Lecture 2. Introduction to FEM. What it is? What we are solving? Potential formulation Why? Boundary conditions
|
|
- Theodora Lynch
- 5 years ago
- Views:
Transcription
1 Introduction to FEM What it is? What we are solving? Potential formulation Why? Boundary conditions Lecture 2
2 Notation Typical notation on the course: Bolded quantities = matrices (A) and vectors (a) Unit vector = e z x = position = (x,y) Operators (click for explanatory material) Gradient of scalar function f = x fe x + y fe y + z fe z Divergence of vector function f = f x e x + f y e y + f y e y f = x f x + y f y + z f z Curl of vector function f = e x e y e z x y z f x f y f z
3 Notation Special cases: Curl of z-directional vector f = fe z f = f y e x f x e y Curl of a 2D vector (xy-plane) f(x, y) = f x e x + f y e y f = f y x f x y e z Matrix transpose A T and inverse A 1 Dot product of two vectors a b Good to know: Curl of any gradient is always zero: φ 0 Divergence of any curl is always zero: w 0 Wikipedia: Vector calculus identities
4 Idea of FEM We want to find a function A(x) that satisfies some partial differential equation on some domain (Ω) + some boundary conditions For (a convenient) example ν x A x = J x Known functions of the position x
5 Idea of FEM ν x A x = J x IDEA: write A(x) as a weighted sum of some known functions N(x) and some unknown coefficients a: n A x a j N j x = A(x) j=1 The functions N are typically called shape functions (or trial functions) Compare to Fourier series The problem of finding an unknown function now reduced to finding unknown scalar coefficients
6 Idea of FEM In other words: we want to find the coefficients a n A x = a j N j x j=1 in such a way that the equation ν A J is satisfied as well as possible Please note that the end result still is a function It s simply defined by a set of scalar coefficients Takeaway = we have an approximation for A(x) at each and any x useful later
7 What we are solving Next: where does the div-grad equation come from? ν A = J
8 What we are solving? Five field variables (E, D, H, B, J) are needed to present a complete electromagnetic field. D B 0 B E t D H J t Maxwell s equations D E J E B H Material equations + Boundary conditions are needed
9 Boundary conditions = some info about the solution on the outer boundary E.q. values, derivatives, some combination Have to know what happens here Why we need these? Because of (rather complex) maths = if we leave part of the boundary free, we get infinitely many solutions satistying the fixed part + Maxwell s eqs inside. Maxwell s equations satisfied in here
10 Interface conditions for magnetics Boundaries inside problem domain 1 The tangential component of field strength is continuous H H t1 H t2 Comes from H = 0 (on the thin boundary) B The normal component of flux density is continuous B = μh B n1 B n2 Comes from B = 0 Takeaway: the normal component of H and tangential component of B often have jumps whenever μ does.
11 Reluctivity In electromechanics, the reluctivity function ν is often used instead of the permeability H = νb ν = 1 μ Easier notation
12 Simplifications related to timedependence Dynamic problems: Complete Maxwell s equations needed (wave equation) Quasi-static problems: Displacement current neglected D 0 t Static problems: Induced electric field is also neglected B 0 t
13 Next Introducing the vector potential Definition Properties
14 Vector potential Problem: Even in static case, we have two equations for one unknown υb = J B = 0 Ampere s law Gauss s law Let s define B by a vector potential A(x) B = A Now Gauss s law is satisfied automatically since any vector A satisfies ( cuz maths) A = 0
15 Vector potential By definition, the flux density B is B Bxex Byey A A x, y ez A Bx By y A x
16 Vector potential Flux lines In 2D, the vector potential is constant in the direction of the flux density B This can be seen by computing the directed derivative (how much A changes in direction of B): A A A B = A A = x A y y A x = 0
17 Vector potential Flux lines In other words: the flux lines correspond to the equipotential lines of A = isolines = contour lines = lines at which A is constant Compare to the contours ( height lines ) in a map Values of A Flux lines
18 Vector potential Note: vector potential is not unique in 3D * * Several A give the same B A Af => A Af Af A A gauge condition is often included to ensure uniqueness A = 0 In 2D, it is satisfied automatically no need for concern
19 Vector potential formulation Let s combine the potential with Maxwell s equations: Partial differential equation for the vector potential H J H B => B J => A J B A Nice to know: called curl-curl equation
20 Formulation in 2D Current density and vector potential point in the same direction J J x, y ez A Ax, ye z Unit vector of z-direction
21 By writing open Formulation in 2D υ A e z = Je z we get or simply A A e x z Je x y y z ( A) J There s our original example, wooo!
