1. Pieter Heres, Aday 2005 1 Error control in Krylov subspace methods for Model Order Reduction Pieter Heres June 21, 2005 Eindhoven

2. Pieter Heres, Aday 2005 2 Overview Application Krylov subspace methods Error control

3. Pieter Heres, Aday 2005 3 Interconnect structures

4. Pieter Heres, Aday 2005 4 Coupled simulation Incorporate passive layout effects in full chip simulation Passive circuit Active circuit Maxwell’s equations

5. Pieter Heres, Aday 2005 5 Model Order Reduction To quickly capture the essential features of passive structure Implementation in Philips layout simulator Fasterix –Preservation of stability (and passivity) Example

6. Pieter Heres, Aday 2005 6 RF Transformer Courtesy to Jos Bergervoet, Philips Research

7. Pieter Heres, Aday 2005 7 System equations Circuit equations Matrices 5202 x 5202, partly full 4 ports Simulated for frequencies up to 30 GHz Defined such that afterward components can be added

8. Pieter Heres, Aday 2005 8 Frequency domain Laplace transform to frequency domain: Transfer function: Approximation for frequency behavior

9. Pieter Heres, Aday 2005 9 Overview Application Krylov subspace methods Error control

10. Pieter Heres, Aday 2005 10 Krylov subspace methods Expand X( s ) : Collect the terms for different powers of s : In general:

11. Pieter Heres, Aday 2005 11 Krylov subspace methods (2) Collecting the moments in one space: gives a Krylov space: In general: Orthonormal basis of space Projecting the space onto the space: V T GV, preserves the first moments of X( s ) :

12. Pieter Heres, Aday 2005 12 Algorithm Solve G W = B V 1 R = W (QR step) for j = 1,2,… Solve G W = –C V j for i = 1,2,…,j H i,j = V i T W W = W – V i H i,j end V j+1 H j+1,j = W (QR step) end Project system matrices

13. Pieter Heres, Aday 2005 13 Reduced system Projected system matrices: Reduced system: Transfer function of reduced system:

14. Pieter Heres, Aday 2005 14 Reduced system (2) Consider the shift-and-inverted system: Or: And the reduced form: where

15. Pieter Heres, Aday 2005 15 Frequency domain simulation

16. Pieter Heres, Aday 2005 16 Time domain simulation

17. Pieter Heres, Aday 2005 17 Overview Application Krylov subspace methods Error control

18. Pieter Heres, Aday 2005 18 When to stop? Approximate There is a closed expression for error function: In-practical: Closed expression (eventually a bound) becomes approximation

19. Pieter Heres, Aday 2005 19 Error We state:

20. Pieter Heres, Aday 2005 20 Updating reduced matrices Projection In every iteration a block is added: The matrices can be cheaply updated in every step: Solve G W = B V 1 R = W Calculate 1 st matrix for j = 1,2,… Solve G W = –C V j Orthogonalize V j+1 H j+1,j = W Update matrices end

21. Pieter Heres, Aday 2005 21 Error definition For practical reasons we define the error: Officially we should take:

22. Pieter Heres, Aday 2005 22 Results

23. Pieter Heres, Aday 2005 23 Results (2)

24. Pieter Heres, Aday 2005 24 Sequence argument The following bounds can easily be derived:

25. Pieter Heres, Aday 2005 25 Sequence argument (2) In terms of errors: This all can be formulated as:

26. Pieter Heres, Aday 2005 26 Sequence argument (3) Finally: Choose the value in between: Finally:

27. Pieter Heres, Aday 2005 27 Conclusion Model Order Reduction techniques have shown to be useful for passive electronic applications Krylov subspace methods for Model Order Reduction can be fully automatic