1. CSE245: Computer-Aided Circuit Simulation and Verification Lecture Note 5 Numerical Integration Spring 2010 Prof. Chung-Kuan Cheng 1

2. Numerical Integration: Outline One-step Method for ODE (IVP) –Forward Euler –Backward Euler –Trapezoidal Rule –Equivalent Circuit Model Convergence Analysis Linear Multi-Step Method Time Step Control 2

3. Ordinary Difference Equaitons N equations, n x variables, n dx/dt. Typically analytic solutions are not available  solve it numerically 3

4. Numerical Integration Forward Euler Backward Euler Trapezoidal 4

5. Numerical Integration: State Equation Forward Euler Backward Euler 5

6. Numerical Integration: State Equation Trapezoidal 6

7. 7 Equivalent Circuit Model-BE Capacitor + C - + - + -

8. 8 Equivalent Circuit Model-BE Inductor + L - + - + -

9. 9 Equivalent Circuit Model-TR Capacitor + C - + - + -

10. 10 Equivalent Circuit Model-TR Inductor + L - + - + -

11. Trap Rule, Forward-Euler, Backward-Euler All are one-step methods x k+1 is computed using only x k, not x k-1, x k-2, x k-3... Forward-Euler is the simplest No equation solution explicit method. Backward-Euler is more expensive Equation solution each step implicit method most stable (FE/BE/TR) Trapezoidal Rule might be more accurate Equation solution each step implicit method More accurate but less stable, may cause oscillation Summary of Basic Concepts 11

12. 12 Stabilities Froward Euler

13. Difference Eqn Stability region 1 Im(z) Re(z) Forward Euler ODE stability region Region of Absolute Stability FE region of absolute stability 13

14. 14 Stabilities Backward Euler

15. Difference Eqn Stability region 1 Im(z) Re(z) Backward Euler Region of Absolute Stability BE region of absolute stability 15

16. 16 Stabilities Trapezoidal

17. Convergence Consistency: A method of order p (p>1) is consistent if Stability: A method is stable if: Convergence: A method is convergent if: Consistency + Stability Convergence 17

18. A-Stable Dahlqnest Theorem: –An A-Stable LMS (Linear MultiStep) method cannot exceed 2nd order accuracy The most accurate A-Stable method (smallest truncation error) is trapezoidal method. 18

19. Convergence Analysis: Truncation Error Local Truncation Error (LTE): –At time point t k+1 assume x k is exact, the difference between the approximated solution x k+1 and exact solution x * k+1 is called local truncation error. –Indicates consistancy –Used to estimate next time step size in SPICE Global Truncation Error (GTE): –At time point t k+1, assume only the initial condition x 0 at time t 0 is correct, the difference between the approximated solution x k+1 and the exact solution x * k+1 is called global truncation error. –Indicates stability 19

20. LTE Estimation: SPICE Taylor Expansion of x n+1 about the time point t n : Taylor Expansion of dx n+1 /dt about the time point t n : Eliminate term in above two equations we get the trapezoidal rule LTE 20

21. Time Step Control: SPICE We have derived the local truncation error the unit is charge for capacitor and flux for inductor Similarly, we can derive the local truncation error in terms of (1) the unit is current for capacitor and voltage for inductor Suppose E D represents the absolute value of error that is allowed per time point. That is together with (1) we can calculate the time step as 21

22. Time Step Control: SPICE (cont ’ d) DD 3 (t n+1 ) is called 3rd divided difference, which is given by the recursive formula 22