1. Maria Ugryumova Model Order Reduction for EM modeling of IC's CASA day, November 2007

2. Contents Introduction Simulation of EM behaviour Boundary Value Problem, Kirchhoff’s equations Reduced Order Modelling for EM Simulations The Approach of EM Simulation used in FASTERIX Transient Analysis Simulation for FULL Model in Time Domain Super Node Algorithm Details of Super Node Algorithm Observations Examples: Transmission line model, Lowpass filter Conclusions on Super Node Algorithm Passivity Future work 2

3. Simulation of EM behaviour Printed Circuit Board Equivalent Circuit Model Simulator Currents through Conductor radiated EM fields 3

4. Simulation of EM behaviour Printed Circuit Board Equivalent Circuit Model Simulator Currents through Conductor radiated EM fields Equivalent Circuit Model 3

5. Simulation of EM behaviour Printed Circuit Board Equivalent Circuit Model Simulator Currents through Conductor radiated EM fields Equivalent Circuit Model Simulator 3

6. Simulation of EM behaviour Printed Circuit Board Equivalent Circuit Model Simulator Currents through Conductor, radiated EM fields 3

7. BC: Boundary Value Problem Discretisation Set of quadrilateral elements: Set of edges (excluding element edges in boundary): 4

8. Kirchhoff’s equations J – collects currents flowing into interconnection system Solved for I – currents over branches and V – potentials at the nodes Linear set of (N edge + N elem) equations Matrix coefficients are integrals. They are frequency independent matrices from FASTERIX 5 FASTERIX – to simulate PCB’s

9. Approach FASTERIX Model Order Reduction Simulation of PCB’s 6

10. Reduced Order Modelling for EM Simulations Super Node Algorithm (SNA) Good results in Frequency Domain SNA produces not passive models SNA delivers models based on max applied frequency 7 Model Order Reduction Preservation of passivity PRIMA, Laguerre SVD FASTERIX - l ayout simulation tool

11. Reduced Order Modelling for EM Simulations Super Node Algorithm (SNA) Good results in Frequency Domain SNA produces not passive models SNA delivers models based on max applied frequency 7 Model Order Reduction Preservation of passivity PRIMA, Laguerre SVD FASTERIX - l ayout simulation tool for EM effects

12. The Approach of EM Simulation used in FASTERIX 1. Subdivision PCB into quadrilateral elements 2. Equivalent Circuit (EQCT) Each finite element corresponds to i-th node; Each pair of neighbour nodes is connected with RL-branch; Such large model is inefficient for simulator; 3. Reduced Equivalent Circuit Built on accessible nodes + some internal nodes from EQCT; The higher user-defined frequency, the more super nodes; Each pair of super nodes has RLGC-branch. 4. Simulation for Transient Analysis 8

13. Transient Analysis Rise time << Time of propagation V(OUT) PSTAR simulator Voltage in the node Rout is measured depending on time. 9

14. Simulation for FULL Model in Time Domain In order to get solution for full model to compare results with Input: Pulse, rise time = 100psOutput: Voltage on the resistor Rout 10

15. Contents Introduction Simulation of EM behaviour Boundary Value Problem, Kirchhoff’s equations Reduced Order Modelling for EM Simulations The Approach of EM Simulation used in FASTERIX Transient Analysis Simulation for FULL Model in Time Domain Super Node Algorithm Details of Super Node Algorithm Observations Examples: Transmission line model, Lowpass filter Conclusions on Super Node Algorithm Passivity Future work 11

16. Super Node Algorithm [Cloux, Maas, Wachters] Geometrical details are small compared with the wavelength of operation The subdivision of the set of nodes: N =N U N’ 12

17. Super Node Algorithm [Cloux, Maas, Wachters] Depends on frequency 12 Geometrical details are small compared with the wavelength of operation The subdivision of the set of nodes: N =N U N’

18. Details of Super Node Algorithm Depends on frequency 12 Geometrical details are small compared with the wavelength of operation The subdivision of the set of nodes: N =N U N’

19. Details of Super Node Algorithm Admittance matrix: 13 Solving Kirchhoff's equation independent on frequency

20. Details of Super Node Algorithm Admittance matrix: Solving Kirchhoff's equation independent on frequency For high frequency range: 13

21. Details of Super Node Algorithm Admittance matrix: Solving Kirchhoff's equation independent on frequency For high frequency range: Admittance matrix: Branch of Reduced Equivalent Circuit 13

22. Observations Problems of modeling in Time Domain V(Rout) Original BEM discretisation leads to passive systems Super Node Algorithm is based on physical principals 14 Lowpass filter Max freq = 10GHz 257 elements 98 supernodes Reduced model by SNA Full model by SNA

23. Observations Problems of modeling in Time Domain V(Rout) Original BEM discretisation leads to passive systems Super Node Algorithm is based on physical principals Lowpass filter Max freq = 10GHz 257 elements 98 supernodes Increasing the number of super nodes? Modified SNA 14 Reduced model by SNA Full model by SNA

24. Lowpass filter Max frequency = 7 GHz 85 nodes 40 nodes full unreduced model FASTERIX: 227 Elements, 40 Super Nodes Fine mesh: 2162 Elements Experiment 16 V(Rout) T

25. Transmission Line Model Max frequency = 3 GHz FASTERIX: 160 Elements, 100 Super Nodes Fine mesh: 1550 Elements 50 nodes 120 nodes full unreduced model Increasing of super nodes on the fine mesh gives more accurate results but has an upper limit. Experiment Increasing of super nodes does not give “right” properties of admittance matrices. 15 Time delay ~ 1.3 ns. V(Rout) T

26. Conclusions on Super Node Algorithm Super node algorithm is motivated by physical and electronic insight. It is worth to modify it; Decreasing the distance between super nodes does not ensure passivity of the reduced model; Increasing of super nodes on the fine mesh gives more accurate results but has an upper limit for simulator; N ecessity of detailed analysis of properties of projection matrix P due to guaranty passivity of the models. 17

27. Passivity Incapable of generating energy; The transfer function H(s) of a passive system is positive real, that is, 18 H(s) is analytic for all s with Re(s) > 0

28. Passivity Incapable of generating energy; Before reduction Eigenvalues of G have non-negative real part, C is symmetricpositive semi-definite ; After reduction by SNA G, C - G, C - indefinite; have the same number of positive and negative eigenvalues; The transfer function H(s) of a passive system is positive real that is After projection: can have diff. number of pos. and neg. eigenvalues. We need to define properties of P to have all positive eigenvalues. 18 H(s) is analytic for all s with Re(s) > 0 will have positive eigenvalues;

29. Future work Investigation of matrix properties and eigenvalues when increasing the number of super nodes; Deriving a criterion for choosing super nodes that guarantees passivity; Implementation in FASTERIX and comparison with MOR algorithms; Making start with EM on IC problem for SiP. 19

30. Thank you for attention