SAPOR: SecondOrder Arnoldi Method for Passive Order Reduction of RCS Circuits

1
SAPOR Second-Order Arnoldi Method for Passive
Order Reductionof RCS Circuits

• Yangfeng Su, Jian Wang, Xuan Zeng (Fudan
  U.)Zhaojun Bai (U. of California, Davis)Charles
  Chiang (Synopsys Inc.)Dian Zhou (Fudan U.)
• ICCAD 2004, San Jose

Microelectronics Department, Fudan University
2
Outline

• Introduction
• MOR of RCS Circuits
• SAPOR
• Numerical Examples
• Conclusions

3
Outline

• Introduction
• MOR of RCS Circuits
• SAPOR
• Numerical Examples
• Conclusions

4
Background

• Chip performance is dominated by interconnects

Interconnect modelingand simulation becomemore
and more important!
5
Interconnect Analysis

• Challenges
• Magnetic coupling effects
• Complex structure large scale
• Modeling
• Inductance based
• Susceptance based
• Simulation
• MOR (Model Order Reduction)

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Magnetic Coupling Modeling

• Inductance based
• Slow mutual inductance dropping as distance
  increasing
• Inductance matrix L is quite dense
• Susceptance based
• Much faster mutual susceptance dropping
• Susceptance matrix S is diagonally dominant, and
  can be effectively sparsified

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RCS Circuit Reformulation

• First-order formulation (MNA)
• Second-order formulation

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Model Order Reduction
First-order formulation
Second-order formulation
AWE
Pioneer Work
ENOR
PRIMA
?
SMOR
Passivity preserving
PVL
Numerical stability
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SAPOR

• SOAR Second-Order ARnoldi method
• First proposed for quadratic eigenproblems, 2003
• SAPOR Second-order Arnoldi method for
• Passive Order Reduction
• A variant of SOAR
• Proven moment-matching
• Guaranteed passivity
• Numerical stable

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Outline

• Introduction
• MOR of RCS Circuits
• SAPOR
• Numerical Examples
• Conclusions

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RCS Circuit Order Reduction

• RCS circuits are best formulated as 2nd-order
  system
• Important properties are preserved
• However, PRIMA to RCS circuits cannot guarantee
  passivity
• Using PRIMA-style orthogonal projection to define
  a reduced model to preserve structure and
  passivity

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ENOR Method Sheehan,1999

• Let and
• Then moment vectors of V given by recursion
• Using Gram-Schmidt to orthonormalize the moment
  vectors
• Unsolved issues
• Simultaneously moments calculation and
  orthonormalization for the recursions cannot
  guarantee exact moment-matching
• Y vectors are not orthonormalized, which could
  cause numerical instability

With
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Interconnect Circuit
8-bit bus with 2 shielding wires (coupling not
shown)
330 nodal voltages 160 susceptance currents
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ENOR Method (cont.)

• Reduced system order 40, 60, 80

Absolute Error
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SMOR Method Zheng and Pileggi,2002

• Eliminate Y vectors in ENOR recursion
• Assume that
• Simplified recurrence relation
• Orthonormalize approximated moment vectors
• Unsolved issue
• Approximated recursion introduces errors

With
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SMOR Method (cont.)

• Reduced system order 40, 60, 80

Absolute Error
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Outline

• Introduction
• MOR of RCS Circuits
• SAPOR
• Numerical Examples
• Conclusions

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Moments Recursion
Shifting with
Tayler expansion

• Recurrence relation for moment vectors

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System Linearization
Introducing variable Z satisfying

• Linearized form

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Linearized System

• q0 and p0 are the first moments of V and Z,
  respectively
• i-th moment of is
• i-th moment of V is

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Generalized SOAR

• The only matrix inversion involved is K?1 (in A)
• A memory-saving variant exists in which p vectors
  are not explicitly saved

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Theory of GSOAR

• Theorem 1 vectors form an
  orthonormal basis of moment space
  of V.
• Structure-preserving reduced order systems
• Theorem 2 moments of the reduced system match
  the first n moments of the original system
  exactly.

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Summary of SAPOR

• Formulating RCS circuit as 2nd-order system
• Using generalized SOAR to compute an orthonormal
  basis Qn of moment vector space of V
• Defining the reduced order system by PRIMA-style
  directly orthogonal projection based on Qn

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Advantages of SAPOR

• SAPOR accurately matches the moments of the
  original system.
• SAPOR is an Arnoldi-like Krylov subspace
  technique and is numerical stable and efficient.
• Computational costs of ENOR, SMOR and SAPOR are
  of the same order.

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Outline

• Introduction
• MOR of RCS Circuits
• SAPOR
• Numerical Examples
• Conclusions

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Interconnect Circuit
8-bit bus with 2 shielding wires (coupling not
shown)
330 nodal voltages 160 susceptance currents
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SAPOR

• Reduced system order 40, 60, 80

Absolute Error
More matched moments lead to wider matched
frequency range
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MOR Accuracy Comparison

• Reduced system order 80

ENORSMORSAPOR
Absolute Error
SAPOR is far more accurate
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Frequency Responses Comparison

• Reduced system order 80

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Outline

• Introduction
• MOR of RCS Circuits
• SAPOR
• Numerical Examples
• Conclusions

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Conclusions

• SAPOR a structure-preserving technique for MOR
  of second order system (with non-homogeneous
  right hand).
• PRIMA-style direct orthogonal projection
  guarantees the passivity.
• Moment-matching is proven.
• Arnoldi-like method, the reduction process is
  numerical stable and efficient.

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Further Reading

• Y. Su et al, SAPORSecond-order Arnoldi method
  for passive order reduction of RCS circuits,
  ICCAD 2004
• Z. Bai and Y. Su, SOARA second-order Arnoldi
  method for the solution of the quadratic
  eigenvalue problem, SIAM J. Matrix Anal. and
  Appl., to appear
• Z. Bai et al, Arnoldi methods for
  structure-preserving dimension reduction of
  second-order dynamical systems, submitted to
  Oberwolfach proceedings of Dimension Reduction,
  2004
• http//www.cs.ucdavis.edu/bai
• 
  Thank You!