Piecewise Approximations of RLCK Circuit Responses using Moment Matching

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Piece-wise Approximationsof RLCK Circuit
Responsesusing Moment Matching

• Chirayu S. Amin, Yehea I. Ismail, and Florentin
  Dartu

ECE Department Northwestern University Evanston
, IL 60208, USA
Intel Corporation, Hillsboro, OR 97124, USA
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Outline

• Motivation/Introduction
• Piece-wise Approximations
• Moments of a Piece-wise Function
• Moment Matching
• Piece-wise Linear Functions (PWL)
• Piece-wise Quadratic Functions (PWQ)
• Hybrid Piece-wise Functions (HPW)
• Extraction of Delay, Transition Time, etc
• Results
• Conclusions

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M2PWFMoments to Piece-wise Functions

• Motivation
• Fast and direct metric for RLCK circuits still
  missing
• Lognormal, Weibull, D2M, S2M, S2P, etc. do not
  workfor RLCK circuits
• Sum of exponentials (SOE) form (AWE, PRIMA, PVL,
  etc) used by model-order reduction techniques is
  very expensive and an overkill

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Shortcomings of SOE
PVL used for SOE
RLCK circuit response (extracted industrial
netlist)

• Requires too many moments ? too much runtime

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Limited propagation of informationto downstream
cells

• Cell library models can only handle limited
  information about input signals

No need to obtain all the details about the input
for STA!
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Example Piece-wise Linear (PWL)
5 pieces only!
RLCK Circuit Response
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Piece-wise functionsfor RC and RLCK circuits

• Main Idea
• y(t) is a piece-wise function
• Match moments to compute y(t)
• Advantages
• y(t) is very general
• Uses very few moments (4 or 5)
• Easy to extract timing parameters
• 50 delay
• Transition time

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Outline

• Motivation/Introduction
• Piece-wise Approximations
• Moments of a Piece-wise Function
• Moment Matching
• Piece-wise Linear Functions (PWL)
• Piece-wise Quadratic Functions (PWQ)
• Hybrid Piece-wise Functions (HPW)
• Extraction of Delay, Rise Time, etc
• Results
• Conclusions

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General piece-wise function
where u(t) is the unit step function
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What should the pieces xk(t) look like?

• Polynomials
• Simple
• Examples ? Linear, Quadratic, Cubic
• Mix-and-match approach
• First piece is quadratic
• Second piece is linear, etc
• Theory tested for
• linear, quadratic, and a hybrid quadratic version

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Piece-wise linear (PWL) function

• PWL
• Simplest piece-wise function

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Piece-wise quadratic (PWQ) function

• More accurate than PWL
• More variation in shapes of the pieces xk(t)

Voltage
1
x(t)
vn-1
xn
v2
x2
v1
x1
v0
t0
t1
t2
tn-1
tn
time
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Hybrid piece-wise (HPW) function(Enhancing
simple polynomials)

• First piece is quadratic with time t
• Remaining pieces are quadratic with 1/t
• Moment matching remains similar tothat for PWQ

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Selection of time-points tk

• Select tn and divide the time 0 to tn into n
  pieces
• tn 10?m1 works well and gives accurate
  results
• Ratio r 1 ? equidistant time-points
• Ratio r gt 1 (r ? 1.15) improves results

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Extracting delay and transition time
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Outline

• Motivation/Introduction
• Piece-wise Approximations
• Moments of a Piece-wise Function
• Moment Matching
• Piece-wise Linear Functions (PWL)
• Piece-wise Quadratic Functions (PWQ)
• Hybrid Piece-wise Functions (HPW)
• Extraction of Delay, Transition Time, etc
• Results
• Conclusions

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Experimental Setup

• Comparison of piece-wise approximations and
  traditional approach (PVL) with SPICE
• For a fixed number of moments
• Circuits
• Extracted netlists
• Industrial circuits
• Clock distribution networks
• Transmission lines, meshes, etc
• Tests cover RC as well as RLCK netlists

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Results Industrial RLCK netlist
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Results RLC transmission line
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Results RC circuit (uniform mesh)
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Delay errors at receiver nodesfor an extracted
netlist
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Outline

• Motivation/Introduction
• Piece-wise Approximations
• Moments of a Piece-wise Function
• Moment Matching
• Piece-wise Linear Functions (PWL)
• Piece-wise Quadratic Functions (PWQ)
• Hybrid Piece-wise Functions (HPW)
• Extraction of Delay, Transition Time, etc
• Results
• Conclusions

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Conclusions

• A new family of piece-wise MOR techniques
• Useful for RLCK and RC responses
• Accurate results with only 4 or 5 moments
• For RLCK circuits, error in delay is less than 5
• Fast Piece-wise Linear (PWL), Piece-wise
  Quadratic (PWQ), and Hybrid Piece-wise (HPW)
  approximations
• Closed form expressions for timing
  parameterssuch as delay and transition time
• Method is general enough to be extended
  easilyfor other types of piece-wise functions

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Q A