22 Formulation in 2D ( A) J This is the so-called div-grad equation Poisson s equation if ν is constant Appears in wildly diverse problems Thermal problem Electric potential Magnetic scalar potential Will take a look on these later Boundary conditions have different meanings in different problems
23 Boundary conditions = have to know something about the solution on the outer boundary Three common conditions for the vector potential formulation Dirichlet condition = A is constant = field is parallel to boundary Homogeneous Neumann condition = ν A perpendicular to boundary (alt. notation A n n = 0 = field is = A n) (Anti)periodic boundary = A 1 = ±A 2 on some parts of the boundary = used for modelling symmetry sectors
24 Interface conditions Interfaces (= boundaries wholly inside problem domain) are automatically included 1 H The conditions H t1 = H t2 and B n1 = B n2 are consequence of the Maxwells s equations Since we are approx. solving the equations, these conditions are also approx. satisfied B Will take a closer look later during the course, in the exercises B = μh
25 Example problem Magnetic field in the air gap and slots of a machine; = 0 (constant) J J J Nice to know = since ν is constant, the div-grad equation is reduced to the Poisson s equation ν 0 A = J ν 0 A = J 2 A = μ 0 J
26 Boundary conditions in the example The permeability of iron is much larger than those of the air and conductors On the air-iron interface Air Fe 1 Air 1 Fe Air Fe H 0 t Ht => Bt Bt => Bt Bt 0 0 Fe Fe Flux is almost perpendicular to the iron surface (Homogeneous) Neumann boundary condition ν A n = 0
27 Boundary conditions in the example Solution region can be reduced because of symmetry A 0 2 A 0 n A 0 Neumann condition on the right boundary No flux goes through left boundary. Dirichlet boundary condition A is constant: often A = 0 A n 0 2 A J 0 A n 0
28 Solution by finite element method 1. The problem domain is divided into small triangles called elements shape functions are generated based on these 2. Boundary conditions are applied 3. Computer solves the problem and draws nice pics for ya 4. Profit!
29 Assignment 1 Levitation melting b.gif Simulate this in FEMM Axisymmetric geometry Harmonic = sinusoidal current supply
30 Today we Conclusion 1. Understood the basic idea of FEM 2. Introduced the vector potential 3. and used it to derive the div-grad equation from the Maxwell s equations 4. Learned the most typical boundary conditions
Chapter 1 Mathematical Foundations
Computational Electromagnetics; Chapter 1 1 Chapter 1 Mathematical Foundations 1.1 Maxwell s Equations Electromagnetic phenomena can be described by the electric field E, the electric induction D, the
More informationAntennas and Propagation. Chapter 2: Basic Electromagnetic Analysis
Antennas and Propagation : Basic Electromagnetic Analysis Outline Vector Potentials, Wave Equation Far-field Radiation Duality/Reciprocity Transmission Lines Antennas and Propagation Slide 2 Antenna Theory
More informationTECHNO INDIA BATANAGAR
TECHNO INDIA BATANAGAR ( DEPARTMENT OF ELECTRONICS & COMMUNICATION ENGINEERING) QUESTION BANK- 2018 1.Vector Calculus Assistant Professor 9432183958.mukherjee@tib.edu.in 1. When the operator operates on
More informationELE3310: Basic ElectroMagnetic Theory
A summary for the final examination EE Department The Chinese University of Hong Kong November 2008 Outline Mathematics 1 Mathematics Vectors and products Differential operators Integrals 2 Integral expressions
More information1 Fundamentals. 1.1 Overview. 1.2 Units: Physics 704 Spring 2018
Physics 704 Spring 2018 1 Fundamentals 1.1 Overview The objective of this course is: to determine and fields in various physical systems and the forces and/or torques resulting from them. The domain of
More information+ f f n x n. + (x)
Math 255 - Vector Calculus II Notes 14.5 Divergence, (Grad) and Curl For a vector field in R n, that is F = f 1, f 2,..., f n, where f i is a function of x 1, x 2,..., x n, the divergence is div(f) = f
More informationChapter 5. Magnetostatics
Chapter 5. Magnetostatics 5.1 The Lorentz Force Law 5.1.1 Magnetic Fields Consider the forces between charges in motion Attraction of parallel currents and Repulsion of antiparallel ones: How do you explain
More informationChapter 7. Time-Varying Fields and Maxwell s Equations
Chapter 7. Time-arying Fields and Maxwell s Equations Electrostatic & Time arying Fields Electrostatic fields E, D B, H =J D H 1 E B In the electrostatic model, electric field and magnetic fields are not
