Seminar_01_Introduction. Maxwell Equations

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sdubitsky

2023-7-06 published at Education&Career category

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1.Electrical Engineering Theoretical FoundationsAdditional Chapters Simon DUBITSKY Higher school of High Voltage Energy QuickFieldFEA Software Week 1. Introduction 1 Spring 2023
2.Week 1: Introduction How to solve the EM field equation using numerical simulation (practical aspects). What math concept do we rely on in this course? The equations we solve: Maxwell equations again. What must be added to Maxwell equation to solve it? Why we prefer to use potential instead of field vectors? Concept of Field Formulation: Do we have to solve all equations at once? How to jump from Maxwell equations to the first practical problem? Summary: the Numerical Field Simulation workflow. Let’s solve our first problem Week 1. Introduction 2
3.Basic concepts of vector calculus Flux of a vector Circulation of a vector What math will we use Week 1. Introduction 3 Divergence Rotor (Curl)
4.Maxwell Equations (SI convention) 𝑆 𝐃⋅𝐝𝐬=𝑞 𝑆 𝐁⋅𝐝𝐬=0 Week 1. Introduction 4
5.It is not a full story. What should be added? 1. Various material models describing the macroscopic field inside the material. Any material law is always an (simplified) approximation Week 1. Introduction 5
6.It is still not a full story. What more should be added? 2. For numerical solution field equation are often written with respect to potential, not field vectors Week 1. Introduction 6 You already know a lot about potentials. Examples are: Scalar potential of electric field: the potential U(x, y, z) is such scalar quantity thatQuestion: Why we are sure that such potential really exists? Scalar potential of magnetic field: the potential UM (x, y, z) is such scalar quantity thatAgain: please think when the UM exists and when it does not exist? Vector magnetic potential: the potential A(x, y, z) is such vector quantity thatQuestion: Does the vector potential A always exists? Why the potential is numerically better than field vectors E, D, H, B ?
7.Simplified subset of Maxwell equations Solving the full set of Maxwell equations is almost impossible (and always impractical) Week 1. Introduction 7 Simplified set of Maxwell equations (examples): Electrostatics: Nothing is changed in time; No magnetic field exists (B=0; H=0) Magnetostatics: Nothing is changed in time; No electric field exists (E=0; D=0) AC Magnetics: Every field quantity changed in time sinusoidal with the same frequency: H = H0cos (ωt + φ0) No displacement current: JD = dD / dt = 0
8.What’s more?Geometry + Mesh + Boundary Conditions + Materials = Result Every numerical problem is solved: In the given geometry (bounded or unbounded) – the geometry is always a simplification In discrete space (by the mesh) With given boundary conditions and properties of materials Week 1. Introduction 8 1. Geometry 2. Mesh
9.What’s more?Boundary Conditions & Field Sources Week 1. Introduction 9 3. Boundary conditions Each field problem is solved with boundary conditions (BC). BC reflects the influence of the world outside our model Sometimes BC serve as a field source (e.g. given potential) BC may reflect the symmetry condition when our model is ½ or ¼ of the real geometry
10.The purpose of computing is insight, not numbers Richard Hamming, 1962 Week 1. Introduction 10 4. Calculation Result What we use to understand the calculated field: Field pictures: color plot, vector plot, field lines Local field values Integral field quantities: flux, energy, force XY-plot over various cross-section Animated pictures
11.Resume: What you need to solve a field problem numerically? Week 1. Introduction 11 1. Choose formulation 2. Draw your geometry 3. Define your materials 4. Set boundary conditions 5. Build the mesh 6. Enjoy your results
12.Example: Week 1. Introduction 12 Finding the best place for an optical cable in overhead power transmission line The communication fiber-optical cable if often mounted along with high voltage overhead power line. The cable itself is self-supported and fully dielectric, i.e., it contains no metal parts.An optical cable should be properly located relative to phase conductors. What is a proper location? According the Russian electrical code this is a place where the average potential U reaches its minimal value, U <=12 kV (25 kV with using special polyethylene). Also, the electric filed E should be checked.
13.Week 1. Introduction 13 Model of Overhead Line: the cross-section
14.Week 1. Introduction 14 Model of Overhead Line: the geometry model The whole model The tower
15.Week 1. Introduction 15 Model of Overhead Line: boundary condition Outer boundary: no field here (U=0) Earth surface:U=0
16.Week 1. Introduction 16 Model of Overhead Line: boundary condition Left Circuit: Momentary phase voltagesUA1, UB1, UC1 Right Circuit: Momentary phase voltagesUA2, UB2, UC2 The Tower: Is grounded: U=0
17.Week 1. Introduction 17 Model of Overhead Line:the solution Good place for an optical cable(the potential U<6 kV)
18.Week 1. Introduction 18 Model of Overhead Line: XY-plot of Potential U(y)
19.Week 1. Introduction 19 Model of Overhead Line: XY-plot of electric field E(y)
20.Week 1. Introduction 20 Model of Overhead Line: Conclusion By numerical simulation we have found a good place for the self-supported dielectric optical cable on the tower: Optimal coordinates are: x=0; y=22 m The potential is this point is: U=2.7 kV The electric field stress there is: E=6.0 kV/m Good place for an optical cable
21.Week 1. Introduction 21 Your first homework Next week we will focus on Electrostatic analysis – the simplest case of EM fields. We will learn how to: Create a QuickField problem, Draw the geometry Set up boundary conditions Manage the mesh density Obtain the problem solution, and Understand and analyze the results Be prepared: order, download, and install your copy of ELCUT 6.6 Professional Edition Learn with introductory videos on https://quickfield.com/seminar/arhive.htm

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