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Chirayu S. Amin†, Yehea I. Ismail†, and Florentin Dartu*
Piece-wise Approximations of RLCK Circuit Responses using 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 Results
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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M2PWF: Moments to Piece-wise Functions
Motivation Fast and direct metric for RLCK circuits still missing Lognormal, Weibull, D2M, S2M, S2P, etc. do not work for 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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(extracted industrial netlist)
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 information to downstream cells
Gate CL 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 functions for 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 Results
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
x1 time xn VDD x(t) Voltage (t1,v1) (t2,v2) (tn-1,vn-1) (tn,vn) (t0,v0) x3 circuit response piece-wise function x(t) tn 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
x1 t1 t2 tn-1 tn v0 v1 v2 x2 time xn vn-1 1 x(t) Voltage 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 vn-1 xn x(t) 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 to that for PWQ
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Selection of time-points tk
Select tn and divide the time 0 to tn in to n pieces tn = 10|m1| works well and gives accurate results Ratio r = 1 equidistant time-points Ratio r > 1 (r 1.15) improves results r r2 r3 ...... time
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Extracting delay and transition time
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Outline Motivation/Introduction Piece-wise Approximations Results
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
RT = 2 , CT = 1.5 F, LT = 7H
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Results: RC circuit (uniform mesh)
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Delay errors at receiver nodes for an extracted netlist
50% delay (% error) at receiver nodes (time is scaled) Node SPICE HPW 4 moments SOE (PVL) 8 moments 1 2.15 2.19 (3.09) 0.67 (-68) 1.48 (-30) 2 2.38 2.34 (-0.36) 0.74 (-68) 1.67 (-29) 3 2.11 2.08 (-0.02) 0.61 (-71) 1.43 (-31) 4 1.91 1.92 (3.02) 0.55 (-70) 1.17 (-37) 5 2.12 2.10 (1.53) 0.64 (-69) 1.37 (-34)
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Outline Motivation/Introduction Piece-wise Approximations Results
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 parameters such as delay and transition time Method is general enough to be extended easily for other types of piece-wise functions
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Q & A
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