Low-Frequency Electromagnetic Modeling for Electrical and Biological Systems Using MATLAB E-Book

Sergey N. Makarov

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 835

Veröffentlichungsjahr: 2015

Beschreibung

Provides a detailed and systematic description of the Method of Moments (Boundary Element Method) for electromagnetic modeling at low frequencies and includes hands-on, application-based MATLAB® modules with user-friendly and intuitive GUI and a highly visualized interactive output.

Includes a full-body computational human phantom with over 120 triangular surface meshes extracted from the Visible Human Project® Female dataset of the National library of Medicine and fully compatible with MATLAB® and major commercial FEM/BEM electromagnetic software simulators.

This book covers the basic concepts of computational low-frequency electromagnetics in an application-based format and hones the knowledge of these concepts with hands-on MATLAB® modules. The book is divided into five parts. Part 1 discusses low-frequency electromagnetics, basic theory of triangular surface mesh generation, and computational human phantoms. Part 2 covers electrostatics of conductors and dielectrics, and direct current flow. Linear magnetostatics is analyzed in Part 3. Part 4 examines theory and applications of eddy currents. Finally, Part 5 evaluates nonlinear electrostatics. Application examples included in this book cover all major subjects of low-frequency electromagnetic theory. In addition, this book includes complete or summarized analytical solutions to a large number of quasi-static electromagnetic problems. Each Chapter concludes with a summary of the corresponding MATLAB® modules.

• Combines fundamental electromagnetic theory and application-oriented computation algorithms in the form of stand alone MATLAB® modules
• Makes use of the three-dimensional Method of Moments (MoM) for static and quasistatic electromagnetic problems
• Contains a detailed full-body computational human phantom from the Visible Human Project® Female, embedded implant models, and a collection of homogeneous human shells

Low-Frequency Electromagnetic Modeling for Electrical and Biological Systems UsingMATLAB® is a resource for electrical and biomedical engineering students and practicing researchers, engineers, and medical doctors working on low-frequency modeling and bioelectromagnetic applications.

