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CSE245: Computer-Aided Circuit Simulation and
VerificationLecture Note 4
Model Order Reduction (2)
Spring 2010
Prof. Chung-Kuan Cheng
1
Model Order Reduction: Overview
• Explicit Moment Matching– AWE, Pade Approximation
• Implicit Moment Matching (Projection Framework)– Krylov Subspace Methods
• PRIMA, SPRIM• Gaussian Elimination
– TICER, Y-Delta Transformation
2
Conventional Design Flow
Function Sepc
RTL
Logic Synth.
Gate-level Net.
Floorplanning
Place & Route
Layout
Front-end
Back-end
Beh. Simul
Stat. Wire Model
Gate-Lev. Sim
Para. Extraction
Parasitics resistance, capacitance and inductance cause noise, energy consumption and power distribution problem
3
Parasitic Extraction
R,L,C Extraction
Model Order Reduction
4
Moment Matching Projection method
• Key ideal of Model Order reduction:
“Moments Matching” and “Projection”• Step1: identify internal state function and
variables.• Step2: Compose moments matching. (Pade,
Taylor expression).• Step3: Project matrix with matching moments.
(Block Arnoldi (PRIMA) or block Lanczos (PVL))• Step4: Get the reduced state function.
5
Explicit V.S. Implicit Moment Matching
• Explicit moment matching methods – Numerically ill-conditioned
• Implicit moment matching methods – construct reduced order models through
projection, or congruence transformation.– Krylov subspaces vectors instead of moments are
used.
6
Congruence Transformation
• Definition:
• Property: Congruence transformation preserves semidefiniteness of the matrix
7
Krylov Subspace
• Given an n x n matrix A and a n x 1 vector r the Krylov subspace is defined as
• Given an n x q matrix Vq whose column vectors are v1, v2, …, vq. The span of Vq is defined as
8
PRIMA• Passive Reduced-order Interconnect
Macromodeling Algorithm.– Krylov subspace based projection method– Reduced model generated by PRIMA is passive
and stable.
where
PRIMA
(system of size n) (system of size q, q<<n)
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PRIMA• step 1. Circuit Formulation
• step 2. Find the projection matrix Vq
– Arnoldi Process to generate Vq
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PRIMA: Arnoldi
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PRIMA• step 3. Congruence Transformation
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PRIMA: Properties• Preserves passivity, and hence stability
• Matches moments up to order q (proof in next slide)
• Original matrices A and C are structured.
• But and do not preserve this structure in general
13
PRIMA: Moment Matching Proof
Used lemma 1
14
PRIMA: Lemma Proof
15
SPRIM• Structure-Preserving Reduced-Order
Interconnect Macromodeling– Similar to PRIMA except that the projection
matrix Vq is different
– Preserves twice as many moments as PRIMA– Preserves structure– Preserves passivity, stability and reciprocity– Matching the same number of moment as
PRIMA, but preserve the structure which can reduced numerical calculation.
16
SPRIM
• Suppose Vq is generated by Arnoldi process as in PRIMA. Partition Vq accordingly
• Recall
• Construct New Projection Matrix
17
SPRIM
• Congruence Transformation
• Now structure is preserved
• Transfer function for the reduced order model
18
Traditional Y- Transformation
• Conductance in series
• Conductance in star-structure
1gn1 n22g 21
2112 gg
ggy
n1 n2
n1n1
1g
2g 3g
n2 n3n2 n3
321
3113 ggg
ggy
n0
n0
321
2112 ggg
ggy
321
3223 ggg
ggy
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TICER (TIme Constant Equilibration Reduction)
• 1) Calculate time constant for each node
• 2) Eliminate quick nodes and slow nodes– Quick node: Eliminate if – Slow node: Eliminate if
• 3) Insert new R’s/C’s between former neighbors of N
– If nodes j and k had been connected to N through gjN and gkN, add a conductance of value gjNgkN/GN between j and k
– If nodes j and k had been connected to N through cjN and gkN, add a capacitor of value cjNgkN/GN between j and k 20
TICER: Issues
• Fill-in– The order that nodes are eliminated matters
• Minimum Degree Ordering can be implemented to reduce fill-in
– May need to limit number of incident resistors to control fill-in
• Error control leads to low reduction ratio
• Accuracy– Matches 0th moment at every node in the
reduced circuit. – Only Correct DC op point guaranteed
21
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