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2D Electrostatic Fields: A Complex Variable Approach [Kõva köide]

(RLC Solutions, USA.)
  • Formaat: Hardback, 362 pages, kõrgus x laius: 234x156 mm, kaal: 453 g, 4 Tables, black and white; 182 Line drawings, black and white; 182 Illustrations, black and white
  • Ilmumisaeg: 17-Sep-2021
  • Kirjastus: CRC Press
  • ISBN-10: 0367769751
  • ISBN-13: 9780367769758
Teised raamatud teemal:
2D Electrostatic Fields: A Complex Variable ApproachSuurem pilt
  • Formaat: Hardback, 362 pages, kõrgus x laius: 234x156 mm, kaal: 453 g, 4 Tables, black and white; 182 Line drawings, black and white; 182 Illustrations, black and white
  • Ilmumisaeg: 17-Sep-2021
  • Kirjastus: CRC Press
  • ISBN-10: 0367769751
  • ISBN-13: 9780367769758
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780367769758.html
Märksõnad:

This book demonstrates how to use functions of a complex variable to solve engineering problems that obey the 2D Laplace equation (and in some cases the 2D Poisson equation). The book is written with the engineer/physicist in mind and the majority of the book focuses on electrostatics.



This book demonstrates how to use functions of a complex variable to solve engineering problems that obey the 2D Laplace equation (and in some cases the 2D Poisson equation). The book was written with the engineer/physicist in mind and the majority of the book focuses on electrostatics. A key benefit of the complex variable approach to electrostatics is the visualization of field lines through the use of field maps. With todays’ powerful computers and mathematical software programs, field maps are easily generated once the complex potential has been determined. Additionally, problems that would have been considered out of scope previously are now easily solved with these mathematical software programs. For example, solutions requiring the use of non-elementary functions such as elliptic and hypergeometric functions would have been viewed as not practical in the past due to the tedious use of look up tables for evaluation. Now, elliptic and hypergeometric functions are built-in functions for most mathematical software programs making their evaluation as easy as a trigonometric function. Key highlights in the book include

  • 2D electrostatics completely formulated in terms of complex variables

  • More than 60 electrostatic field maps

  • Comprehensive treatment for obtaining Green’s functions with conformal mapping

  • Fully worked Schwarz-Christoffel transformations to more than usual number of problems

  • A full chapter devoted to solving practical problems at an advanced level

  • Detailed solutions to all end of chapter problems available on book’s website

Although the text is primarily self-contained, the reader is assumed to have taken differential and integral calculus and introductory courses in complex variables and electromagnetics.