More informationMathematical Notes for E&M Gradient, Divergence, and Curl
Mathematical Notes for E&M Gradient, Divergence, and Curl In these notes I explain the differential operators gradient, divergence, and curl (also known as rotor), the relations between them, the integral
More informationTransmission Lines and E. M. Waves Prof. R. K. Shevgaonkar Department of Electrical Engineering Indian Institute of Technology, Bombay
Transmission Lines and E. M. Waves Prof. R. K. Shevgaonkar Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture 18 Basic Laws of Electromagnetics We saw in the earlier lecture
More informationRadiation Integrals and Auxiliary Potential Functions
Radiation Integrals and Auxiliary Potential Functions Ranga Rodrigo June 23, 2010 Lecture notes are fully based on Balanis [?]. Some diagrams and text are directly from the books. Contents 1 The Vector
More information4 Finite Element Analysis of a three-phase PM synchronous machine
Assignment 4 1-1 4 Finite Element Analysis of a three-phase PM synchronous machine The goal of the third assignment is to extend your understanding on electromagnetic analysis in FEM. This assignment is
More informationChapter 7. Time-Varying Fields and Maxwell s Equation
Chapter 7. Time-Varying Fields and Maxwell s Equation Electrostatic & Time Varying Fields Electrostatic fields E, D B, H =J D H 1 E B In the electrostatic model, electric field and magnetic fields are
More informationINTRODUCTION to the DESIGN and FABRICATION of IRON- DOMINATED ACCELERATOR MAGNETS
INTRODUCTION to the DESIGN and FABRICATION of IRON- DOMINATED ACCELERATOR MAGNETS Cherrill Spencer, Magnet Engineer SLAC National Accelerator Laboratory Menlo Park, California, USA Lecture # 1 of 2 Mexican
More informationr r 1 r r 1 2 = q 1 p = qd and it points from the negative charge to the positive charge.
MP204, Important Equations page 1 Below is a list of important equations that we meet in our study of Electromagnetism in the MP204 module. For your exam, you are expected to understand all of these, and
More informationElectromagnetic Waves For fast-varying phenomena, the displacement current cannot be neglected, and the full set of Maxwell s equations must be used
Electromagnetic Waves For fast-varying phenomena, the displacement current cannot be neglected, and the full set of Maxwell s equations must be used B( t) E = dt D t H = J+ t D =ρ B = 0 D=εE B=µ H () F
More informationElectromagnetic Theory Prof. D. K. Ghosh Department of Physics Indian Institute of Technology, Bombay
Electromagnetic Theory Prof. D. K. Ghosh Department of Physics Indian Institute of Technology, Bombay Module - 4 Time Varying Field Lecture - 30 Maxwell s Equations In the last lecture we had introduced
More informationUnit-1 Electrostatics-1
1. Describe about Co-ordinate Systems. Co-ordinate Systems Unit-1 Electrostatics-1 In order to describe the spatial variations of the quantities, we require using appropriate coordinate system. A point
More informationElectromagnetic Theory Prof. D. K. Ghosh Department of Physics Indian Institute of Technology, Bombay
Electromagnetic Theory Prof. D. K. Ghosh Department of Physics Indian Institute of Technology, Bombay Lecture -1 Element of vector calculus: Scalar Field and its Gradient This is going to be about one
More informationCHAPTER 2. COULOMB S LAW AND ELECTRONIC FIELD INTENSITY. 2.3 Field Due to a Continuous Volume Charge Distribution
CONTENTS CHAPTER 1. VECTOR ANALYSIS 1. Scalars and Vectors 2. Vector Algebra 3. The Cartesian Coordinate System 4. Vector Cartesian Coordinate System 5. The Vector Field 6. The Dot Product 7. The Cross
More informationChapter 5. Magnetostatics
Chapter 5. Magnetostatics 5.4 Magnetic Vector Potential 5.1.1 The Vector Potential In electrostatics, E Scalar potential (V) In magnetostatics, B E B V A Vector potential (A) (Note) The name is potential,
More informationKINGS COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING QUESTION BANK
KINGS COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING QUESTION BANK SUB.NAME : ELECTROMAGNETIC FIELDS SUBJECT CODE : EC 2253 YEAR / SEMESTER : II / IV UNIT- I - STATIC ELECTRIC
More informationTime-Varying Systems; Maxwell s Equations
Time-Varying Systems; Maxwell s Equations 1. Faraday s law in differential form 2. Scalar and vector potentials; the Lorenz condition 3. Ampere s law with displacement current 4. Maxwell s equations 5.