Leseprobe

CONTENTS

COVER

TITLE PAGE

PREFACE

SUBJECT OF THE TEXT

DISTINCT FEATURES

AUDIENCE

NUMERICAL ALGORITHM

APPLICATION EXAMPLES

ANALYTICAL SOLUTIONS

ORGANIZATION OF THE TEXT

OTHER COMPUTATIONAL SOFTWARE

ACKNOWLEDGMENTS

ABOUT THE COMPANION WEBSITE

PART I: LOW-FREQUENCY ELECTROMAGNETICS. COMPUTATIONAL MESHES. COMPUTATIONAL PHANTOMS

1 CLASSIFICATION OF LOW-FREQUENCY ELECTROMAGNETIC PROBLEMS. POISSON AND LAPLACE EQUATIONS IN INTEGRAL FORM

INTRODUCTION

1.1 CLASSIFICATION OF LOW-FREQUENCY ELECTROMAGNETIC PROBLEMS

PROBLEMS

1.2 POISSON AND LAPLACE EQUATIONS, BOUNDARY CONDITIONS, AND INTEGRAL EQUATIONS

PROBLEMS

REFERENCES

2 TRIANGULAR SURFACE MESH GENERATION AND MESH OPERATIONS

INTRODUCTION

2.1 TRIANGULAR MESH AND ITS QUALITY

PROBLEMS

2.2 DELAUNAY TRIANGULATION. 3D VOLUME AND SURFACE MESHES

PROBLEMS

2.3 MESH OPERATIONS AND TRANSFORMATIONS

PROBLEMS

2.4 ADAPTIVE MESH REFINEMENT AND MESH DECIMATION

2.5 SUMMARY OF MATLAB® SCRIPTS

PROBLEMS

REFERENCES

3 TRIANGULAR SURFACE HUMAN BODY MESHES FOR COMPUTATIONAL PURPOSES

INTRODUCTION

3.1 REVIEW OF AVAILABLE COMPUTATIONAL HUMAN BODY PHANTOMS AND DATASETS

3.2 TRIANGULAR HUMAN BODY SHELL MESHES INCLUDED WITH THE TEXT

PROBLEMS

3.3 VHP-F WHOLE-BODY MODEL INCLUDED WITH THE TEXT

PROBLEMS

REFERENCES

PART II: ELECTROSTATICS OF CONDUCTORS AND DIELECTRICS. DIRECT CURRENT FLOW

4 ELECTROSTATICS OF CONDUCTORS. FUNDAMENTALS OF THE METHOD OF MOMENTS. ADAPTIVE MESH REFINEMENT

INTRODUCTION

4.1 ELECTROSTATICS OF CONDUCTORS. MoM (SURFACE CHARGE FORMULATION)

PROBLEMS

4.2 GAUSSIAN QUADRATURES. POTENTIAL INTEGRALS. ADAPTIVE MESH REFINEMENT

PROBLEMS

4.3 SUMMARY OF MATLAB® MODULES

REFERENCES

5 THEORY AND COMPUTATION OF CAPACITANCE. CONDUCTING OBJECTS IN EXTERNAL ELECTRIC FIELD

INTRODUCTION

5.1 CAPACITANCE DEFINITIONS: SELF-CAPACITANCE

PROBLEMS

5.2 CAPACITANCE OF TWO CONDUCTING OBJECTS

PROBLEMS

5.3 SYSTEMS OF THREE CONDUCTING OBJECTS

PROBLEMS

5.4 ISOLATED CONDUCTING OBJECT IN AN EXTERNAL ELECTRIC FIELD

PROBLEMS

5.5 SUMMARY OF MATLAB® MODULES

REFERENCES

6 ELECTROSTATICS OF DIELECTRICS AND CONDUCTORS

INTRODUCTION

6.1 DIELECTRIC OBJECT IN AN EXTERNAL ELECTRIC FIELD

PROBLEMS

6.2 COMBINED METAL–DIELECTRIC STRUCTURES

PROBLEMS

6.3 APPLICATION EXAMPLE: MODELING CHARGES IN CAPACITIVE TOUCHSCREENS

PROBLEMS

6.4 SUMMARY OF MATLAB® MODULES

REFERENCES

7 TRANSMISSION LINES: TWO-DIMENSIONAL VERSION OF THE METHOD OF MOMENTS

INTRODUCTION

7.1 TRANSMISSION LINES: VALUE OF THE ELECTROSTATIC MODEL—ANALYTICAL SOLUTIONS

PROBLEMS

7.2 THE 2D VERSION OF THE MOM FOR TRANSMISSION LINES

PROBLEMS

7.3 SUMMARY OF MATLAB® MODULES

REFERENCES

8 STEADY-STATE CURRENT FLOW

INTRODUCTION

8.1 BOUNDARY CONDITIONS. INTEGRAL EQUATION. VOLTAGE AND CURRENT ELECTRODES

PROBLEMS

8.2 ANALYTICAL SOLUTIONS FOR DC FLOW IN VOLUMETRIC CONDUCTING OBJECTS

PROBLEMS

8.3 MoM ALGORITHM FOR DC FLOW. CONSTRUCTION OF ELECTRODE MESH

PROBLEMS

8.4 APPLICATION EXAMPLE: EIT

PROBLEMS

8.5 APPLICATION EXAMPLE: tDCS

PROBLEMS

8.6 SUMMARY OF MATLAB®MODULES

REFERENCES

PART III: LINEAR MAGNETOSTATICS

9 LINEAR MAGNETOSTATICS

INTRODUCTION

9.1

INTEGRAL EQUATION OF MAGNETOSTATICS: SURFACE CHARGE METHOD

PROBLEMS

9.2 ANALYTICAL VERSUS NUMERICAL SOLUTIONS: MODELING MAGNETIC SHIELDING

PROBLEMS

9.3 SUMMARY OF MATLAB® MODULES

REFERENCES

10 INDUCTANCE. COUPLED INDUCTORS. MODELING OF A MAGNETIC YOKE

INTRODUCTION

10.1 INDUCTANCE

PROBLEMS

10.2 MUTUAL INDUCTANCE AND SYSTEMS OF COUPLED INDUCTORS

PROBLEMS

10.3 MODELING OF A MAGNETIC YOKE

PROBLEMS

10.4 SUMMARY OF MATLAB® MODULES

REFERENCES

PART IV: THEORY AND APPLICATIONS OF EDDY CURRENTS

11 FUNDAMENTALS OF EDDY CURRENTS

INTRODUCTION

11.1 THREE TYPES OF EDDY CURRENT APPROXIMATIONS

PROBLEMS

11.2 EXACT SOLUTION FOR EDDY CURRENTS WITHOUT SURFACE CHARGES CREATED BY HORIZONTAL LOOPS OF CURRENT

PROBLEMS

11.3 EXACT SOLUTION FOR A SPHERE IN AN EXTERNAL AC MAGNETIC FIELD

PROBLEMS

11.4 A SIMPLE APPROXIMATE SOLUTION FOR EDDY CURRENTS IN A WEAKLY CONDUCTING MEDIUM

PROBLEMS

11.5 SUMMARY OF MATLAB® MODULES

REFERENCES

12 COMPUTATION OF EDDY CURRENTS VIA THE SURFACE CHARGE METHOD

INTRODUCTION

12.1 NUMERICAL SOLUTION IN A WEAKLY CONDUCTING MEDIUM WITH EXTERNAL MAGNETIC FIELD

PROBLEMS

12.2 COMPARISON WITH FEM SOLUTIONS FROM MAXWELL 3D OF ANSYS: SOLUTION CONVERGENCE

PROBLEMS

12.3 EDDY CURRENTS EXCITED BY A COIL

PROBLEMS

12.4 SUMMARY OF MATLAB® MODULES

REFERENCES

PART V: NONLINEAR ELECTROSTATICS

13 ELECTROSTATIC MODEL OF A pn-JUNCTION: GOVERNING EQUATIONS AND BOUNDARY CONDITIONS

INTRODUCTION

13.1 BUILT-IN VOLTAGE OF A pn-JUNCTION

PROBLEMS

13.2 COMPLETE ELECTROSTATIC MODEL OF A pn-JUNCTION

PROBLEMS

REFERENCES

14 NUMERICAL SIMULATION OF pn-JUNCTION AND RELATED PROBLEMS

INTRODUCTION

14.1 ITERATIVE SOLUTION FOR ZERO BIAS VOLTAGE

PROBLEMS

14.2 NUMERICAL SOLUTION FOR THE ELECTRIC FIELD REGION

PROBLEMS

14.3 ANALYTICAL SOLUTION FOR THE DIFFUSION REGION: SHOCKLEY EQUATION

PROBLEMS

14.4 SUMMARY OF MATLAB® MODULES

REFERENCES

INDEX

END USER LICENSE AGREEMENT

List of Tables

Chapter 01

TABLE 1.1 Schematic classification of low-frequency electromagnetic numerical problems

TABLE 1.2 Boundary conditions for Maxwell’s equations [32, 47–50]

Chapter 03

TABLE 3.1 Computational models suggested by IEEE ICES [4, 5]

TABLE 3.2 Surface body meshes for electromagnetic modeling

TABLE 3.3 The major functions within the mesh processing toolbox for the VHP-Female model

TABLE 3.4 Individual tissues of the VHP-Female model as of 11/14/2014

Chapter 04

TABLE 4.1 List of available/tested Gaussian formulas on triangles [19] used in the text—see the MATLAB® script

tri.m

for more information

TABLE 4.2 Accurate capacitance values for a parallel-plate plate capacitor versus the approximation given by Equation 4.31

TABLE 4.3 Relative error percentage of the numerical solution as a function of the radius of the neighboring sphere and the number of integration points in the Gaussian quadrature

Chapter 05

TABLE 5.1 Comparison of numerical and analytical solutions for a conducting sphere

TABLE 5.2 Comparison of numerical and analytical solutions for a conducting circle

TABLE 5.3 Comparison between analytical (Eqs. 5.21 and 5.22 with 10

6

terms) and numerical solutions for the capacitance,

C

, of two conducting spheres of radius

and with 2048 triangles per sphere

TABLE 5.4 Comparison with the analytical solution for a conducting sphere: the total field is computed at the radial distance 1.05

R

along the

z

-axis given

TABLE 5.5 Relative error percentage (vs. the magnitude of the incident field) of the numerical solution as a function of the radius of the neighboring sphere and the number of integration points in the Gaussian quadrature

Chapter 06

TABLE 6.1 Values of integral(6.13)obtained using three different methods

TABLE 6.2 Comparison between numerical and analytical solutions for a dielectric sphere with

:The total vertical field is compared at

given

TABLE 6.3 Relative error percentage of the numerical solution as a function of the radius of the neighboring sphere and the number of integration points in the Gaussian quadrature

TABLE 6.4 Data on charge (pC) or capacitance (pF) for each pad

TABLE 6.5 Data on charge (pC) or capacitance (pF) for each sensing column

Chapter 07

TABLE 7.1 Characteristic impedances of some common wire transmission lines [2]

TABLE 7.2 Characteristic impedances of common printed transmission lines

TABLE 7.3 MoM convergence for the parallel-plate line from Ref. [19]

Chapter 08

TABLE 8.1 Electrode current error and current density error at the cylinder center according to Equation 8.62 using various mesh sizes for the uniform electrode mesh

TABLE 8.2 Solution data for less precise integration accuracy

TABLE 8.3 Solution data for single/double precision of the MoM matrix and for the direct or iterative matrix solver

TABLE 8.4 Electrical properties employed during simulation [73]

TABLE 8.5 Solution data for single/double precision of the MoM matrix and for the direct or iterative matrix solver

Chapter 09

TABLE 9.1 Relative permeabilities for some ferromagnetic materials [16]

TABLE 9.2 Relative error percentage of the numerical solution as a function of the radius of the neighboring sphere and the number of integration points in the Gaussian quadrature

TABLE 9.3 Relative error percentage of the numerical solution as a function of the radius of the neighboring sphere and the number of integration points in the Gaussian quadrature

