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# Fundamentals of Electromagnetics with Matlab - Lonngren & Savov

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v Contents Preface ix Chapter 1 MATLAB and Vectors 2 1.1 MATLAB and a Review of Vectors 3 1.2 Coordinate Systems 16 1.3 Integral Relations for Vectors 29 1.4 Differential Relations for Vectors 39 1.5 Phasors 56 1.6 Conclusion 60 1.7 Problems 60 Chapter 2 Static Electric and Magnetic Fields 70 2.1 Coulomb’s Law 71 2.2 Electric Field 76 2.3 Superposition Principles 78 2.4 Gauss’s Law 87 2.5 Potential Energy and Electric Potential 95 2.6 Numerical Integration 111 2.7 Dielectric Materials 121 2.8 Capacitance 126 2.9 Electrical Currents 130 2.10 Fundamentals of Magnetic Fields 135 2.11 Magnetic Vector Potential and the Biot-Savart Law 146 2.12 Magnetic Forces 155 2.13 Magnetic Materials 166 2.14 Magnetic Circuits 172 2.15 Inductance 175 2.16 Boundary Conditions 181 2.17 Conclusion 189 2.18 Problems 190 LonngrenSavovTOC.fm Page v Wednesday, January 12, 2005 12:05 PM vi Contents Chapter 3 Boundary Value Problems Using MATLAB 204 3.1 Poisson’s and Laplace’s Equations 205 3.2 Analytical Solution in One Dimension—Direct Integration Method 211 3.3 Numerical Solution of a One-Dimensional Equation— Finite Difference Method 222 3.4 Analytical Solution of a Two-Dimensional Equation— Fourier Series Expansion 233 3.5 Finite Difference Method Using MATLAB 243 3.6 Finite Element Method Using MATLAB 249 3.7 Method of Moments Using MATLAB 263 3.8 Conclusion 273 3.9 Problems 274 Chapter 4 Time-Varying Electromagnetic Fields 282 4.1 Faraday’s Law of Induction 283 4.2 Equation of Continuity 296 4.3 Displacement Current 301 4.4 Maxwell’s Equations 307 4.5 Poynting’s Theorem 312 4.6 Time-Harmonic Electromagnetic Fields 318 4.7 Conclusion 322 4.8 Problems 322 Chapter 5 Electromagnetic Wave Propagation 330 5.1 Wave Equation 331 5.2 One-Dimensional Wave Equation 336 5.3 Time-Harmonic Plane Waves 352 5.4 Plane Wave Propagation in a Dielectric Medium 361 5.5 Reflection and Transmission of an Electromagnetic Wave 373 5.6 Waveguide—Propagation with Dispersion 388 5.7 Conclusion 398 5.8 Problems 398 Chapter 6 Transmission Lines 408 6.1 Equivalent Electrical Circuits 409 6.2 Transmission Line Equations 412 6.3 Sinusoidal Waves 418 6.4 Terminations 423 6.5 Impedance on the Transmission Line and Matching 431 6.6 Smith Chart 438 LonngrenSavovTOC.fm Page vi Wednesday, January 12, 2005 12:05 PM vii 6.7 Transient Effects and the Bounce Diagram 449 6.8 Pulse Propagation 457 6.9 Lossy Transmission Lines 462 6.10 Dispersion and Group Velocity 468 6.11 Conclusion 476 6.12 Problems 477 Chapter 7 Radiation of Electromagnetic Waves 484 7.1 Radiation Fundamentals 485 7.2 Short Electric Dipole Antenna 493 7.3 Long Dipole Antenna 501 7.4 Antenna Parameters 506 7.5 Magnetic Dipole Antenna 517 7.6 Aperture Antennas, Diffraction of Waves 522 7.7 Antenna Arrays 535 7.8 Conclusion 546 7.9 Problems 547 Appendix A Mathematical Formulas 552 A.1 Vector Identities 553 A.2 Vector Operations in the Three Coordinate Systems 554 A.3 Summary of the Transformations Between Coordinate Systems 555 A.4 Integral Relations 557 Appendix B Mathematical Foundation of the Finite Element Method 558 B.1 Minimum Energy Condition 559 B.2 Linear Interpolation Coefficients 560 B.3 S-matrix Elements 560 B.4 Decoupled and Coupled Node Potentials 561 B.5 The Matrix Equation for the Unknown Potentials 563 Appendix C Material Parameters 564 Appendix D Transmission Line Parameters of Two Parallel Wires 568 Appendix E Plasma Evolution Adjacent to a Metallic Surface 574 Appendix F References 578 Appendix G Answers 582 Index 635 LonngrenSavovTOC.fm Page vii Wednesday, January 12, 2005 12:05 PM LonngrenSavovTOC.fm Page viii Wednesday, January 12, 2005 12:05 PM iv Preface Electromagnetic Field Theory is one of the fundamental courses that an electrical and computer engineering student is required