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

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

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

(extracted industrial netlist) Shortcomings of SOE *PVL used for SOE RLCK circuit response (extracted industrial netlist) Requires too many moments  too much runtime

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!

Example: Piece-wise Linear (PWL) 5 pieces only! RLCK Circuit Response

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

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

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

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

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

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

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

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

Extracting delay and transition time v tv

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

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

Results: Industrial RLCK netlist

Results: RLC transmission line RT = 2 , CT = 1.5 F, LT = 7H

Results: RC circuit (uniform mesh)

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)

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

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

Q & A