Chapter 10

TABLE 10.1 Comparison between analytical and numerical solutions for the inductance (H) for one loop of current without a magnetic core from Exercise 10.7

TABLE 10.2 Inductance computations for Exercise 10.6—a comparison of two numerical solutions for the coil inductance (in H). ANSYS MAXWELL 3D solutions are obtained with a final energy error of less than 0.01% with an integration box of 200 × 200 × 200 mm, and final mesh sizes of 2 × 10

6

tetrahedra and higher

TABLE 10.3 Inductance computations for exercise 10.8—a comparison of two numerical solutions for the coil inductance (in H). ANSYS MAXWELL 3D solutions are obtained with a final energy error less than 0.01%, an integration box of 200 × 200 × 200 mm, and final mesh sizes of 0.5 × 10

6

tetrahedra and higher

TABLE 10.4 Relative error percentage of the numerical inductance solution as compared to Equation 10.10

TABLE 10.5 Relative error percentage of the numerical inductance solution as compared to Equation 10.14

TABLE 10.6 Standard frequencies for RFID tags (RFID TX/RX systems). Circuits with coupled inductors are highlighted

TABLE 10.7 Mutual inductance values

TABLE 10.8 Magnetic circuit parameters versus electric circuit parameters (see, for example, [10], Ch. 1)

Chapter 11

TABLE 11.1Skin depth

δ

for some common materials

Chapter 12

Table 12.1 Simulated data for the eddy current density at the observation point

Table 12.2 FEM convergence data for observation point #1. All results are given in [A/m

2

]

Table 12.3 BEM convergence data for observation point #1. All results are given in [A/m

2

]

Table 12.4 FEM convergence data for observation point #2. All results are given in [A/m

2

]

Table 12.5 BEM convergence data for observation point #2. All results are given in [A/m

2

]

Table 12.6 FEM convergence data for observation point #1. All results are given in [A/m

2

]

Table 12.7 BEM convergence data for observation point #1. All results are given in [A/m

2

]

Chapter 13

Table 13.1 Data on concentrations for doped and undoped Si (see [1–5] and early references [6–8])

Table 13.2 Physical constants present in Equations 13.37−13.40 and some necessary properties of Si, GaAs, and Ge

List of Illustrations

Chapter 01

FIGURE 1.1 Physical model of an electric circuit depicting (a) Electrostatics and (b) Magnetostatics scenarios produced by direct current flow. Note that the electric field between the two wires decreases when moving from the source to the load. This is not the case when the wires have the infinite conductivity resulting in zero potential drop. This figure was generated using numerical modeling tools developed in the text.

FIGURE 1.2 Illustration of electrostatic and magnetostatic approximations.

FIGURE 1.3 Illustration of eddy current (quasistatic) approximation.

FIGURE 1.4 (a) Eddy current approximation in a highly conducting medium and (b) eddy current approximation in a weakly conducting medium. Oscillating curves outline the total magnetic field within a conductor.

FIGURE 1.5 Junction structure of a 1N4148 Si switching diode.

FIGURE 1.6 Electrostatic FEM modeling of (a) the geometry and (b), (c) the response of a 345kV power tower.

FIGURE 1.7 A vector field.

FIGURE 1.8 Derivation of integral equations in terms of surface charge density for different media. The normal vector to surface

S

is pointing from inside to outside (the outer normal vector).

FIGURE 1.9 An infinite periodic structure.

FIGURE 1.10 (a) Theory of a corner reflector and (b) the application of the method of images. Image (b) is the profile of a metallic backplane.

Chapter 02

FIGURE 2.1 Mesh generation for a planar rectangle.

FIGURE 2.2 Mesh generation for a planar rectangle with nonuniform nodes.

FIGURE 2.3 (a) A 3D triangular mesh for a sphere with 512 triangles—a 2-manifold mesh; (b) manifold edge; (c) nonmanifold edge; (d) nonmanifold node.

FIGURE 2.4 Radii of the inscribed circle (the largest circle contained in the triangle) and the circumscribed circle (the smallest circle containing the triangle), respectively, for a right-angled isosceles triangle.

r

in

is called the

inradius

and

r

out

is the

circumradius

.

FIGURE 2.5 A 2D mesh for triangle ABC.

FIGURE 2.6 A tetrahedron.

FIGURE 2.7 A 3D cross created using a rectangular mesh.

FIGURE 2.8 A mesh for a brick made by merging planar meshes shown in Figure 2.2.

FIGURE 2.9 A human head mesh. Mesh edges are not shown.

FIGURE 2.10 A human torso mesh. Mesh edges are not shown.

FIGURE 2.11 (a) Delaunay triangulation of a set of four nodes; (b) non-Delaunay triangulation of the same node set.

FIGURE 2.12 Delaunay triangulation for a circle obtained by edge flips: (a) the creation of a new node; (b–d) a series of edge flips that results in high quality elements.

FIGURE 2.13 In parts (a) and (b), we see the results of unconstrained Delaunay triangulation of a non-convex polygon and elements crossing the geometric boundaries; this situation in rectified using constrained Delaunay triangulation with (c) boundary edges included into the mesh and (d) removal of unnecessary triangles.

FIGURE 2.14 Illustration of three-dimensional surface mesh generation for a pelvic bone from the stack of images for Visible Human Project using MATLAB tools. A portion of the original point cloud is shown in (a) with the resulting surface mesh given in (b).

FIGURE 2.15 Two methods of three-dimensional mesh generation. (a) The original dataset is surrounded by additional points with tetrahedra removed from a convex hull to extract the desired shape. (b) The Ball-Pivoting Algorithm [25] is shown.

FIGURE 2.16 (a) Node set 1, (b) node set 2, and (c) node set 3 for Delaunay triangulation in Problem 2.2.1.

FIGURE 2.17 Three Delaunay meshes for use in Problem 2.2.2—(a) the original mesh, (b) the desired mesh of Problem 2.2.2a and (c) the desired mesh for Problem 2.2.2b.

FIGURE 2.18 Two meshes with constrained edges—non-convex geometry for use in Problem 2.2.3. (a) is the output of the MATLAB script

constrained.m

; (b) is desired mesh after script modification.

FIGURE 2.19 Problem geometry with (a) non-intersecting boundaries and (b, c) low quality initial meshes.

FIGURE 2.20 Concept of Laplacian smoothing.

FIGURE 2.21 The (a) initial mesh and (b) the results of Laplacian smoothing with algorithm

WCC

after 9th iteration. Run the script

test_combcircle.m

to this section to see the dynamics of Laplacian smoothing for different methods.

FIGURE 2.22 Non-uniform implicit high-quality mesh generation for simple shapes using DISTMESH [6, 36]. Minimum triangle quality is (a) 0.68 for a disk and (b) 0.63 for the square plate, respectively.

FIGURE 2.23 Boolean operations with simple surface meshes using DISTMESH [6, 36].

FIGURE 2.24 (a) Ray–triangle intersection for a human eye; (b) segment–triangle intersection.

FIGURE 2.25 Three types of intersection of a triangle from a master mesh

X

with various triangles of a slave mesh

Y

. Cases #1 and #3 are equivalent if we treat the master and slave meshes as one set of triangles.

