Stability and Passivity of the Super Node Algorithm for EM modelling of ICs

1. Stability and Passivity of the Super Node Algorithmfor EM modelling of ICs

4. Smaller feature sizes but increasing complexity Smaller feature sizes but increasing complexity

9. Inefficient: because of it contains many nodes, many RLC elements (order of ten thousand)Inefficient: because of it contains many nodes, many RLC elements (order of ten thousand)

15. RLC matrices � positive definite. P � incidence matrix PROOF is AVAILABLE!!!!!RLC matrices � positive definite. P � incidence matrix PROOF is AVAILABLE!!!!!

16. Important: 1. ports are preserved. 2. Every terminal (port) is a super nodeImportant: 1. ports are preserved. 2. Every terminal (port) is a super node

19. Y1 describes the reduced model linear relation: Jn=YnVn Y1 describes the reduced model linear relation: Jn=YnVn

22. Solution of (3) depends on the frequency range of interest Solution of (3) depends on the frequency range of interest

30. Each branch characterizes by m stable poles, but it does not mean that the whole Circuit will have the same poles to guarantee stability: realization must be done correctly ------------? more finite poles redundancy (each branch characterizes by m stable poles)Each branch characterizes by m stable poles, but it does not mean that the whole Circuit will have the same poles to guarantee stability: realization must be done correctly ------------? more finite poles redundancy (each branch characterizes by m stable poles)

31. Now the question is: how to guarantee that reduced circuit will be described by EXACTLY the SAME poles????????? Does the way of choosing sk influence at stability? (Open question) New poles appear in RHP Redundancy Now the question is: how to guarantee that reduced circuit will be described by EXACTLY the SAME poles????????? Does the way of choosing sk influence at stability? (Open question) New poles appear in RHP Redundancy

32. Capacitance does not influence at the poles of the circut stable if pi>0 and both ports: grounded / voltage / current sources SNA realization: not all super nodes play a role of the ports! poles are defined by pi=Ri/Li Capacitance does not influence at the poles of the circut stable if pi>0 and both ports: grounded / voltage / current sources SNA realization: not all super nodes play a role of the ports! poles are defined by pi=Ri/Li

36. Why modal approximation and not another passivity preserving tehnique??? The answer: we do not need to calculate ALL generalized eigenvalues. But in fact we can use any passivity preserving reduction techniques.Why modal approximation and not another passivity preserving tehnique??? The answer: we do not need to calculate ALL generalized eigenvalues. But in fact we can use any passivity preserving reduction techniques.

38. Optimal choice of match frequencies sk? (point of interest only for stable models)Optimal choice of match frequencies sk? (point of interest only for stable models)