CSE245: Computer-Aided
Circuit Simulation and
Verification
Lecture Note 4
Model Order Reduction (2)
Spring 2010
Prof. Chung-Kuan Cheng
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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
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Conventional Design Flow
Function Sepc
Front-end
Back-end
Parasitics resistance, capacitance
and inductance cause noise
, energy consumption and
power distribution problem
RTL
Beh. Simul
Logic Synth.
Stat. Wire Model
Gate-level Net.
Gate-Lev. Sim
Floorplanning
Para. Extraction
Place & Route
Layout
3
Parasitic Extraction
R,L,C Extraction
Model Order Reduction
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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.
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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.
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Congruence Transformation
• Definition:
• Property: Congruence transformation
preserves semidefiniteness of the matrix
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Krylov Subspace
• Given an n x q matrix Vq whose column
vectors are v1, v2, …, vq. The span of Vq is
defined as
• Given an n x n matrix A and a n x 1 vector r
the Krylov subspace is defined as
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PRIMA
• Passive Reduced-order Interconnect
Macromodeling Algorithm.
– Krylov subspace based projection method
– Reduced model generated by PRIMA is passive
and stable.
PRIMA
(system of size n)
(system of size q, q<<n)
where
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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
general
do not preserve this structure in
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PRIMA: Moment Matching Proof
Used lemma 1
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PRIMA: Lemma Proof
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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.
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SPRIM
• Recall
• Suppose Vq is generated by Arnoldi process as
in PRIMA. Partition Vq accordingly
• Construct New Projection Matrix
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SPRIM
• Congruence Transformation
• Now structure is preserved
• Transfer function for the reduced order
model
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Traditional Y- Transformation
•
Conductance in series
g1
n1
•
g2
n0
g1 g 2
y12 
g1  g 2
n1
n2
Conductance in star-structure
n1
y12 
g1
g2
n2
n0
g1 g 2
g1  g 2  g 3
n1
y13 
g3
n3
n2
n2
g 2 g3
y23 
g1  g 2  g 3
g1 g 3
g1  g 2  g 3
n3
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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
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between j and k
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
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