More informationDIVERGENCE AND CURL THEOREMS
This document is stored in Documents/4C/Gausstokes.tex. with LaTex. Compile it November 29, 2014 Hans P. Paar DIVERGENCE AND CURL THEOREM 1 Introduction We discuss the theorems of Gauss and tokes also
More informationTheory of Electromagnetic Nondestructive Evaluation
EE 6XX Theory of Electromagnetic NDE: Theoretical Methods for Electromagnetic Nondestructive Evaluation 1915 Scholl Road CNDE Ames IA 50011 Graduate Tutorial Notes 2004 Theory of Electromagnetic Nondestructive
More informationThe Magnetic Field
4-50 4. The Magnetic Field 4.9 Mathematical Field Theory A field is defined as a special domain in which a physical quantity is depicted as a function of space and time. Mathematics supports the analytical
More informationUNIT I ELECTROSTATIC FIELDS
UNIT I ELECTROSTATIC FIELDS 1) Define electric potential and potential difference. 2) Name few applications of gauss law in electrostatics. 3) State point form of Ohm s Law. 4) State Divergence Theorem.
More informationHomogenization of the Eddy Current Problem in 2D
Homogenization of the Eddy Current Problem in 2D K. Hollaus, and J. Schöberl Institute for Analysis and Scientific Computing, Wiedner Hauptstr. 8, A-14 Vienna, Austria E-mail: karl.hollaus@tuwien.ac.at
More informationA Brief Revision of Vector Calculus and Maxwell s Equations
A Brief Revision of Vector Calculus and Maxwell s Equations Debapratim Ghosh Electronic Systems Group Department of Electrical Engineering Indian Institute of Technology Bombay e-mail: dghosh@ee.iitb.ac.in
More informationLecture 10: Vector Calculus II
Lecture 10: Vector Calculus II 1. Key points Vector fields Field Lines/Flow Lines Divergence Curl Maple commands VectorCalculus[Divergence] VectorCalculus[Curl] Student[VectorCalculus][FlowLine] Physics[Vector]
More informationMotor-CAD combined electromagnetic and thermal model (January 2015)
Motor-CAD combined electromagnetic and thermal model (January 2015) Description The Motor-CAD allows the machine performance, losses and temperatures to be calculated for a BPM machine. In this tutorial
More informationGauss Law. In this chapter, we return to the problem of finding the electric field for various distributions of charge.
Gauss Law In this chapter, we return to the problem of finding the electric field for various distributions of charge. Question: A really important field is that of a uniformly charged sphere, or a charged
More informationElements of Vector Calculus : Scalar Field & its Gradient
Elements of Vector Calculus : Scalar Field & its Gradient Lecture 1 : Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay Introduction : In this set of approximately 40 lectures
More informationTake-Home Exam 1: pick up on Thursday, June 8, return Monday,
SYLLABUS FOR 18.089 1. Overview This course is a review of calculus. We will start with a week-long review of single variable calculus, and move on for the remaining five weeks to multivariable calculus.
More informationMATH 308 COURSE SUMMARY
MATH 308 COURSE SUMMARY Approximately a third of the exam cover the material from the first two midterms, that is, chapter 6 and the first six sections of chapter 7. The rest of the exam will cover the
More informationElectromagnetic wave propagation. ELEC 041-Modeling and design of electromagnetic systems
Electromagnetic wave propagation ELEC 041-Modeling and design of electromagnetic systems EM wave propagation In general, open problems with a computation domain extending (in theory) to infinity not bounded
More informationIntroduction to Electromagnetic Theory
Introduction to Electromagnetic Theory Lecture topics Laws of magnetism and electricity Meaning of Maxwell s equations Solution of Maxwell s equations Electromagnetic radiation: wave model James Clerk
More informationScientific Computing
Lecture on Scientific Computing Dr. Kersten Schmidt Lecture 4 Technische Universität Berlin Institut für Mathematik Wintersemester 2014/2015 Syllabus Linear Regression Fast Fourier transform Modelling
More informationEECS 117 Lecture 19: Faraday s Law and Maxwell s Eq.
