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Fields and Waves

Fields and Waves. Lesson 3.6. ELECTROSTATICS - Numerical Simulation. Direct Computation of V. If we can express entire problem in terms of V then:. we can solve directly for V derive all other quantities e.g. E-field, D-field, C and r.

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Fields and Waves

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  1. Fields and Waves Lesson 3.6 ELECTROSTATICS - Numerical Simulation Darryl Michael/GE CRD

  2. Direct Computation of V If we can express entire problem in terms of V then: • we can solve directly for V • derive all other quantities e.g. E-field, D-field, C and r This approach can be used if conductor defines Outer Boundary • can be SYMMETRIC or NON-SYMMETRIC systems Why is this a useful approach?? • V is a scalar field - easier to manipulate than E-field • We can control V on conductors • Can apply numerical methods to solve problem

  3. Use of Laplace and Poisson’s Equations Start with 2 of MAXWELL’s equations: & In rectangular coordinates:

  4. Use of Laplace and Poisson’s Equations Poisson’s equation: Laplace’s equation: (when r = 0) Do Problem 1

  5. h h Numerical Solution: Finite Difference Method Use the FINITE DIFFERENCE Technique for solving problems Solve for approximate V on the Grid - for 2-D Problem Vtop Vright Vcenter Vleft Vcenter at (x,y) = (0,0) Vbottom

  6. h h Vleft Vcenter Vbottom Numerical Solution: Finite Difference Method At (x,y) = (h/2,0) Vtop Vright At (x,y) = (-h/2,0)

  7. Numerical Solution: Finite Difference Method 0 Now, Can get similar expression for

  8. Numerical Solution: Finite Difference Method Finally we obtain the following expression: Rearrange the equation to solve for Vcenter : Poisson Equation Solver Laplace Equation Solver

  9. 100V 10V 30V 60V Numerical Solution: Example Solution Technique - by Iteration Guess a solution : V=0 everywhere except boundaries V1 V2 V4 V3 V1= V2 = V3 = V4 = 0 Put new values back Start:

  10. Numerical Solution - use of EXCEL Spreadsheet Do Problem 2 • To get an accurate solution, need lots of points - one way is to use a SPREADSHEET In spreadsheet, A1 A1 to A31 set boundary voltage = 0Volts Set these cells to 100 Copy B2 formula to rest of cells A31

  11. Numerical Solution: Problems Do Problem 3a and 3b Helpful Hints for Problems 3c - 3e 3c. At point P, what is rs ? Get rs from Boundary Conditions: Approximate 3d. Use spreadsheet to add columns: 3e. Use C=Q/V

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