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CSE245: Computer-Aided Circuit Simulation and Verification

CSE245: Computer-Aided Circuit Simulation and Verification. Lecture Note 6 Numerical Integration Spring 2008 Prof. Chung-Kuan Cheng. Numerical Integration: Outline. One-step Method for ODE (IVP) Forward Euler Backward Euler Trapezoidal Rule Equivalent Circuit Model Convergence Analysis

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CSE245: Computer-Aided Circuit Simulation and Verification

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  1. CSE245: Computer-Aided Circuit Simulation and Verification Lecture Note 6 Numerical Integration Spring 2008 Prof. Chung-Kuan Cheng

  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

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

  4. Numerical Integration • Forward Euler • Backward Euler • Trapezoidal

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

  6. Numerical Integration: State Equation • Trapezoidal

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

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

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

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

  11. Summary of Basic Concepts Trap Rule, Forward-Euler, Backward-Euler All are one-step methods xk+1 is computed using only xk, not xk-1, xk-2, xk-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 UCSD CSE 245 Notes – SPRING 2006

  12. Stabilities Froward Euler

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

  14. Stabilities Backward Euler

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

  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: Convergence Consistency + Stability

  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.

  19. Convergence Analysis: Truncation Error • Local Truncation Error (LTE): • At time point tk+1 assume xk is exact, the difference between the approximated solution xk+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 tk+1, assume only the initial condition x0 at time t0 is correct, the difference between the approximated solution xk+1 and the exact solution x*k+1 is called global truncation error. • Indicates stability

  20. LTE Estimation: SPICE • Taylor Expansion of xn+1 about the time point tn: • Taylor Expansion of dxn+1/dt about the time point tn: • Eliminate term in above two equations we get the trapezoidal rule LTE

  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 ED represents the absolute value of error that is allowed per time point. That is together with (1) we can calculate the time step as

  22. Time Step Control: SPICE (cont’d) • DD3(tn+1) is called 3rd divided difference, which is given by the recursive formula

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