University of California, Berkeley EECS 117 Lecture 19 p. 1/2 EECS 117 Lecture 19: Faraday s Law and Maxwell s Eq. Prof. Niknejad University of California, Berkeley University of California, Berkeley EECS
More informationUNIT-I INTRODUCTION. 1. State the principle of electromechanical energy conversion.
UNIT-I INTRODUCTION 1. State the principle of electromechanical energy conversion. The mechanical energy is converted in to electrical energy which takes place through either by magnetic field or electric
More informationfiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES
Content-ELECTRICITY AND MAGNETISM 1. Electrostatics (1-58) 1.1 Coulomb s Law and Superposition Principle 1.1.1 Electric field 1.2 Gauss s law 1.2.1 Field lines and Electric flux 1.2.2 Applications 1.3
More informationGeneral review: - a) Dot Product
General review: - a) Dot Product If θ is the angle between the vectors a and b, then a b = a b cos θ NOTE: Two vectors a and b are orthogonal, if and only if a b = 0. Properties of the Dot Product If a,
More informationMATH 2400: Calculus III, Fall 2013 FINAL EXAM
MATH 2400: Calculus III, Fall 2013 FINAL EXAM December 16, 2013 YOUR NAME: Circle Your Section 001 E. Angel...................... (9am) 002 E. Angel..................... (10am) 003 A. Nita.......................
More informationPrinciples of Mobile Communications
Communication Networks 1 Principles of Mobile Communications University Duisburg-Essen WS 2003/2004 Page 1 N e v e r s t o p t h i n k i n g. Wave Propagation Single- and Multipath Propagation Overview:
More informationElectromagnetic Forces on Parallel Current-
Page 1 of 5 Tutorial Models : Electromagnetic Forces on Parallel Current-Carrying Wires Electromagnetic Forces on Parallel Current- Carrying Wires Introduction One ampere is defined as the constant current
More informationAntenna Theory (Engineering 9816) Course Notes. Winter 2016
Antenna Theory (Engineering 9816) Course Notes Winter 2016 by E.W. Gill, Ph.D., P.Eng. Unit 1 Electromagnetics Review (Mostly) 1.1 Introduction Antennas act as transducers associated with the region of
More informationPHY752, Fall 2016, Assigned Problems
PHY752, Fall 26, Assigned Problems For clarification or to point out a typo (or worse! please send email to curtright@miami.edu [] Find the URL for the course webpage and email it to curtright@miami.edu
More informationMultiple Integrals and Vector Calculus (Oxford Physics) Synopsis and Problem Sets; Hilary 2015
Multiple Integrals and Vector Calculus (Oxford Physics) Ramin Golestanian Synopsis and Problem Sets; Hilary 215 The outline of the material, which will be covered in 14 lectures, is as follows: 1. Introduction
More informationMixed Finite Elements Method
Mixed Finite Elements Method A. Ratnani 34, E. Sonnendrücker 34 3 Max-Planck Institut für Plasmaphysik, Garching, Germany 4 Technische Universität München, Garching, Germany Contents Introduction 2. Notations.....................................
More informationELECTRO MAGNETIC FIELDS
SET - 1 1. a) State and explain Gauss law in differential form and also list the limitations of Guess law. b) A square sheet defined by -2 x 2m, -2 y 2m lies in the = -2m plane. The charge density on the
More informationVector Calculus, Maths II
Section A Vector Calculus, Maths II REVISION (VECTORS) 1. Position vector of a point P(x, y, z) is given as + y and its magnitude by 2. The scalar components of a vector are its direction ratios, and represent
More informationPhysics 6303 Lecture 2 August 22, 2018
Physics 6303 Lecture 2 August 22, 2018 LAST TIME: Coordinate system construction, covariant and contravariant vector components, basics vector review, gradient, divergence, curl, and Laplacian operators
More informationField and Wave Electromagnetic
Field and Wave Electromagnetic Chapter7 The time varying fields and Maxwell s equation Introduction () Time static fields ) Electrostatic E =, id= ρ, D= εe ) Magnetostatic ib=, H = J, H = B μ note) E and
More informationDifferential Operators and the Divergence Theorem
1 of 6 1/15/2007 6:31 PM Differential Operators and the Divergence Theorem One of the most important and useful mathematical constructs is the "del operator", usually denoted by the symbol Ñ (which is
More informationNotes 19 Gradient and Laplacian
ECE 3318 Applied Electricity and Magnetism Spring 218 Prof. David R. Jackson Dept. of ECE Notes 19 Gradient and Laplacian 1 Gradient Φ ( x, y, z) =scalar function Φ Φ Φ grad Φ xˆ + yˆ + zˆ x y z We can
More informationWeighted Regularization of Maxwell Equations Computations in Curvilinear Polygons
Weighted Regularization of Maxwell Equations Computations in Curvilinear Polygons Martin Costabel, Monique Dauge, Daniel Martin and Gregory Vial IRMAR, Université de Rennes, Campus de Beaulieu, Rennes,
More informationBrief Review of Vector Algebra
APPENDIX Brief Review of Vector Algebra A.0 Introduction Vector algebra is used extensively in computational mechanics. The student must thus understand the concepts associated with this subject. The current
More informationApplied'&'Computa/onal'Electromagne/cs (ACE) Part/III Introduc8on/to/the/Finite/Element/Technique/for/ Electromagne8c/Modelling
Applied'&'omputa/onal'Electromagne/cs (AE) Part/III Introduc8on/to/the/Finite/Element/Technique/for/ Electromagne8c/Modelling 68 lassical and weak formulations Partial differential problem lassical formulation
More information43.1 Vector Fields and their properties
Module 15 : Vector fields, Gradient, Divergence and Curl Lecture 43 : Vector fields and their properties [Section 43.1] Objectives In this section you will learn the following : Concept of Vector field.
