Realizable Parasitic Reduction Using Generalized Y Transformation

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Realizable Parasitic Reduction Using
Generalized Y-? Transformation

• Jeffrey Z. Qin, Chung-Kuan Cheng
• Computer Science and Engineering
• University of California, San Diego

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Agenda

• Motivation
• Generalize Y-? Transformation
• Elements
• Order of Reduced Model
• Order of Picking Reducible Nodes
• Numerical Problem
• Realize ?-Admittance Using Templates
• Experimental Results
• Conclusion

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Motivation

• Existing model order reduction methods
• Matrix-based reduction
• PACT, PRIMA
• Topology-based reduction
• AWE ? tree based, not realizable
• TICER ? 1st order realizable model
• Features needed ? more general topology,
  higher-order realizable model

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Overview Y-? Transformation

• Advantage of traditional Y-? transformation
• Topology-based method
• Handles linear circuits of any topology
• Transforms with fidelity transfer function
  evaluated by Y-? is equal to the one evaluated
  by Cramers rule
• Plus, after being generalized
• stable models
• realizable parasitic reduction

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Traditional Y-? Transformation

• Conductance in series
• Conductance in star-structure

n0
n1
n2
n1
n2
n1
n1
n0
n3
n2
n2
n3
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Fidelity of Y-? Transformation
Cramers rule
Y-? Transformation
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Generalized Y-? Reduction

• Improve traditional Y-? transformation

generalized
traditional
Handle RCLK-VJ? No
YES
Transformed model order? High
REDUCED
Optimize priority of picking
nodes?
No YES
Resolve common-factor numerical problems?
No YES
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Handle RCLK-VJ Elements
Handle R, C, L, and J
all coefficients are positive! because no
subtraction ops.
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Handle RCLK-VJ Elements (2)
Handle K
Nodal equations are the same.
e. g.
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Handle RCLK-VJ Elements (3)

• Handle V
• voltage sources (V) can be transformed into
  current sources (J).

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Order Reduced by Truncation
Admittance truncated after certain order terms
Fidelity of Y- ? transformation is still
preserved.
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Optimize Node Ordering

• Y-? transformation Gauss elimination
• priority of picking nodes pivoting in LU
  factorization

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Common Factors
Type I Common Factors
n1
not recognize
recognize type-I
n2
n3
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Common Factors (2)
Type II Common Factors
n2
n3
n2
n3
should be canceled out.
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Common Factors (3)
Why is it harmful?

• 1st order coefficients grow at about 2n rate
• 2nd order coefficients grow at about 3n rate,
• Worse when circuit is larger, preserved order is
  higher.
• Scaling doesnt work!

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Common Factors (4)

• Theoretical Contribution
• common-factor effects generally exist in Y-?
  transformation for linear circuits of any
  topology
• devise algorithm to treat common factors
• ? admittance is in the simplest form, or has no
  redundancy, after the treatment.

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Realizable Parasitic Reduction

• Why realize?
• Realizable reduced order models preferred by
  standard parasitic extraction tools, for
  stability, compatibility concerns.
• Hard problem!
• No tool is able to realize 1st-order admittance
  matching 1st order moments from arbitrary linear
  circuit
• ? admittance may not be realizable in nature
• Y-? all coefficients are non-negative!

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1st order Realization

• Y-? reduction 1st-order reduced admittance
  
• can be realized, because

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High Order Realization Template

• solution
• Formulate the realization problem as an
  optimization problem using geometric programming
  (GP).
• 0th and 1st order templates

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High-Order Realization Template (2)

• Higher-order templates

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Realization Formulation
Pij and qij are the j-th product term of
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Realization Formulation (2)

• Then, given a low-order admittance
  where ai and bj are positive constants, we
  have
• GP formulation

Objective
subject to
and
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Waveform Evaluation
Comparison with AWE and PRIMA
SPICE PRIMA (8th order) AWE (3rd order) Y-? (8th
order)
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Waveform Evaluation (2)
Comparison with TICER
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Pole Estimation
Exact transfer function
dominant poles
Y-? matches less moments, but more accurate!
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Natural frequency Estimation
Pole plot
Waveform plot
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Common-Factor Impacts
Growth of powers of s
Without cancellation For tree-like circuits,
the order of the input admittance grows
quadratically with the size of LC elements.
With cancellation The orders are the same as
those of original circuits.
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Common-Factor Impacts (2)
Growth of coefficients
Scaling doesnt work, as coefficients of
different orders grow at different
rate. Common-factor cancellation works great!
1st order coefficient
2nd order coefficient
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Conclusion

• A generalized Y-? reduction
• handles RCLK-VJ circuits of any topology
• Yields order-reduced models with fidelity
• Common factor cancellations
• remove fake poles/zeros
• make the reduction numerically stable.
• Parasitic realization
• Stability of the reduced model is guaranteed
• Realizable models good for parasitic reduction.