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Circuits Theory Examples

Circuits Theory Examples. Newton-Raphson Method. Formula for one-dimensional case : . Series of successive solutions :. If the iteration process is converged , the limit is the solution of the equation f(x)=0. Multidimensional case :. where :. JACOBI AN MATRIX. ALGOR ITHM.

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Circuits Theory Examples

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  1. Circuits TheoryExamples Newton-Raphson Method

  2. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged , the limit is the solution of the equationf(x)=0.

  3. Multidimensional case: where: JACOBIAN MATRIX

  4. ALGORITHM STARTING POINT STEP 0 STEP 1 Calculate STEP 2 Solve the equation: find STEP 3 check STOP conditions If the current solution is not acceptable: GO TO 1

  5. EXAMPLE of STOP PROCEDURE k=k+1 GOTO 1 No Yes No Yes STOP

  6. Stop condition parameter • Stop condition parameter

  7. Numerical EXAMPLES Example 1

  8. Solve the following set of nonlinearequation using the Newton’s Method:

  9. Starting point (first approximation): Calculate:

  10. where:

  11. (1a) (1b) (1c)

  12. (1a) (1b) (1c) Let us assume (1a) (1b) (1c)

  13. Gauss elimination computer scheme STEP 1 ELIMINATE y1from b i c: Multiply by and add to 1b

  14. Multiply by and add to 1c

  15. New set : (2a) (2b) (2c) (2a) (2b) (2c)

  16. Elimination scheme repeat for equations 2b i 2c: (2a) Multiply by add o 2c (2b) (2c)

  17. (3a) (3b) (3c) (3a) (3b) (3c)

  18. Back substitution part: Setting y3 to 3b: Multiply by add to 3b

  19. Because It is the first calculated approximation of the solution. Next iterations form a converged series:

  20. Example 2 Nonlinear circuit having two variables (node voltages)

  21. e1 e2

  22. Data:

  23. Nodal equations: 1 2

  24. Jacobian matrix:

  25. We choose starting vector: Calculate:

  26. Applying N-R scheme: where: hence:

  27. STOP CRITERIA not satisfied: k=k+1:

  28. Second NR iteration where: hence:

  29. for k=7: where: hence:

  30. Because:

  31. Briefly about: Iterative models of nonlinear elements

  32. Iterative NR model of nonlinear resistor (voltage controled)

  33. From NR method: circuit

  34. Model iterowany opornika (6)

  35. Example3 Newton-Raphson Iterative model method

  36. e1 e2

  37. Data:

  38. Scheme for (k+1) iteration 1 2

  39. 1 2 1

  40. 1 2 2

  41. 1 2

  42. 1 2

  43. For starting vector: • We calculate parameters of the models:

  44. For nonlinear element g6:

  45. Linear equations for the first approximation: Solution for k=1

  46. Second step Solution for k=2

  47. Briefly about: Forward Euler Method (Explicit) Backward Euler Method (Implicit)

  48. Forward Euler Method (Explicit) Backward Euler Method (Explicit)

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