α) . (B.46)

By substituting (B.44) into (B.43), and evaluating the resulting integral we obtain

[H(p)

](n,n′)

=

Jn′−n(kd)Jn′(kap) ej(n−n

′)θd for n > 0

(−1)nJn′−n(kd)Jn′(kap) ej(n−n′)θ0 for n < 0

. (B.47)

Bibliography

[1] B. Gustavsen, A. Bruaset, J. Bremnes, and A. Hassel, “A finite element approach for

calculating electrical parameters of umbilical cables,” IEEE Trans. Power Del., vol. 24,

no. 4, pp. 2375–2384, Oct. 2009.

[2] S. Carrell and T. Macalister, “Scottish minister gives green light to controversial 137-mile

power line,” The Guardian, 2010.

[3] U. R. Patel, B. Gustavsen, and P. Triverio, “An Equivalent Surface Current Approach

for the Computation of the Series Impedance of Power Cables with Inclusion of Skin and

Proximity Effects,” IEEE Trans. Power Del., vol. 28, pp. 2474–2482, 2013.

[4] COMSOL Multiphysics. COMSOL, Inc. [Online]. Available: https://www.comsol.com/

[5] U. R. Patel, B. Gustavsen, and P. Triverio, “Application of the MoM-SO Method for

Accurate Impedance Calculation of Single-Core Cables Enclosed by a Conducting Pipe,”

in 10th International Conference on Power Systems Transients (IPST 2013), Vancouver,

Canada, July 18–20 2013.

[6] U. R. Patel, and P. Triverio, “MoM-SO: a Complete Method for Computing the Impedance

of Cable Systems Including Skin, Proximity, and Ground Return Effects,” IEEE Trans.

Power Del., submitted.

[7] U. R. Patel, B. Gustavsen, and P. Triverio, “Proximity-Aware Calculation of Cable Series

Impedance for Systems of Solid and Hollow Conductors,” IEEE Trans. Power Del., 2013,

in press.

138

https://www.comsol.com/

Bibliography 139

[8] T. West and V. Jandhyala, “Universal Serial Bus Signal Integrity Analysis for High-Speed

Signaling,” Technical Report, University of Washington, Dec 2001.

[9] CEA, “Key canadian electricity statistics,” Canadian Electricity Association, Tech. Rep.,

2013. [Online]. Available: www.electricity.ca

[10] J. D. Glover, M. S. Sarma, and T. J. Overbye, Power System Analysis & Design. Cengage

Learning, 2011.

[11] “Vision 2050: The Future of Canada’s Electricity,” Canadian Electricity Association,

Tech. Rep., 2013. [Online]. Available: www.electricity.ca

[12] U.S. Energy Information Administration, “Annual Energy Outlook 2014 with projections

to 2040,” U.S. Energy Information Administration, Tech. Rep., 2014.

[13] ENTSOE, “Scenario Outlook and Adequacy Forecast 2013-2030,” European Network of

Transmission System Operators for Electricity, Tech. Rep., 2013. [Online]. Available:

https://www.entsoe.eu/

[14] ——, “Statistical Factsheet 2013,” European Network of Transmission System Operators

for Electricity, Tech. Rep., 2014. [Online]. Available: https://www.entsoe.eu/

[15] Powerworld simulator. PowerWorld Corporation. [Online]. Available: http://www.

powerworld.com/

[16] U.S.-Canada Power System Outage Task Force, “Final Report on the August 14, 2003

Blackout in the United States and Canada: Causes and Recommendations,” Tech. Rep.,

2004.

[17] J. Edmiston, “Toronto weather creates havoc in city as ice storm leaves 400,000 in the

dark across Ontario,” National Post, Dec. 2013.

[18] J. Martinez-Velasco, Power System Transients. Parameter Determination. CRC Press,

2010.

[19] W. S. Meyer and T. Liu. Alternative Transients Program. [Online]. Available:

http://www.emtp.org/

www.electricity.ca

www.electricity.ca



http://www.powerworld.com/

http://www.powerworld.com/

http://www.emtp.org/

Bibliography 140

[20] EMTP-RV. POWERSYS Corporation. [Online]. Available: http://emtp.com/

[21] PSCAD Homepage. Manitoba Hydro International Limited. [Online]. Available:

https://hvdc.ca/pscad/

[22] B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by

vector fitting,” IEEE Trans. Power Del., vol. 14, pp. 1052–1061, 1999.