to take in order to gain a physical understanding of the foundations and the heritage of the field that will occupy his or her professional life for the several decades following graduation. The acquiring of an appreciation for the laws of nature that govern and limit the speed of the smallest computer chip continue to be crucial as this speed approaches the ultimate limit. With the many changes that are occurring in undergraduate curriculums due to the rapid development of new technologies and hence additional courses, it is common to find that only one course in electromagnetic theory is now required for students. However, most of the students are “computer savvy” and have been introduced to and have used MATLAB in their previous courses and are motivated by its ability to create pictures on a computer screen that can help illustrate complicated physical phenomena. Our Approach The underlying philosophy of this one semester undergraduate text is to combine the student’s computer/MATLAB ability that has been gained in earlier courses with an introduction to electromagnetic theory in a coherent fashion in order to stimulate the physical understanding of this difficult topic. Where two terms of Electromagnetic Theory were once required, the challenge of squeezing study into one term can at least be partially met with the use of MATLAB to diminish the drudgery of numerical computations while enhancing understanding of concepts. Therefore, in this text numerous examples are solved using MATLAB along with the creation of several figures throughout the text, and all of the “.m” files are made available for the reader to examine and to modify. We therefore believe that it is possible to take this seemingly abstract material and make it understandable and interesting to the student. This belief has been confirmed by using the material in classes for six years and continually using student feedback to improve it. ch00d_preface.fm Page iv Wednesday, December 22, 2004 9:36 AM v Preface Organization of the Text We review essential features of MATLAB immediately in Chapter 1 in order to sat- isfy the novice’s initial trepidations and incorporate its MATLAB’s capabilities throughout the entire text. After an initial review in Chapter 1 of MATLAB, vector calculus, and phasors, we follow in the footsteps of the giants who have preceded us in and summarize the fundamentals of static electromagnetic fields, including sev- eral examples that the reader may have encountered previously. We discuss analyti- cal and MATLAB techniques in order to illustrate the spatial behavior of a static field in a finite boundary in Chapter 3. The majority of the text is directed toward the presentation of time varying electromagnetic fields and Maxwell’s equations in Chapter 4. From these equations we derive a wave equation that can be most easily understood using a diverse selection of examples from other disciplines. A study of plane electromagnetic waves directly follows this review of waves in Chapter 5. In Chapter 6 the subject of transmission lines is emphasized, owing to its importance in modern technology. This includes MATLAB programs for the creation of a Smith chart and its application. Finally, in Chapter 7 the subject of radiation of electro- magnetic waves is explained, first from a very simple physical interpretation, and then summarizing many of the important parameters associated with antennas. Anticipating the student’s further study of modern topics in electrical engineer- ing, we have tried to present a somewhat broader look in numerical methods than most introductory electromagnetics texts. The Finite Element Method, Method of Moments, and Finite Time Difference are all examples of this effort. With MAT- LAB, we believe most students can handle this material well and will be better pre- pared for their application later. Aids to Learning The Appendices and page layout are designed to