FIGURE 2.26 Checking in/out status for a 2-manifold mesh. Only mesh cross-section is shown. In (a), the observation point is outside the shell and in (b), the observation point is within the shell.

FIGURE 2.27 A combined 3D mesh with three circles.

FIGURE 2.28 A combined mesh for Problem 2.3.3.

FIGURE 2.29 Initial mesh for a rectangle with random nodes subject to Laplacian smoothing. No inner boundaries exist in this case.

FIGURE 2.30 Example (a) sphere and (b) spheroid created with DISTMESH.

FIGURE 2.31 A surface human head mesh (corresponding to a human head phantom from Phantom Laboratory, NY). Mesh edges are shown.

FIGURE 2.32 Two scenarios of edge subdivisions for triangles with the largest error: (a) new nodes at edges of the triangle in question and (b) new nodes on the boundary and adjacent edges only.

FIGURE 2.33 Iterative meshes in the adaptive mesh refinement process. Left column shows the error plot at the previous iteration step; right column the mesh refined according to this error.

FIGURE 2.34 Iterative meshes in the adaptive mesh refinement process for the square plate.

FIGURE 2.35 (a) Edge collapse method; (b) vertex removal.

FIGURE 2.36 Interface outline for module

E00.m.

FIGURE 2.37 Mesh for a circle adaptively refined at its center.

FIGURE 2.38 Mesh for a circle adaptively refined at a selected boundary.

Chapter 03

FIGURE 3.1 Axial cryosection images depicting internal organs within the (a) head and (b) abdomen of the visible female. .

FIGURE 3.2 Human body numerical models developed at the NICT. An image of the surface of each model is shown. A volume rendering image of the pregnant woman model is also shown, providing the shape of the fetus. The block size in these models is 2 mm. There are 51 different tissue types in the adult male, female, and child models and a total of 56 different tissue types, including gestational tissues (fetus, fetal brain, fetal eyes, amniotic fluid, and placenta) in the pregnant woman model [18].

FIGURE 3.3 The Virtual Family of IT’IS—Duke (34-year-old male), Ella (26-year-old female), Billie (11-year-old female), and Thelonious (6-year-old male). Reproduced with permission from The Virtual Family of IT’IS Foundation, Zurich, Switzerland.

FIGURE 3.4 Cyberware’s Model WB4 whole-body color 3D scanner.

FIGURE 3.5 The process of transforming a 3D color scan into a surface triangular mesh: left, the original color scan in MeshLab; center, the surface mesh resulting from the Poisson surface reconstruction in MeshLab; right, the postprocessed mesh of sufficient size and element quality suitable for numerical modeling.

FIGURE 3.6 At the left, the raw surface mesh with holes. At the right, the processed surface mesh.

FIGURE 3.7 Process of generating triangular surface shells from data acquisition via scanning, iterative data processing, and final product.

FIGURE 3.8 Two views (a, b) of the same “crowd” created by inserting multiple human body shells and rotating or translating them to create the desired configuration.

FIGURE 3.9 Workflow utilized in the construction of the VHP-Female computational model.

FIGURE 3.10 (a) Segmentation highlighted for the skull region. (b) Triangular surface mesh constructed for the skull from segmentation.

FIGURE 3.11 Partial VHP-Female model created, processed, and visualized using MATLAB® platform: (a) skeleton bones, (b) anterior view of organs and muscles, and (c) posterior view of organs and muscles. All meshes are processed and displayed within the MATLAB shell.

FIGURE 3.12 Cranium model of the VHP-Female phantom. Cross sections of triangulated surfaces are shown. The CSF shell follows the skull.

FIGURE 3.13 Part separation of the cut VHP-Female phantom in MATLAB as viewed from the (a) left side and (b) lower right. The final result for the head and shoulders is shown in two distinct views.

FIGURE 3.14 Visual mesh processing tool for mesh operations such as (a) the stitching of two separate meshes into (b) a single mesh.

FIGURE 3.15 Triangular mesh intersection algorithm with constrained 2D Delaunay triangulation for individual triangles.

FIGURE 3.16 Three types of triangle intersections between a master mesh X and various triangles of a slave mesh Y.

FIGURE 3.17 Mesh intersection with and without edge collapse: (a) the two intersecting meshes; (b) resulting low quality mesh without edge collapse; (c) resulting high quality mesh with edge collapse.

FIGURE 3.18 Schematic mesh expansion in the normal direction.

FIGURE 3.19 Deformable fat shell of the VHP-Female computational model for simulations of (a) high, (b) medium, (c) and low BMI values.

FIGURE 3.20 (a) Normal and (b) pathological femur bones.

Chapter 04

FIGURE 4.1 Potential calculation for the given electric field. (a) Meaning of the line integral, (b) potential in a uniform electric field, (c) electric potential (or voltage) along a conductor.

FIGURE 4.2 Piecewise-constant basis function on surface triangles of metal conductors.

FIGURE 4.3 A uniform electric field.

FIGURE 4.4 Electric field across a semiconductor pn-junction.

FIGURE 4.5 A nonuniform electric field.

FIGURE 4.6 Parallel-plate capacitor with four triangles.

FIGURE 4.7 Example of barycentric triangle subdivision for (a) the first and (b) second iterations. New nodes are added at the edge centers.

FIGURE 4.8 Geometric representation of the variables in the analytical formulas [21].

FIGURE 4.9 (a) Parallel-plate capacitor including surface charge distribution. Darker colors correspond to a higher absolute charge density. (b) Equipotential lines and lines of force for a capacitor with

in the central cross-sectional plane (the plates are at ±1 V). Simulations were done with MATLAB® module

E21.m.

FIGURE 4.10 Process of adaptive mesh generation starting with a uniform mesh. The capacitor plates have a separation to length ratio (

d

/

a

) of 0.2. Equation 4.29 has been used with

.

FIGURE 4.11 (a) Minimum triangle quality and (b) solution convergence as a function of the number of faces during adaptive meshes for the square-plate capacitor with

. Equation 4.29 has been used with

. Note that the convergence curve for an adaptive mesh refinement process will not necessarily have the smooth and monotonic character shown in Figure 4.11b. Simulations have been done using MATLAB module

E21a.m.

FIGURE 4.12 Curve 1—ratio of the accurate capacitance values (found numerically) to the values predicted by Equation 4.31 calculated via MATLAB module

E21a.m

. Curve 2—earlier result of less accurate simulations without adaptive mesh refinement reported in Ref. [30].

FIGURE 4.13 Triangle for potential integral calculation.

FIGURE 4.14 Interface outline for module

E20a.m.

FIGURE 4.15 Interactive simulation plot from module

E20a.m

after five adaptive steps. The plot updates on every step.

FIGURE 4.16 Interactive simulation plot from module

E21a.m

after five adaptive steps. The plot updates after every step.

FIGURE 4.17 Interface outline for module

E21.m.

FIGURE 4.18 Typical simulation results for module

E21.m

. (a) Parallel-plate capacitor, (b) capacitor made of two arbitrary metal plates, and (c) capacitor made of two perpendicular and shifted plates.