[23] N. Hadjsaid, J. C. Sabonnadiere, P. Van Hove, and S. Henry, “European Electric System:

Driving its Modernization [Guest Editorial],” IEEE Power and Energy Magazine, vol. 12,

pp. 18–25, 2014.

[24] “Electricity T&D White Paper,” NRGExpert Energy Intelligence, Tech. Rep., 2013.

[25] D. K. Cheng, Field and Wave Electromagnetics (2nd Edition). New Jersey: Prentice Hall,

1989.

[26] L.M. Wedephol, and D.J. Wilcox, “Transient analysis of underground power-transmission

systems. System-model and wave-propagation characteristics,” Proc. IEEE, vol. 120, no. 2,

pp. 253–260, Feb. 1973.

[27] S. Schelkunoff, “The electromagnetic theory of coaxial transmission lines and cylindrical

shields,” Bell System Technical Journal, vol. 13, pp. 532–579, 1934.

[28] A. Ametani and K. Fuse, “Approximate method for calculating the impedances of mul-

ticonductors with cross section of arbitrary shapes,” Elect. Eng. Jpn., vol. 112, no. 2,

1992.

[29] Y. Yin and H. W. Dommel, “Calculation of frequency-dependent impedances of under-

ground power cables with finite element method,” IEEE Trans. Magn., vol. 25, no. 4, pp.

3025–3027, 1989.

[30] S. Cristina and M. Feliziani, “A finite element technique for multiconductor cable param-

eters calculation,” IEEE Trans. Magn., vol. 25, no. 4, pp. 2986–2988, 1989.

http://emtp.com/

https://hvdc.ca/pscad/

Bibliography 141

[31] E. Comellini, A. Invernizzi, G. Manzoni., “A computer program for determining electrical

resistance and reactance of any transmission line,” IEEE Trans. Power App. Syst., no. 1,

pp. 308–314, 1973.

[32] P. de Arizon and H. W. Dommel, “Computation of cable impedances based on subdivision

of conductors,” IEEE Trans. Power Del., vol. 2, no. 1, pp. 21–27, 1987.

[33] A. Pagnetti, A. Xemard, F. Paladian and C. A. Nucci, “An improved method for the

calculation of the internal impedances of solid and hollow conductors with the inclusion of

proximity effect,” IEEE Trans. Power Del., vol. 27, no. 4, pp. 2063 –2072, Oct. 2012.

[34] R. A. Rivas, and J. R. Martı, “Calculation of frequency-dependent parameters of power

cables: Matrix partitioning techniques,” IEEE Trans. Power Del., vol. 17, no. 4, pp. 1085–

1092, 2002.

[35] K. M. Coperich, A. E. Ruehli, and A. Cangellaris, “Enhanced skin effect for partial-

element equivalent-circuit (peec) models,” IEEE Transactions on Microwave Theory and

Techniques, vol. 48, no. 9, pp. 1435–1442, 2000.

[36] J. D. Morsey, V. I. Okhmatovski, and A. C. Cangellaris, “Finite-thickness conductor models

for full-wave analysis of interconnects with a fast integral equation method,” IEEE Trans.

Adv. Packag., vol. 27, no. 1, pp. 24–33, 2004.

[37] C. Yang and V. Jandhyala, “A time-domain surface integral technique for mixed electro-

magnetic and circuit simulation,” IEEE Trans. Adv. Packag., vol. 28, no. 4, pp. 745–753,

2005.

[38] D. De Zutter, and L. Knockaert, “Skin Effect Modeling Based on a Differential Surface

Admittance Operator,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 8, pp. 2526 – 2538,

Aug. 2005.

[39] J. R. Carson et al., “Wave propagation in overhead wires with ground return,” Bell Syst.

Tech. J, vol. 5, no. 4, pp. 192–6, 1926.

Bibliography 142

[40] M. D’Amore and M. Sarto, “A new formulation of lossy ground return parameters for

transient analysis of multiconductor dissipative lines,” IEEE Trans. Power Del., vol. 12,

no. 1, pp. 303–314, 1997.

[41] A. Semlyen, “Ground return parameters of transmission lines an asymptotic analysis for

very high frequencies,” IEEE Transactions on Power Apparatus and Systems, no. 3, pp.

1031–1038, 1981.

[42] A. Deri, G. Tevan, A. Semlyen, and A. Castanheira, “The complex ground return plane a

simplified model for homogeneous and multi-layer earth return,” IEEE Trans. Power App.

Syst., no. 8, pp. 3686–3693, 1981.