More informationMultiple Integrals and Vector Calculus: Synopsis
Multiple Integrals and Vector Calculus: Synopsis Hilary Term 28: 14 lectures. Steve Rawlings. 1. Vectors - recap of basic principles. Things which are (and are not) vectors. Differentiation and integration
More informationDetermination of a Synchronous Generator Characteristics via Finite Element Analysis
SERBIAN JOURNAL OF ELECTRICAL ENGINEERING Vol. 2, No. 2, November 25, 157-162 Determination of a Synchronous Generator Characteristics via Finite Element Analysis Zlatko Kolondzovski 1, Lidija Petkovska
More informationCONSIDER a simply connected magnetic body of permeability
IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 1, JANUARY 2014 7000306 Scalar Potential Formulations for Magnetic Fields Produced by Arbitrary Electric Current Distributions in the Presence of Ferromagnetic
More information8.03 Lecture 12. Systems we have learned: Wave equation: (1) String with constant tension and mass per unit length ρ L T v p = ρ L
8.03 Lecture 1 Systems we have learned: Wave equation: ψ = ψ v p x There are three different kinds of systems discussed in the lecture: (1) String with constant tension and mass per unit length ρ L T v
More informationINTRODUCTION ELECTROSTATIC POTENTIAL ENERGY. Introduction. Electrostatic potential energy. Electric potential. for a system of point charges
Chapter 4 ELECTRIC POTENTIAL Introduction Electrostatic potential energy Electric potential for a system of point charges for a continuous charge distribution Why determine electic potential? Determination
More informationElectromagnetic Fields. Lecture 2. Fundamental Laws
Electromagnetic Fields Lecture 2 Fundamental Laws Laws of what? Electric field... is a phenomena that surrounds electrically charged objects or that which is in the presence of a time-varying magnetic
More informationAN EXAMPLE OF SYSTEM WHICH CAN BE USED TO EXPLICITLY SHOW THE SYMMETRY BETWEEN THE ELECTRIC AND MAGNETIC FIELDS. Arief Hermanto
P - 17 AN EXAMPLE OF SYSTEM WHICH CAN BE USED TO EXPLICITLY SHOW THE SYMMETRY BETWEEN THE ELECTRIC AND MAGNETIC FIELDS Arief Hermanto Physics Department, Gadjah Mada University Abstract Electric and magnetic
More informationELECTROMAGNETIC FIELD
UNIT-III INTRODUCTION: In our study of static fields so far, we have observed that static electric fields are produced by electric charges, static magnetic fields are produced by charges in motion or by
More information(Refer Slide Time: 03: 09)
Computational Electromagnetics and Applications Professor Krish Sankaran Indian Institute of Technology Bombay Lecture No 26 Finite Volume Time Domain Method-I Welcome back in the precious lectures we
More informationChap. 1 Fundamental Concepts
NE 2 Chap. 1 Fundamental Concepts Important Laws in Electromagnetics Coulomb s Law (1785) Gauss s Law (1839) Ampere s Law (1827) Ohm s Law (1827) Kirchhoff s Law (1845) Biot-Savart Law (1820) Faradays
More informationFoundations of Geomagnetism
Foundations of Geomagnetism GEORGE BACKUS University of California, San Diego ROBERT PARKER University of California, San Diego CATHERINE CONSTABLE University of California, San Diego m.m CAMBRIDGE UNIVERSITY
More informationUNIT-III Maxwell's equations (Time varying fields)
UNIT-III Maxwell's equations (Time varying fields) Faraday s law, transformer emf &inconsistency of ampere s law Displacement current density Maxwell s equations in final form Maxwell s equations in word
More informationElectromagnetic Field Theory Chapter 9: Time-varying EM Fields
Electromagnetic Field Theory Chapter 9: Time-varying EM Fields Faraday s law of induction We have learned that a constant current induces magnetic field and a constant charge (or a voltage) makes an electric
More informationOverview in Images. S. Lin et al, Nature, vol. 394, p , (1998) T.Thio et al., Optics Letters 26, (2001).