Preface xv
Author xix
Symbols xxi
Chapter 1 Functions of a Complex Variable 1(40)
1.1 Complex Numbers and Variables
1(5)
1.2 Conjugate Coordinates
6(1)
1.3 Analytic Functions
6(2)
1.4 Real and Imaginary Parts of Analytic Functions
8(1)
1.5 Taylor Series
9(1)
1.6 Multi-valued Functions
9(3)
1.7 2D Vectors and Vector Operators
12(6)
1.8 Line Integrals
18(3)
1.9 Divergence Theorem in 2D
21(2)
1.10 Curl Theorem in 2D
23(2)
1.11 Divergence and Curl Theorems in Conjugate Coordinates
25(1)
1.12 Cauchy's First Integral Theorem
25(1)
1.13 Cauchy's Second Integral Theorem
26(3)
1.14 Laurent Series
29(2)
1.15 Classification of Singularities
31(1)
1.16 The Residue Theorem
32(3)
1.17 Green's Identities in 2D
35(4)
References
39(2)
Chapter 2 Electrostatics 41(50)
2.1 Coulomb's Law
41(1)
2.2 Electric Field Intensity
42(1)
2.3 Electric Fields of Dipoles and Multipoles
43(4)
2.4 Continuous Charge Distributions
47(2)
2.5 Gauss's Law in 2D
49(3)
2.6 Polarization
52(3)
2.7 Maxwell's Equations
55(1)
2.8 Boundary Conditions
56(1)
2.9 Electrostatic Potential
57(2)
2.10 Complex Potential
59(2)
2.11 Complex Potential for a Dipole
61(1)
2.12 Complex Potential for a Double Layer
62(2)
2.13 Transforming Poisson's Equation into Laplace's Equation
64(2)
2.14 Equipotential Contours
66(1)
2.15 Lines of Force
67(1)
2.16 Field Maps
68(2)
2.17 Gauss's Law for lnhomogeneous Mediums
70(2)
2.18 Dielectric Boundary Conditions for Φ
72(1)
2.19 Uniqueness Theorem
73(1)
2.20 Conductors and Insulators
74(1)
2.21 Capacitance
75(3)
2.22 Method of Curvilinear Squares
78(3)
2.23 Energy in the Electrostatic Field
81(2)
2.24 Green's Reciprocation Theorem
83(2)
2.25 Induced Charges on Grounded Conductors
85(4)
References
89(2)
Chapter 3 Line Charges 91(46)
3.1 The Complex Potential Plane
91(2)
3.2 Single Line Charge
93(3)
3.2.1 Coaxial Circular Cylinders
93(1)
3.2.2 Two Conductive Plates That Meet at the Origin
94(2)
3.3 Two Line Charges
96(8)
3.3.1 Three Coplanar Plates
97(2)
3.3.2 Two Noncentric Circular Cylinders
99(4)
3.3.3 Conductive Cylinder and a Conductive Plane
103(1)
3.4 Φ for Conductor Boundary in Parametric Form
104(3)
3.5 Green's Function
107(4)
3.6 Method of Images and Green's Functions
111(1)
3.7 Green's Function for a Conductive Cylinder
111(2)
3.8 Green's Function for a Conductive Plane
113(1)
3.9 Green's Function for Two Conducting Planes
114(1)
3.10 Ray Tracing for Planar Dielectric Boundaries
115(4)
3.11 Ray Tracing for Planar Conductor Boundaries
119(1)
3.12 Ray Tracing for Planar Line of Force Boundaries
120(1)
3.13 Ray Tracing for Multiple Planar Boundaries
121(2)
3.14 1D Array of Line Charges
123(3)
3.15 2D Array of Line Charges
126(5)
3.16 Line Charge Between a Grounded Cylinder and a Floating Cylinder
131(2)
3.17 Line Charge Between Two Grounded Concentric Cylinders
133(3)
References
136(1)
Chapter 4 Conformal Mapping I 137(30)
4.1 Defining Conformal Transformations
137(1)
4.2 Transforming Complex Potentials
138(2)
4.3 Translation
140(1)
4.4 Magnification and Rotation
140(1)
4.5 Complex Inversion and Inversion
141(3)
4.6 Inversion of a Point
144(1)
4.7 Inversion of a Triangle with Vertex at zc
144(1)
4.8 Inversion of a Line
145(1)
4.9 Inversion of a Circle
146(2)
4.10 Inversion of Orthogonal Circles
148(1)
4.11 Symmetry Preservation with Inversion
149(2)
4.12 Mobius Transform
151(2)
4.13 Logarithm Transformation
153(1)
4.14 Riemann Sphere
154(2)
4.15 Charges at Infinity
156(2)
4.16 Dielectric Cylinder and Line Charge
158(2)
4.17 Floating Conductive Cylinder and Line Charge
160(1)
4.18 Line Charge Between Two Concentric Conductive Cylinders Revisited
161(1)
4.19 Nonconcentric Cylinders to Concentric Cylinders
162(3)
References
165(2)
Chapter 5 Conformal Mapping II 167(74)
5.1 Riemann Mapping Theorem
167(2)
5.2 Symmetry of Conformal Maps
169(4)
5.3 van der Pauw Theorem
173(2)
5.4 Thompson-Lampard Theorem
175(1)
5.5 Schwarz-Christoffel Transformation
175(5)
5.6 S-C Transformation with bn = infinity
180(1)
5.7 S-C Transformation onto a Unit Disk
181(1)
5.8 Phase of A1
181(1)
5.9 Exterior Angle for a Vertex at Infinity
182(1)
5.10 Boundary Condition for Parallel Lines that Meet at Infinity
182(8)
5.10.1 φs,infinity = π
182(2)
5.10.2 φn,infinity = π