[43] F. Pollaczek, “On the field produced by an infinitely long wire carrying alternating cur-

rent,” Elektrische Nachrichtentechnik, vol. 3, pp. 339–359, 1926.

[44] O. Saad, G. Gaba, and M. Giroux, “A closed-form approximation for ground return

impedance of underground cables,” IEEE Trans. Power Del., vol. 3, pp. 1536–1545, 1996.

[45] E. D. Sunde, Earth conduction effects in transmission systems. Dover Publications New

York, 1968, vol. 7.

[46] M. Nakagawa and K. Iwamoto, “Earth-return impedance for the multi-layer case,” IEEE

Trans. Power App. Syst., vol. 95, no. 2, pp. 671–676, 1976.

[47] A. Morched, B. Gustavsen, M. Tartibi, “A universal model for accurate calculation of

electromagnetic transients on overhead lines and underground cables,” IEEE Trans. Power

Del., vol. 14, no. 3, pp. 1032–1038, 1999.

[48] T. Noda, N. Nagaoka and A. Ametani, “Phase domain modeling of frequency-dependent

transmission lines by means of an ARMA model,” IEEE Trans. Power Del., vol. 11, no. 1,

pp. 401–411, 1996.

[49] C. Paul, Transmission Lines in Digital and Analog Electronic Systems:Signal Integrity and

Crosstalk. Wiley-IEEE Press, 2010.

Bibliography 143

[50] C. A. Balanis, Advanced Engineering Electromagnetics. New York: John Wiley & Sons,

1989.

[51] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas,

Graphs, and Mathematical Tables. New York: Dover, 1964.

[52] A. Konrad, “Integrodifferential finite element formulation of two-dimensional steady-state

skin effect problems,” IEEE Trans. on Magn., vol. 18, no. 1, pp. 284–292, Jan 1982.

[53] A.R. Djordjevic, T.K. Sarkar, and S.M. Rao, “Analysis of Finite Conductivity Cylindrical

Conductors Excited by Axially-Independent TM Electromagnetic Field,” IEEE Trans.

Microw. Theory Tech., vol. 33, no. 10, pp. 960–966, 1985.

[54] C. R. Paul, Analysis of Multiconductor Transmission Lines, 2nd ed. Hoboken, N.J: Wiley,

2007.

[55] F.M. Tesche, “A Simple Model for the Line Parameters of a Lossy Coaxial Cable Filled

With a Nondispersive Dielectric,” IEEE Trans. Electromagn. Compat., vol. 49, pp. 12–17,

2007.

[56] M. Kamon, M. J. Tsuk, and J. K. White, “Fasthenry: A multipole-accelerated 3-d in-

ductance extraction program,” IEEE Transactions on Microwave Theory and Techniques,

vol. 42, no. 9, pp. 1750–1758, 1994.

[57] C. Yang and T. Wang, “A moment method solution for TMz and TEz waves illuminating

two-dimensional objects above a lossy half space,” IEEE Trans. Electromagn. Compat.,

vol. 38, no. 3, pp. 433–440, 1996.

[58] N. Fache, F. Olyslager, and D. De Zutter, Electromagnetic and circuit modelling of multi-

conductor transmission lines. New York: Oxford University Press, 1993.

[59] F. M. Tesche and T. Karlsson, EMC analysis methods and computational models. New

York: John Wiley & Sons, 1997.

[60] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. Academic

Press, 2007.

Bibliography 144

[61] Electromagnetic transients Program, Reference manual (EMTP Theory book). Boneville

Power Administration, 1986.

[62] A. Ametani, “A general formulation of impedance and admittance of cables,” IEEE Trans.

Power App. Syst., no. 3, pp. 902–910, 1980.

[63] U. R. Patel, B. Gustavsen, and P. Triverio, “MoM-SO: a Fast and Fully-Automated Method

for Resistance and Inductance Computation in High-Speed Cable,” in 17th IEEE Workshop

on Signal and Power Integrity, Paris, France, May 12–15 2013.

[64] D. G. Zill, Advanced Engineering Mathematics. Jones & Bartlett Publishing Co, 2009.

[65] G. N. Watson, A treatise on the theory of Bessel functions. Cambridge university press,

1995.

A Surface Admittance Approach For Fast Calculation of the ... · Ee(p) n, and Je (p) n nth-Fourier...

Documents

Transcript of A Surface Admittance Approach For Fast Calculation of the ... · Ee(p) n, and Je (p) n nth-Fourier...