Chapter 05

FIGURE 5.1 Conductor geometry for capacitance definitions: (a) capacitance of two conductors, (b) self-capacitance, (c) capacitance to ground of a conductor, and (d) capacitance of two equal conductors separated by large distances.

FIGURE 5.2 Two geometries with analytical solutions for the self-capacitance. (a) Sphere of radius

R

and (b) thin metal disk of radius

R

.

FIGURE 5.3 Typical self-capacitance values for a 177 cm tall male person. Simulations are done with MATLAB® module

E40.m

.

FIGURE 5.4 (a) The physical model and (b) equivalent circuit for understanding ESD and its effect on a

device under test

(DUT).

FIGURE 5.5 Surface charge distribution over the human body based on an applied voltage of 1 V. The image shows the distinct distribution for (a) standing and (b) kneeling male subjects. Simulations were accomplished using MATLAB® module

E40.m.

FIGURE 5.6 A combined human shell mesh.

FIGURE 5.7 Geometry combination with an analytical solution for the capacitance: two nonconcentric spheres.

FIGURE 5.8 Electric field, potential, and charge distribution for two closely spaced spheres. Simulations are performed with MATLAB® module

E23.m

.

FIGURE 5.9 Typical body to ground capacitance values for a 177 cm tall male person. The ground plane size is 2 m × 2 m (shown not to scale) and is located exactly 0.5 cm below the feet. Simulations are done with MATLAB® module

E43.m

.

FIGURE 5.10 Two human shells in close proximity.

FIGURE 5.11 Two scenarios for a conducting object placed between two capacitor plates: (a) the floating conductor and (b) the grounded conductor.

FIGURE 5.12 Sketch of enforcing the charge conservation conditions for the system with three conductors.

FIGURE 5.13 Test of a capacitive sensor.

FIGURE 5.14 Comparison between experimental data (circles) of Ref. [30] and MoM computations with MATLAB® module

E24.m

(solid curves) for (a) a conducting sphere and (b) a conducting cylinder inserted between the plates of a parallel plate capacitor.

FIGURE 5.15 Problem geometry for Example 5.2.

FIGURE 5.16 Geometry for Problem 5.3.4 with (a) two large plates and (b) two tall but narrow plates.

FIGURE 5.17 The analytical solution for the conducting sphere—charge separation.

FIGURE 5.18 (a) Standing and (b) kneeling human body shells subjected to an applied vertical electric field of 1 V/m: the surface charge distribution on the body surface and the electric potential distribution in the observation plane are shown, and all computations were conducted with MATLAB module

E42.m

.

FIGURE 5.19 Default human body mesh for Problem 5.4.5.

FIGURE 5.20 Interface outline for modules (a)

E20.m

and (b)

E40.m

from Section 5.1.

FIGURE 5.21 Snapshots for typical simulations results: modules

E20.m

(left) and

E40.m

(right) from Section 5.1.

FIGURE 5.22 Snapshots for some simulations results: (a) module

E23.m

and (b)

E43.m

from Section 5.2. Surface charge distribution, electric potential, and electric field distributions are shown.

FIGURE 5.23 Interface outline for module

E24.m

from Section 5.3.

FIGURE 5.24 Snapshots for typical simulation results. (a) Module

E24.m

for a conducting object of zero charge, (b) module

E24.m

for the same conducting object at zero potential, and (c) module

E44.m

for an uncharged body over a finite ground plane below a power conductor.

Chapter 06

FIGURE 6.1 Collection of bound surface charges on dielectric–dielectric interfaces.

FIGURE 6.2 Dielectric sphere in an external electric field.

FIGURE 6.3 Triangle for potential integral calculations.

FIGURE 6.4 Problem geometry in terms of surface charges—free charges on the metal surface and polarization charges on the dielectric surface/interface.

FIGURE 6.5 Structure of the direct MoM solution for a metal–dielectric structure with noncoincident faces.

FIGURE 6.6 Usage of the MoM matrix for all dielectric faces. All metal faces are touching dielectric faces; the total number of metal faces is

N

M

, and the total number of dielectric faces is

.

FIGURE 6.7 Capacitor geometry for testing analytical results and the MoM algorithm.

FIGURE 6.8 Capacitor geometries describing (a) thin metal conductors on opposite sides of a dielectric substrate and (b) a metal capacitor embedded within a dielectric brick.

FIGURE 6.9 Self-capacitance method for a capacitive touchscreen. The (a) human hand model and (b) The touchscreen is not to scale, but are provided for illustration purposes only.

FIGURE 6.10 Mutual-capacitance method for a capacitive touchscreen. Surface charge distribution is illustrated when the driven row is subject to an applied voltage. Finger projection is a circle.

FIGURE 6.11 Realistic finger models: (a) male finger and (b) female finger. Each mesh consists of exactly 1000 triangles.

FIGURE 6.12 Interface outline for module

E31.m

from Section 6.1.

FIGURE 6.13 Snapshots for typical simulation results for module

E31.m

from Section 6.1. A dielectric sphere in air (a) and an air bubble in a dielectric material (b) are studied.

FIGURE 6.14 Interface outline for module

E32.m

from Section 6.2.

FIGURE 6.15 Snapshots for some simulations results for module

E32.m

from Section 6.2.

FIGURE 6.16 Interface outline for module

E33.m

for Section 6.2.

FIGURE 6.17 Interface outline for module

E34.m

.

FIGURE 6.18 Interface outline for module

E35.m

.

FIGURE 6.19 (a) Snapshots for typical simulations results for module

E34.m

and (b) example results from module

E35.m

.

Chapter 07

FIGURE 7.1 Sketch of a transmission line cross section. Electric and magnetic fields are perpendicular to the direction of propagation of the signal on the line. The wave number is given by the vectork.

FIGURE 7.2 Reduction of a transmission line problem to a 2D electrostatic formulation and illustration of Equation 7.14.

FIGURE 7.3 (a) Direct capacitances of the coupled microstrip line; (b) odd-/even-mode capacitances.

FIGURE 7.4 2D line cross section from Figure 7.2 and the (uniform) MoM edge subdivision. Edge centers are marked by white circles.

FIGURE 7.5 (a) Geometry of the parallel-plate waveguide; (b) geometry of the microstrip line.

FIGURE 7.6 Illustration of the problem geometry for Equations 7.33 and 7.34.

FIGURE 7.7 Concept of adaptive mesh refinement with approximately equal total charge per cell. The mesh is “adapted” to the solution behavior providing finer resolution for larger charge densities—the area or total charge for every edge stays approximately

the same

.

FIGURE 7.8 (a) Microstrip transmission line; (b) coplanar-waveguide (CPW) transmission line; and (c) stripline in a finite enclosure.

FIGURE 7.9 Interface outline for module

struct2d.m

.

FIGURE 7.10 Some planar meshes generated by

struct2d.m

: (a) coupled microstrip; (b) quadline; and (c) a birdcage coil model.

FIGURE 7.11 Snapshots for differential line fields from Example 7.3. (a) Electric field for the coupled microstrip line in the even mode; (b) electric field for the coupled microstrip line in the odd mode; and (c) surface charge distribution on the individual microstrip and air–dielectric boundary.