Overview in Images 5 nm K.S. Min et al. PhD Thesis K.V. Vahala et al, Phys. Rev. Lett, 85, p.74 (000) J. D. Joannopoulos, et al, Nature, vol.386, p.143-9 (1997) T.Thio et al., Optics Letters 6, 197-1974
More informationFinite Element Modeling of Electromagnetic Systems
Finite Element Modeling of Electromagnetic Systems Mathematical and numerical tools Unit of Applied and Computational Electromagnetics (ACE) Dept. of Electrical Engineering - University of Liège - Belgium
More informationEngineering Electromagnetic Fields and Waves
CARL T. A. JOHNK Professor of Electrical Engineering University of Colorado, Boulder Engineering Electromagnetic Fields and Waves JOHN WILEY & SONS New York Chichester Brisbane Toronto Singapore CHAPTER
More informationElectromagnetic Theory Prof. D. K. Ghosh Department of Physics Indian Institute of Technology, Bombay
Electromagnetic Theory Prof. D. K. Ghosh Department of Physics Indian Institute of Technology, Bombay Module -4 Time Varying Lecture - 29 Faraday s Law and Inductance In the previous lecture, we had started
More information2577. The analytical solution of 2D electromagnetic wave equation for eddy currents in the cylindrical solid rotor structures
2577. The analytical solution of 2D electromagnetic wave equation for eddy currents in the cylindrical solid rotor structures Lale T. Ergene 1, Yasemin D. Ertuğrul 2 Istanbul Technical University, Istanbul,
More informationLecture 13: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay. Poisson s and Laplace s Equations
Poisson s and Laplace s Equations Lecture 13: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay We will spend some time in looking at the mathematical foundations of electrostatics.
More information09 The Wave Equation in 3 Dimensions
Utah State University DigitalCommons@USU Foundations of Wave Phenomena Physics, Department of --2004 09 The Wave Equation in 3 Dimensions Charles G. Torre Department of Physics, Utah State University,
More informationxkcd.com It IS about physics. It ALL is.
xkcd.com It IS about physics. It ALL is. Introduction to Space Plasmas The Plasma State What is a plasma? Basic plasma properties: Qualitative & Quantitative Examples of plasmas Single particle motion
More informationMath Divergence and Curl
Math 23 - Divergence and Curl Peter A. Perry University of Kentucky November 3, 28 Homework Work on Stewart problems for 6.5: - (odd), 2, 3-7 (odd), 2, 23, 25 Finish Homework D2 due tonight Begin Homework
More informationFebruary 27, 2019 LECTURE 10: VECTOR FIELDS.
