Model Order Reduction for EM Modeling of IC's

1. Model Order Reduction for EM modeling of IC's Maria Ugryumova 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 Currents through Conductor radiated EM fields Simulator 3

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

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

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

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

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

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

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

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

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 PSTAR simulator V(OUT) Voltage in the node Routismeasured depending on time. Rise time << Time of propagation 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 = 100ps Output: 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] • Geometrical details are small compared with the wavelength of operation • The subdivision of the set of nodes:N=N U N’ Depends on frequency 12

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

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

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) Lowpass filter Max freq = 10GHz 257 elements 98 supernodes Reduced model by SNA Full model by SNA • Original BEM discretisation leads to passive systems • Super Node Algorithm is based on physical principals 14

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

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

25. Transmission Line Model Max frequency = 3 GHz Time delay ~ 1.3 ns. FASTERIX: 160 Elements, 100 Super Nodes V(Rout) Experiment 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. • Increasing of super nodes • does not give “right” properties • of admittance matrices. T 15

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; • Necessity 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, H(s) is analytic for all s with Re(s) > 0 18

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

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