Chapter 08

FIGURE 8.1 Identification of boundary conditions for a conducting body with attached electrodes.

FIGURE 8.2 Three electrode models: (a) Voltage sources - Dirichlet boundary conditions, (b) current sources - Neumann boundary conditions, and (c) mixed sources - mixed boundary conditions.

FIGURE 8.3 Model of a current electrode.

FIGURE 8.4 Conducting cylinder with attached electrodes.

FIGURE 8.5 Conducting cylinder with attached electrodes.

FIGURE 8.6 Voltage electrode above a conducting half-space.

FIGURE 8.7 Problem geometry of a single current electrode on an infinite space with an intermediate layer.

FIGURE 8.8 Problem geometry including source and sink current electrodes.

FIGURE 8.9 Electrode geometry for Problem 8.2.8.

FIGURE 8.10 Chart of MoM matrix calculations for voltage electrodes with Neumann boundary conditions on the conducting interface(s) and Dirichlet boundary conditions on voltage electrodes.

FIGURE 8.11 Human head phantom mesh with small embedded electrodes. The electrode surface is strictly planar.

FIGURE 8.12 Construction of large conformal electrodes. The electrode surface precisely follows the conducting object surface. (a) Edge subdivision, (b) edge swap, and (c) re-triangulation and the final result.

FIGURE 8.13 MoM method “per hand.” A conducting cube with attached electrodes.

FIGURE 8.14 A heterogeneous conducting body with attached electrodes.

FIGURE 8.15 Three steady-state current setups, each with two voltage electrodes. The main object has a conductivity of

σ

; the inner or buried object (if present) has a conductivity of

σ

1

. The first case (a) is used as a test scenario; cases (b) and (c) illustrate the basic concepts behind electric impedance tomography.

FIGURE 8.16 (a) Schematic drawing of electrode positions suited for tDCS—the alphanumeric nomenclature [8, 53]and (b) separation of an FEM mesh of the cerebral cortex into computational subregions including the frontal lobe, the occipital lobe, the parietal lobe, the primary motor cortex, the somatosensory cortex, and the temporal lobe [54].

FIGURE 8.17 Depictions of the contralateral supraorbital (i), extracephalic contralateral shoulder (ii), and extracephalic ipsilateral shoulder (iii) cathode montages [54]. Total current density normalized by the input current density at the electrodes is shown as projected onto three sagittal planes that progressively travel from the left to right on the model using a logarithmic scale.

FIGURE 8.18 Contralateral supraorbital (i), extracephalic contralateral shoulder (ii), and extracephalic ipsilateral shoulder (iii) cathode montages [54]. Total current density normalized by the input current density at the electrodes is shown with surrounding body structures using a logarithmic scale.

FIGURE 8.19 Human head with multiple current or voltage electrodes.

FIGURE 8.20 Interface outline for module

E51.m

.

FIGURE 8.21 Interface outline for module

E52.m

.

FIGURE 8.22 Output of module

E51.m

. (a and b) Solution data for electric potential and current density in the observation plane for a conducting cylinder with two attached electrodes. (c and d) Geometry and solution data (current density and electric potential) for a sphere within a conducting cylinder. (e and f) Identical to (c and d) but the sphere has either zero conductivity (e) or a conductivity 10 times the ambient value (f).

FIGURE 8.23 Output of module

E52.m

. (a) Solution data for a set of voltage electrodes placed on a homogeneous head mesh; (b) solution data for the set of voltage electrodes placed on an inhomogeneous head. Electric potential distribution (curves) and current density distribution (cones) in the observation plane are plotted. Furthermore, the surface charge distribution on tissues boundaries is shown. The observation point is the same in both cases.

Chapter 09

FIGURE 9.1 Magnetostatic boundary conditions.

FIGURE 9.2 Approaching the interface between two media from two different directions.

FIGURE 9.3 A large permanent magnet.

FIGURE 9.4 Magnetic sphere in an external magnetic field.

FIGURE 9.5 Hollow magnetic sphere in an external incident magnetic field.

FIGURE 9.6 Interface outline for modules

E61.m/E62.m.

FIGURE 9.7 Snapshots of typical simulation results for modules

E61.m/E62.m:

magnetic field and scalar magnetic potential.

Chapter 10

FIGURE 10.1 Magnetic flux density generated by circuit #1 (or coil #1).

FIGURE 10.2 Three types of solenoids: (a) an air-filled coil, (b) a coil with a magnetic toroid core, and (c) a coil with a straight cylindrical core.

FIGURE 10.3 (a) Exciter coil, (b) calculation of the flux linkage

λ

via surface integrals.

FIGURE 10.4 Default integration surface for one turn of the coil. The dashed curve is the physical centerline of a bent wire forming a loop of radius

R

. The wire radius is 2

a

.

FIGURE 10.5 Two coupled inductors. Note that we do not use the familiar transformer symbol. Also note the passive reference configuration for each inductance.

FIGURE 10.6 Circuit with two coupled inductors in the phasor form.

FIGURE 10.7 An array of coupled inductors (small coils) located on top of a human-head phantom. This hypothetical setup was tested for applications related to non-invasive brain stimulation.

FIGURE 10.8 Conversion of two coupled inductors to the T-network of three inductances.

FIGURE 10.9 Different configurations of two coupled inductors with coupling coefficent values of (a) approximately 1, (b) less than 1, and (c) much less than one.

FIGURE 10.10 A 120 W inductive power transfer system powered by a 12 V battery. Reproduced with permission from Mesa Systems Co., Medfield, MA.

FIGURE 10.11 Coupling between two coils in a typical near-field wireless-link configuration.

FIGURE 10.12 Received voltage amplitude in coil #2 when the current amplitude in coil #1 is 100 mA. Two coaxial ceramic-core coils with

,

, and a coil length of 10 cm are considered at 1 MHz.

FIGURE 10.13 Modeling mutual coupling of two transmit coils.

FIGURE 10.14 Equivalent circuit with uncoupled inductors.

FIGURE 10.15 Two coil arranged in (a) echelon and (b) row configurations.

FIGURE 10.16 A coupled-inductor circuit.

FIGURE 10.17 A coupled-inductor circuit.

FIGURE 10.18 The coupled-inductor circuit for problem 10.2.14

FIGURE 10.19 An inductor in the form of a solenoid wound around a magnetic core.

FIGURE 10.20 A magnetic circuit with an air gap—a solenoid.

FIGURE 10.21 (a) Generic polygonal shape for the magnetic circuit with an air gap, (b) planar mesh, (c) volumetric mesh obtained by extrusion.

FIGURE 10.22 Effect of magnetic yoke on magnetic field concentration at the central point. (a) Wide yoke, (b) conformal yoke, (c) narrow yoke with penetrating magnets, and (d) narrow yoke with nearly embedded magnets.

FIGURE 10.23 A magnetic circuit.

FIGURE 10.24 A magnetic circuit.

FIGURE 10.25 A magnetic circuit.

FIGURE 10.26 A magnetic circuit.

FIGURE 10.27 Interface outline for modules

E63.m/E64.m.

