Seminar_01_Introduction. Maxwell Equations


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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