EE 616 Computer Aided Analysis of Electronic Networks Lecture 5

1. EE 616 Computer Aided Analysis of Electronic NetworksLecture 5 Instructor: Dr. J. A. Starzyk, Professor School of EECS Ohio University Athens, OH, 45701 09/19/2005 Note: materials in this lecture are from the notes of EE219A UC-berkeley http://www- cad.eecs.berkeley.edu/~nardi/EE219A/contents.html

2. Outline • Nonlinear problems • Iterative Methods • Newton’s Method • Derivation of Newton • Quadratic Convergence • Examples • Convergence Testing • Multidimensonal Newton Method • Basic Algorithm • Quadratic convergence • Application to circuits

3. 1 I1 Id Ir 0 DC Analysis of Nonlinear Circuits - Example Need to Solve

4. Nonlinear Equations • Given g(V)=I • It can be expressed as: f(V)=g(V)-I  Solve g(V)=I equivalent to solve f(V)=0 Hard to find analytical solution for f(x)=0 Solve iteratively

5. Nonlinear Equations – Iterative Methods • Start from an initial value x0 • Generate a sequence of iterate xn-1, xn, xn+1 which hopefully converges to the solution x* • Iterates are generated according to an iteration function F: xn+1=F(xn) • Ask • When does it converge to correct solution ? • What is the convergence rate ?

6. Newton-Raphson (NR) Method Consists of linearizing the system. Want to solve f(x)=0  Replace f(x) with its linearized version and solve. Note: at each step need to evaluate f and f’

7. Newton-Raphson Method – Graphical View

8. Newton-Raphson Method– Algorithm Define iteration Dok = 0 to ….? • How about convergence? • An iteration {x(k)} is said to converge with order q if there exists a vector norm such that for each k  N: until convergence

9. Mean Value theorem truncates Taylor series Newton-Raphson Method– Convergence But by Newton definition

10. Newton-Raphson Method– Convergence Subtracting Dividing Convergence is quadratic

11. Newton-Raphson Method– Convergence Local Convergence Theorem If Then Newton’s method converges given a sufficiently close initial guess (and convergence is quadratic)

12. Newton-Raphson Method– Convergence

13. f(x) X Newton-Raphson Method– Convergence Check False convergence since increment in f small but with large increment in X

14. f(x) X Newton-Raphson Method– Convergence Check False convergence since increment in x small but with large increment in f

15. Newton-Raphson Method– Convergence

16. f(x) X Newton-Raphson Method– Local Convergence Convergence Depends on a Good Initial Guess

17. Newton-Raphson Method– Local Convergence Convergence Depends on a Good Initial Guess

18. + - + + - - Nonlinear Problems –Multidimensional Example Nodal Analysis Nonlinear Resistors Two coupled nonlinear equations in two unknowns

19. Multidimensional Newton Method

20. Multidimensional Newton Method – Computational Aspects Each iteration requires: • Evaluation of F(xk) • Computation of J(xk) • Solution of a linear system of algebraic equations whose coefficient matrix is J(xk) and whose RHS is -F(xk)

21. Multidimensional Newton Method – Algorithm

22. Multidimensional Newton Method – Convergence Local Convergence Theorem If Then Newton’s method converges given a sufficiently close initial guess (and convergence is quadratic)

23. Application of NR to Circuit EquationsCompanion Network • Applying NR to the system of equations we find that at iteration k+1: • all the coefficients of KCL, KVL and of BCE of the linear elements remain unchanged with respect to iteration k • Nonlinear elements are represented by a linearization of BCE around iteration k  This system of equations can be interpreted as the STA of a linear circuit (companion network) whose elements are specified by the linearized BCE.

24. Application of NR to Circuit EquationsCompanion Network • General procedure: the NR method applied to a nonlinear circuit (whose eqns are formulated in the STA form) produces at each iteration the STA eqns of a linear resistive circuit obtained by linearizing the BCE of the nonlinear elements and leaving all the other BCE unmodified • After the linear circuit is produced, there is no need to stick to STA, but other methods (such as MNA) may be used to assemble the circuit eqns

25. Application of NR to Circuit EquationsCompanion Network – MNA templates Note: G0 and Id depend on the iteration count k  G0=G0(k) and Id=Id(k)

26. Application of NR to Circuit EquationsCompanion Network – MNA templates

27. Modeling a MOSFET(MOS Level 1, linear regime) d

28. gds Modeling a MOSFET(MOS Level 1, linear regime)

29. DC Analysis Flow Diagram For each state variable in the system

30. Implications • Device model equations must be continuous with continuous derivatives and derivative calculation must be accurate derivative of function • Watch out for floating nodes (If a node becomes disconnected, then J(x) is singular) • Give good initial guess for x(0) • Most model computations produce errors in function values and derivatives. • Want to have convergence criteria || x(k+1) - x(k) || <  such that  > than model errors.

31. Summary • Nonlinear problems • Iterative Methods • Newton’s Method • Derivation of Newton • Quadratic Convergence • Examples • Convergence Testing • Multidimensional Newton Method • Basic Algorithm • Quadratic convergence • Application to circuits

32. Improving convergence • Improve Models (80% of problems) • Improve Algorithms (20% of problems) Focus on new algorithms: Limiting Schemes Continuations Schemes

33. Outline • Limiting Schemes • Direction Corrupting • Non corrupting (Damped Newton) • Globally Convergent if Jacobian is Nonsingular • Difficulty with Singular Jacobians • Continuation Schemes • Source stepping • More General Continuation Scheme • Improving Efficiency • Better first guess for each continuation step

34. Multidimensional Newton MethodConvergence Problems – Local Minimum Local Minimum

35. X Multidimensional Newton MethodConvergence Problems – Nearly singular f(x) Must Somehow Limit the changes in X

36. Multidimensional Newton MethodConvergence Problems - Overflow f(x) X Must Somehow Limit the changes in f(x)

37. Newton Method with Limiting

38. Newton Method with LimitingLimiting Methods • Direction Corrupting • NonCorrupting Scales differently different dimensions Scales proportionally In all dimensions Heuristics, No Guarantee of Global Convergence

39. Newton Method with LimitingDamped Newton Scheme General Damping Scheme Key Idea: Line Search Method Performs a one-dimensional search in Newton Direction

40. Newton Method with LimitingDamped Newton – Convergence Theorem If Then Every Step reduces F-- Global Convergence!

41. Newton Method with LimitingDamped Newton – Singular Jacobian Problem X Damped Newton Methods “push” iterates to local minimums Finds the points where Jacobian is Singular

42. Newton with Continuation schemes Basic Concepts - General setting Newton converges given a close initial guess  Idea: Generate a sequence of problems, such that a problem is a good initial guess for the following one  Starts the continuation Ends the continuation Hard to insure!

43. Newton with Continuation schemes Basic Concepts – Template Algorithm

44. Newton with Continuation schemes Basic Concepts – Source Stepping Example

45. R Vs + - Newton with Continuation schemes Basic Concepts – Source Stepping Example Diode Source Stepping Does Not Alter Jacobian

46. Newton with Continuation schemes Jacobian Altering Scheme Observations Problem is easy to solve and Jacobian definitely nonsingular. Back to the original problem and original Jacobian (1),1)

47. Summary • Newton’s Method works fine: • given a close enough initial guess • In case Newton does not converge: • Limiting Schemes • Direction Corrupting • Non corrupting (Damped Newton) • Globally Convergent if Jacobian is Nonsingular • Difficulty with Singular Jacobians • Continuation Schemes • Source stepping • More General Continuation Scheme • Improving Efficiency • Better first guess for each continuation step