FIGURE 10.28 Simulation plots from modules

E63.m/E64.m

for helical coils with cylindrical cores. (a and b) Magnetic field distribution with air (ceramic) core, (c and d) magnetic field distribution with a magnetic core.

FIGURE 10.29 Snapshots of the simulation results for modules

E63.m/E64.m:

magnetic field distribution and bound “magnetic” charge density for a magnetic yoke with a wide (a) and narrow (b) gap. The leakage flux is clearly observed in the first case.

FIGURE 10.30 Interface outline for module

E65.m.

FIGURE 10.31 Simulation plots from module

E65.m

for helical coils with cylindrical cores. (a) Mutual coupling with air (ceramic) cores and the resulting magnetic field distribution, (b) mutual coupling with the magnetic cores and the resulting magnetic field.

FIGURE 10.32 Simulation plots from module

E65.m

for helical coils with arbitrary polygonal cores: mutual coupling with the magnetic cores and the resulting magnetic field.

Chapter 11

FIGURE 11.1 Typical geometries without surface charges. (a) A line current that excites eddy currents in a semi-infinite or layered conducting specimen. (b) A loop of current that excites eddy current in a semi-infinite or layered conducting specimen. (c) A rotating uniform magnetic field that excites eddy currents in an infinite conducting cylinder (the cross section of the cylinder is shown).

FIGURE 11.2 Homogeneous conducting half-space with the eddy current densityJ.

FIGURE 11.3 Surface charges arising at the interfaces.

FIGURE 11.4 Problem geometry for rotating magnetic field.

FIGURE 11.5 Approximate solution for the 2D eddy current problem. (a) Rotating magnetic field, (b) rotating eddy current density, and (c) Lorentz force density.

FIGURE 11.6 An eddy current problem.

FIGURE 11.7 An eddy current problem.

FIGURE 11.8 Problem geometry for a single loop.

FIGURE 11.9 Eddy current density generated by a single loop of current.

FIGURE 11.10 Spatial translation of a single loop in Cartesian coordinates.

FIGURE 11.11 Eddy current density generated by a figure-eight coil with two adjacent loops.

FIGURE 11.12 Essence of current behavior in the figure-eight coil.

FIGURE 11.13 Eddy currents in conducting sphere subject to an AC uniform magnetic field.

FIGURE 11.14 Eddy current density magnitude in the great circle normalized to the maximum current density at different ratios

a

/

δ

.

FIGURE 11.15 Eddy currents in a hollow conducting sphere.

FIGURE 11.16 Predictions of Equation 11.87 vs. FEM simulations for a realistic figure-eight coil located above a highly-conducting half-space and tilted by 20° [3, 4]. (a) Problem geometry and (b) maximum eddy current density. Maximum eddy current density is plotted along line

a

1

a

2

in Figure 11.16 as a function of the normal distance

d

, which as indicated in Figure 11.16a. The value

corresponds to the interface.

FIGURE 11.17 Simulation results for a single loop: (a)

xz

-plane and (b)

yz

-plane.

FIGURE 11.18 Simulation results for a figure-eight coil: (a)

xz

-plane and (b)

yz

-plane.

FIGURE 11.19 Simulation results for a small-size coil array: (a)

xz

-plane and (b)

yz

-plane.

FIGURE 11.20 Simulation results for a figure-eight coil.

FIGURE 11.21 Simulation results for a figure-eight coil.

Chapter 12

Figure 12.1 Approaching the interface between two media with different conductivity values.

Figure 12.2 Problem geometry for an external magnetic field generated by a wire.

Figure 12.3 Problem geometry for eddy current computations.

Figure 12.4 Eddy current excitation in a human body: (a) External incident magnetic field and cut plane for eddy current evaluation (sagittal plane view); (b) cut plane for eddy current evaluation and observation points #1 and #2 (coronal plane view); (c) contour plot for eddy current density magnitude in the cut plane in A/m

2

.

Figure 12.5 A conducting cube for eddy current computations.

Figure 12.6 Excitation coil geometry including its magnetic field and magnetic vector potential.

Figure 12.7 Eddy current excitation by a coil in a human body: (a) external incident coil magnetic field and cut plane for eddy current evaluation (sagittal plane view), (b) cut plane for eddy current evaluation and observation point #1 (coronal plane view), (c) contour plot for eddy current density magnitude in the cut plane measured in A/m

2

.

Figure 12.8 Calculation of the magnetic vector potential for a coil.

Figure 12.9 Excitation of eddy currents by a vertical loop of current above a conducting slab.

Figure 12.10 A cross-coil exciter.

Figure 12.11 Interface outline for MATLAB® module

E74.m

. Example menu options are shown.

Figure 12.12 Eddy currents generated in and surface charge distribution residing on a conducting cube in an external AC magnetic field at normal incidence. (a) Geometry of the observation plane and of the external field, (b) surface charge distribution, (c) eddy current flow in the observation plane.

Figure 12.13 (a) Problem initialization and (b) simulated eddy currents in a human body shell and electric potential distribution outside the body excited by an external AC magnetic field.

Figure 12.14 Interface outline for MATLAB® module

E75.m

. Example menu options are shown.

Figure 12.15 Eddy currents and surface charge distribution generated in a conducting cube by a vertical loop of AC current. (a) Geometry of the observation plane and the external field magnetic field generated by a loop, (b) surface charge distribution, (c) eddy current flow and the electric potential in the observation plane.

Figure 12.16 Simulation results for eddy currents excited by a coil as viewed over an opaque (a) and transparent (b) human body shell.

Chapter 13

Figure 13.1 The idea of the “one-way current valve” using two different charge carrier types in one conducting object—the pn-junction. (a) Carrier concentration in equilibrium; (b) effect of reverse bias voltage; (c) effect of forward bias voltage.

Figure 13.2 The Si atom. The four valence electrons are at the highest orbit.

Figure 13.3 Crystal structure of Si and Ge. The valence electrons are shown as small circles. Another important semiconductor, gallium arsenide (GaAs), has a similar crystal lattice [1, 2]. (a) Three-dimensional crystal sketch; (b) two-dimensional bond structure.

Figure 13.4 Simplified energy band diagram for a direct bandgap (

E

G

) semiconductor—after [1, 3].

E

V

is the energy edge of the valence band;

E

C

is the energy edge of the conduction band;

E

G

is the energy bandgap (forbidden gap).

Figure 13.5 (a) Doping Si crystal with a donor impurity (phosphorus); (b) doping Si crystal with an acceptor impurity (boron). The donor atom loses one valence electron and becomes a positively charged ion. Conversely, the acceptor atom becomes the negatively charged ion. Possible electron and hole movement is shown.

Figure 13.6 Donor and acceptor levels in a semiconductor.

E

D

is the energy of the state introduced by a donor atom;

E

A

is the energy of the state introduced by an acceptor atom.

Figure 13.7 Doping profiles in a pn-junction. The length of the junction is

R

.

Figure 13.8 (a) Donor and acceptor concentrations and (b) appearance of the depletion region at thermal equilibrium. (c) shows the electric potential distribution along the junction (black curve) and schematically outlines carrier concentrations (left for holes and right for electrons). (d) shows the conduction and valence bands, respectively.

