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

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

6. 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 STEP 1 ELIMINATE ELIMINATE y1 y1 y1 y1 from from b i cc:cc: Multiply by and add to 1b

15. Multiply by and add to 1c

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

17. (2a) (2b) (2c) Elimination scheme repeat for equations 2b i 2c: Multiply by add o 2c

18. (3a) (3b) (3c) (3a) (3b) (3c)

19. Back substitution part: Setting y 3 to 3b: Multiply by add to 3b

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

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

23. e1e1 e2e2

24. Data:

25. Nodal equations: 1 2

26. Jacobian matrix:

27. We choose starting vector: Calculate:

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

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

30. Second NR iteration where: hence:

31. for k=7: where: hence:

32. Because:

33. Briefly about: Iterative models of nonlinear elements

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

35. circuit  From NR method:

36. Model iterowany opornika (6)

37. Example 3 Newton-Raphson Newton-Raphson Iterative model method

38. e1e1 e2e2

39. Data:

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

41. 1 1 2

42. 2 1 2

43. 1 2

44. 1 2

45. For starting vector: We calculate parameters of the models:

46. For nonlinear element g 6 :

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

48. Second step Solution for k=2

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

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

51. Backward Euler Method (Explicit) is based on the following Taylor series expansion

52. v (t) C vsvs

63. v c (t k )

65. Example with nonlinear capacitor FEM

66. FEM steps

67. BEM step 1

68. Using N-R method with starting point

69. BEM step 2 after N-R procedure with new starting point

70. Using N-R method with starting point