February 27, 209 LECTURE 0: VECTOR FIELDS 02 HONORS MULTIVARIABLE CALCULUS PROFESSOR RICHARD BROWN Synopsis Vector fields, as geometric objects and/or functions, provide a backbone in which all of physics
More informationfree space (vacuum) permittivity [ F/m]
Electrostatic Fields Electrostatic fields are static (time-invariant) electric fields produced by static (stationary) charge distributions. The mathematical definition of the electrostatic field is derived
More informationElectromagnetic waves in free space
Waveguide notes 018 Electromagnetic waves in free space We start with Maxwell s equations for an LIH medum in the case that the source terms are both zero. = =0 =0 = = Take the curl of Faraday s law, then
More informationMATH Max-min Theory Fall 2016
MATH 20550 Max-min Theory Fall 2016 1. Definitions and main theorems Max-min theory starts with a function f of a vector variable x and a subset D of the domain of f. So far when we have worked with functions
More informationUNIT-I INTRODUCTION TO COORDINATE SYSTEMS AND VECTOR ALGEBRA
SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road 517583 QUESTION BANK (DESCRIPTIVE) Subject with Code : EMF(16EE214) Sem: II-B.Tech & II-Sem Course & Branch: B.Tech - EEE Year
More informationModule 3: Electromagnetism
Module 3: Electromagnetism Lecture - Magnetic Field Objectives In this lecture you will learn the following Electric current is the source of magnetic field. When a charged particle is placed in an electromagnetic
More informationNitsche-type Mortaring for Maxwell s Equations
Nitsche-type Mortaring for Maxwell s Equations Joachim Schöberl Karl Hollaus, Daniel Feldengut Computational Mathematics in Engineering at the Institute for Analysis and Scientific Computing Vienna University
More informationHaus, Hermann A., and James R. Melcher. Electromagnetic Fields and Energy. Englewood Cliffs, NJ: Prentice-Hall, ISBN:
MIT OpenCourseWare http://ocw.mit.edu Haus, Hermann A., and James R. Melcher. Electromagnetic Fields and Energy. Englewood Cliffs, NJ: Prentice-Hall, 1989. ISBN: 9780132490207. Please use the following
More informationMagnetic circuits - electromagnet
www.e-lee.net Magnetic circuits - electromagnet Theme: Electrical Machines Chapter: Electromagnetic convertion Section: Type ressource: Lecture Virtual laboratory / Exercice MCQ This lecture deals with
More informationAntennas and Propagation
Antennas and Propagation Ranga Rodrigo University of Moratuwa October 20, 2008 Compiled based on Lectures of Prof. (Mrs.) Indra Dayawansa. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation
More informationMagnetostatic fields! steady magnetic fields produced by steady (DC) currents or stationary magnetic materials.
ECE 3313 Electromagnetics I! Static (time-invariant) fields Electrostatic or magnetostatic fields are not coupled together. (one can exist without the other.) Electrostatic fields! steady electric fields
More informationMagnetostatic Fields. Dr. Talal Skaik Islamic University of Gaza Palestine
Magnetostatic Fields Dr. Talal Skaik Islamic University of Gaza Palestine 01 Introduction In chapters 4 to 6, static electric fields characterized by E or D (D=εE) were discussed. This chapter considers
More informationDHANALAKSHMI SRINIVASAN INSTITUTE OF RESEARCH AND TECHNOLOGY
DHANALAKSHMI SRINIVASAN INSTITUTE OF RESEARCH AND TECHNOLOGY SIRUVACHUR-621113 ELECTRICAL AND ELECTRONICS DEPARTMENT 2 MARK QUESTIONS AND ANSWERS SUBJECT CODE: EE 6302 SUBJECT NAME: ELECTROMAGNETIC THEORY
More informationSimulation and Visualization of Safing Sensor
American Journal of Applied Sciences 2 (8): 1261-1265, 2005 ISSN 1546-9239 2005 Science Publications Simulation and Visualization of Safing Sensor haled M. Furati, Hattan Tawfiq and Abul Hasan Siddiqi
More informationMaxwell s Equations. In the previous chapters we saw the four fundamental equations governging electrostatics and magnetostatics. They are.
Maxwell s Equations Introduction In the previous chapters we saw the four fundamental equations governging electrostatics and magnetostatics. They are D = ρ () E = 0 (2) B = 0 (3) H = J (4) In the integral
More informationLimits to Statics and Quasistatics
Limits to Statics and Quasistatics Reading Haus and Melcher - Ch. 3 Outline Limits to Statics Quasistatics Limits to Quasistatics 1 Electric Fields Magnetic Fields GAUSS FARADAY GAUSS AMPERE For Statics
More informationAnalytical and numerical computation of the no-load magnetic field in induction motors
Analytical and numerical computation of the no-load induction motors Dan M. Ionel University of Glasgow, Glasgow, Scotland, UK and Mihai V. Cistelecan Research Institute for Electrical Machines, Bucharest
More informationHIGH VOLTAGE TECHNIQUES REVİEW: Electrostatics & Magnetostatics
HIGH VOLTAGE TECHNIQUES REVİEW: Electrostatics & Magnetostatics Zap You walk across the rug, reach for the doorknob and...zap!!! In the winter, when you change your pullover you hear and/or see sparks...
More information3 rd ILSF Advanced School on Synchrotron Radiation and Its Applications
3 rd ILSF Advanced School on Synchrotron Radiation and Its Applications September 14-16, 2013 Electromagnets in Synchrotron Design and Fabrication Prepared by: Farhad Saeidi, Jafar Dehghani Mechanic Group,Magnet
More information