Figure 13.9 Schematic of penetration of the potential hill by holes (and electrons) due to thermal diffusion.

Figure 13.10 Electric potential distribution across the pn-junction (the potential hill) at different values of the applied bias voltage. (a) The potential is centered about the middle of the junction; (b) it is set to zero in the p-side. Both figures are equivalent since the electric potential is defined to within a constant.

Figure 13.11 A 1N4148 Si switching diode.

Figure 13.12 Doping profile for problem 13.1.8.

Figure 13.13 Three different electric potential distributions (a–c) for problem 13.1.9.

Figure 13.14 Doping profile for problem 13.1.11.

Figure 13.15 Doping profile for problem 13.1.12.

Figure 13.16 Doping profile for problem 13.1.13.

Figure 13.17 Doping profile for problem 13.1.14.

Figure 13.18 Doping profiles

N

D

(

x

) and

N

A

(

x

) and the total impurity concentration

N

E

(

x

) predicted by Equations 13.31 and 13.32.

Figure 13.19 Concept of the diffusion current for holes as a result of the concentration gradient. (a) Conduction current; (b) diffusion current. A similar concept holds for electrons.

Figure 13.20 The concept of voltage across the pn-junction.

Figure 13.21 The complete set of boundary conditions for the pn-junction.

Chapter 14

FIGURE 14.1 Doping profiles, concentration profiles, electric potential

φ

, charge density

ρ

, and electric field

E

through the specimen including the depletion region. The initial guess—the approximate solution of

n

S

(

x

),

p

S

(

x

),

φ

S

(

x

) utilizing the space-charge neutrality condition, is shown by dashed curves. A grayed rectangle shows the width of the depletion region as described in the next section.

FIGURE 14.2 Doping profiles, concentration profiles, electric potential

φ

, and the electric field

E

through the p+pn+ specimen with a total length of 30 µm. The base doping is

. The initial guess—an approximate solution

n

S

(

x

),

p

S

(

x

),

φ

S

(

x

) at space-charge neutrality—is shown by dashed curves.

FIGURE 14.3 Doping profiles, concentration profiles, electric potential

φ

, and the electric field

E

through the p+pn+ specimen with a total length of 30 µm. The base doping is

. The initial guess—an approximate solution given by Equation 14.17 utilizing the space-charge neutrality condition—is very close to the final solution. Therefore, this approximate solution is omitted from the figure.

FIGURE 14.4 Doping profiles for problem 14.1.7.

FIGURE 14.5 Doping profiles for problem 14.1.8.

FIGURE 14.6 Electric field region of the pn-junction. Outside this region, there are no significant variations of the built-in electric potential,

φ

; charge density,

ρ

; or the built-in electric field,

E

, so that the specimen becomes approximately electrically neutral.

FIGURE 14.7 Electric field region (white rectangle) versus the diffusion region (shadowed rectangles). The diffusion region is approximately electrically neutral, and mechanical diffusion of holes and electrons without the built-in electric field dominates.

FIGURE 14.8 Qualitative behavior of

φ

n

(

x

) (bottom curve) and

φ

p

(

x

) (top curve) through the pn-junction for a forward-bias voltage.

FIGURE 14.9 (a) Carrier concentrations, (b) electric potential, (c) charge density, and (d) electric field for Example 13.5 found using the approximation of the electric field region (Eqs. 14.42–14.44). The five different curves correspond to the five bias voltages given as

.

FIGURE 14.10 Depletion-layer capacitance as a function of the applied bias voltage for the silicon pn-junction with

and the exponential doping profiles given by Equations 13.31 and 13.32. The dashed curve is the capacitance found according to Equation 14.49 and the numerical solution for the electric field region; the solid curve is the analytical formula given by Equations 14.52 and 14.53.

FIGURE 14.11 Doping profiles for problem 14.2.7.

FIGURE 14.12 Doping profiles for problem 14.2.8.

FIGURE 14.13 Junction structure of a 1N4148 Si switching diode for problem 14.2.11.

FIGURE 14.14 Dependence of recombination and generation currents upon bias. (a) Thermal equilibrium. (b) Reverse bias. (c) Forward bias..

FIGURE 14.15 Diffusion region and electric field region of the pn-junction. Current densities along the junction are shown. Edges of the electric field region are denoted by

x

P

and

x

N

.

FIGURE 14.16 Dynamics of the minority-carrier concentrations Δ

n

(

x

), Δ

p

(

x

) (at left) and the total junction current densities

J

n

(

x

),

J

p

(

x

) (at right) to scale at different values of the bias voltage.

Guide

Cover

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LOW-FREQUENCY ELECTROMAGNETIC MODELING FOR ELECTRICAL AND BIOLOGICAL SYSTEMS USING MATLAB®

SERGEY N. MAKAROV

GREGORY M. NOETSCHER

ARA NAZARIAN

Copyright © 2016 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New JerseyPublished simultaneously in Canada

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Library of Congress Cataloging-in-Publication Data:

Makarov, Sergey N.Low-frequency electromagnetic modeling for electrical and biological systems using MATLAB® /  Sergey N. Makarov, Gregory M. Noetscher, Ara Nazarian.  pages cm Includes bibliographical references and index.

 ISBN 978-1-119-05256-2 (cloth)1. ELF electromagnetic fields--Mathematical models. 2. Electromagnetic devices--Computer simulation. 3. Electromagnetism--Computer simulation. 4. Bioelectromagnetism–Computer simulation. 5. MATLAB. I. Noetscher, Gregory M., 1978– II. Nazarian, Ara, 1971– III. Title. TK7867.2.M34 2016 621.382′24028553–dc23    2015004420

Cover image courtesy of the authors.

To Natasha, Yen, and Rosalynn

PREFACE

SUBJECT OF THE TEXT

This text provides a systematic, detailed, and design-oriented course on electromagnetic modeling at low frequencies for electrical and biological systems. Low-frequency electromagnetic modeling, which is also known as a static or quasistatic approximation, is a well-established theoretical subject. Today, the role of low-frequency electromagnetic modeling in system design and testing is dominant in many disciplines. Examples include capacitive touchscreens in cellphones, the near-field wireless link between two cellphones or in implanted devices, power electronics, various bioelectromagnetic stimulation setups, modern biomolecular electrostatics, and many others. The text is divided into five parts:

Part I

Low-Frequency Electromagnetics. Computational Meshes. Computational Phantoms

Part II

Electrostatics of Conductors and Dielectrics. Direct Current Flow

Part III

Linear Magnetostatics

Part IV

Theory and Applications of Eddy Currents

Part V

Nonlinear Electrostatics

DISTINCT FEATURES

A unique feature of this text is the combination of fundamental electromagnetic theory and application-oriented computation algorithms realized in the form of distinct MATLAB® modules. The modules are stand alone open-source simulators, which have a user-friendly and intuitive GUI and a highly visualized interactive output. They are accessible to all MATLAB users. No additional MATLAB toolboxes are necessary. The modules may be either employed along with this text or used and modified independently, for both research and demonstration purposes.