184(1)
5.10.3 φs,infinity = 2π
185(3)
5.10.4 φn,infinity = 2π
188(1)
5.10.5 Example
189(1)
5.11 Polygons with Both Vertices at Infinity
190(3)
5.12 Polygons with One Finite Vertex and One Vertex at Infinity
193(3)
5.13 Polygons with One Finite Vertex and Two Vertices at Infinity
196(5)
5.13.1 General S-C Integral
196(2)
5.13.2 Case 1: ρ = β = 0
198(1)
5.13.3 Case 2: ρ = 1/2, β = 0
199(1)
5.13.4 General Solution
200(1)
5.14 Polygons with Two Finite Vertices and One Vertex at Infinity
201(4)
5.14.1 General S-C Integral
201(1)
5.14.2 Case 1: α = β = 1/2
202(1)
5.14.3 Case 2: α = -1/2, β = 1/2
202(1)
5.14.4 Case 3: β = -1
203(1)
5.14.5 General Solution
204(1)
5.15 Polygons with Two Finite Vertices and Two Vertices at Infinity
205(6)
5.15.1 Case 1
205(2)
5.15.2 Case 2
207(2)
5.15.3 Case 3
209(2)
5.16 The Joukowski Transformation
211(2)
5.17 Polygons with Three Finite Vertices
213(2)
5.18 Polygons with Four Finite Vertices
215(23)
5.18.1 General Rectangle Transformation
216(1)
5.18.2 Two Equal Finite Coplanar Plates
217(1)
5.18.3 Coplanar Center Conductor Between Grounds
218(2)
5.18.4 Coplanar Finite Plate and Semi-infinite Plate
220(1)
5.18.5 Two Coplanar Unequal Finite Plates
221(1)
5.18.6 Quadrilateral with Reflection Symmetry
222(3)
5.18.7 Quadrilateral with One Right Angle
225(3)
5.18.8 Line Charge and Two Finite Coplanar Grounded Plates
228(10)
References
238(3)
Chapter 6 Case Studies with Conformal Mapping 241(70)
6.1 Parallel Plate Capacitor
241(18)
6.1.1 Introduction
241(3)
6.1.2 Zero Thickness, Finite Plate Width Model
244(3)
6.1.3 Zero Thickness, Semi-infinite Plate Width Model
247(2)
6.1.4 Comparison of Zero Plate Thickness Models
249(3)
6.1.5 Finite Thickness, Semi-infinite Plate Width Model
252(3)
6.1.6 Comparison of Finite Plate Thickness Model and FEM Simulations
255(2)
6.1.7 Key Findings and Summary
257(2)
6.2 Characteristic Impedance of Lossless Transmission Lines
259(4)
6.2.1 Introduction
259(1)
6.2.2 Small Diameter Wire Approximation
260(3)
6.2.3 Key Findings and Summary
263(1)
6.3 Charge Imaging on Infinite Plate
263(3)
6.4 Field Plates
266(12)
6.4.1 Introduction
266(3)
6.4.2 Conformal Mapping to a Flat Plate
269(1)
6.4.3 Complex Potential Analysis
270(2)
6.4.4 Electric Field Analysis
272(2)
6.4.5 Induced Charge Density Analysis
274(3)
6.4.6 Key Findings and Summary
277(1)
6.5 Trigate FinFETs
278(13)
6.5.1 Introduction
278(1)
6.5.2 Conformal Mapping to a Flat Plate
279(1)
6.5.3 Electrostatic Potential, ρ not specified
280(1)
6.5.4 Electrostatic Potential for Sheets of Charge
280(3)
6.5.5 Pinch-off Voltage for a Single Sheet of Charge
283(2)
6.5.6 Pinch-off Voltage for Multiple Sheets of Charge
285(1)
6.5.7 Electrostatic Potential for a Region of Charge
286(2)
6.5.8 Pinch-off Voltage for a Region of Charge
288(1)
6.5.9 Infinite Depth Assumption
289(1)
6.5.10 Key Findings and Summary
290(1)
6.6 Uniform Electric Field
291(1)
6.7 Circular Conducting or Dielectric Cylinder in a Uniform Electric Field
292(3)
6.8 Elliptic Dielectric Cylinder in Uniform Electric Field
295(5)
6.9 Limitations for Conformal Mapping
300(4)
6.10 Conclusions
304(3)
References
307(4)
Chapter 7 Other Fields of Physics 311(22)
7.1 Translating to Other Areas of Physics
311(1)
7.2 Steady Electric Current
311(6)
7.3 Magnetostatics
317(4)
7.4 Steady Heat Power Flow
321(3)
7.5 Fluid Dynamics
324(7)
References
331(2)
Appendix A Differentiating An Integral 333(2)
Appendix B Dirac 6-Function 335(4)
Appendix C Elliptic Integrals 339(4)
Appendix D Jacobi's Elliptic Functions 343(2)
Appendix E Gamma And Beta Functions 345(4)
Appendix F Gauss's Hypergeometric Function 349(4)
Appendix G Dilogarithm And Trilogarithm Functions 353(2)
References 355
Index 35
Robert L. Coffie is the Founder and President of RLC Solutions, a semiconductor/microelectronics consulting company. He has designed, developed and matured AlGaN/GaN high electron mobility transistor (HEMT) technologies for RF applications from L-band to Q-band at Northrop Grumman and TriQuint Semiconductor (now Qorvo). He also developed the first JEDEC qualified AlGaN/GaN HEMTs for 600 V power switching applications at Transphorm where he served as Director of Device Engineering.