Constructing Current-Based Gate Models Based on Existing Timing Library

Constructing Current-Based Gate Models Based on Existing Timing Library

Andrew Kahng, Bao Liu, Xu Xu

UC San Diego

http://vlsicad.ucsd.edu

OutlineOutline

Gate Modeling Background

Problem Formulation

Approximation and Regression

Applications

Experiments

Conclusion

Gate ModelsGate Models

K-factor lookup tables

1. Dg = f(Cload, Tr)

2. Trout = g(Cload, Tr)

Efficient capacitance Ceff for distributed load capacitance To achieve identical gate delay

(and output signal transition time at the same time!)

E.g., by achieving the same average gate output current

Calculating Effective CapacitanceCalculating Effective Capacitance

1. If (Ceff > Cload || Ceff < 0)

2. Return Cload

3. Else if(Ceff < )

4. Return Ceff

5. Else

6. Continue iteration

May not converge

No equivalent gate delay and Trout at the same time

Waveforms are not ramp functions!

Trout = g(Ceff, Tr)

Ceff s.t. Iout(Ceff)=Iout(load)

Ceff = Cload

Current-Based Transistor ModelCurrent-Based Transistor Model MOSFET is a voltage-controlled current source,

e.g., as in the alpha-power-law model

For a simple inverter, gate output current is given by one of the transistors

An equivalent inverter macro-model for an inverting complex gate

current-based gate modeling

Ids

Vgs V tW

Leff

P c

PvVgs V t Vds Pv Vgs V t

W

LeffP c Vgs V t Vds Pv Vgs V t

02( ) ( )

( ) ( )

/

Current-Based Gate ModelingCurrent-Based Gate Modeling Consists of a lookup table I(Vi, Vo) and C(Vi, Vo)

Transient analysis for output signal waveform

Vi I(Vi, Vo)

C

RVoVi

Voltage-Based Current-Based

Gate Pre-CharacterizationGate Pre-Characterization Current-based gate models need additional pre-

characterization, e.g., I(Vi, Vo), given by SPICE DC sweep analysis

Cadence Effective Current Source Model (ECSM)

Synopsys Composite Current Source Model (CCS)

Rise_transition (template) { index_1: // slew rate index_2: // load cap values: // output Tr ecsm_waveform (name1) { index_3: // output voltage values: // time point }}

OutlineOutline

Gate Modeling Background

Problem Formulation

Approximation and Regression

Applications

Experiments

Conclusion

Constructing Current-Based Gate Model From Existing Timing LibrariesConstructing Current-Based Gate Model From Existing Timing Libraries

Given gate delays and output slew rates for load caps and input slew rates, find an equivalent current-based gate model, e.g., I(Vi, Vo) and C

Vi I(Vi, Vo)

C

1. Dg = f(Cload, Tr)

2. Trout = g(Cload, Tr)

Tr

Clo

ad

Dg

Tr

Clo

ad

TroutVo

Vi

I

C

To find an unknown underlying physical process by a set of measurements

Q = C V

Inhomogeneous Fredholm integral equations of the first kind

Inverse ProblemInverse Problem

outTrinTrDg

outTrinTrDg

ddL

inTrDg

ddL

VCCdtVoViI

VCCdtVoViI

5.05.0

5.05.0

5.0

0

))(2.08.0(),(

)(5.0),(

Integral equations differential equations

Apply interpolation to reduce variables to those in the I(Vi, Vo) lookup table

Inverse problem solutions are extremely sensitive to input data perturbations!

Inverse problem Optimization w/ objective A + S (A: accuracy, S: smoothness, : weighting factor)

Solving an Inverse ProblemSolving an Inverse Problem

a I C C V

b I C C V

i i L ddi

i i L ddi

0 5

0 8 0 2

. ( )

( . . )( )

OutlineOutline

Gate Modeling Background

Problem Formulation

Solution: Approximation and Regression

Applications

Experiments

Conclusion

Polynomial Regression of I(Vi,Vo)Polynomial Regression of I(Vi,Vo)

A priori knowledge:

Approximate I(Vi, Vo) by a quadratic polynomial

9+1 coefficients in a limited range

Polynomial Regression of I(Vi,Vo)Polynomial Regression of I(Vi,Vo)

I V V a a V a V

a V a V V a V V

a V a V V a V V

i o o o

i i o i o

i i o i o

( , )

0 0 0 1 0 22

1 0 11 1 22

2 02

2 12

2 22 2

I

I

I

I

I V V

I V VV i i o

Vo i o

( , )

( , )

( , )

( , )

( , )

( , )

0 0

0 1

1 0

1 1

0

0

0

0

0

0

a a

a a a

a a a

a a a

a a a

a a a

0 0 2 0

0 1 0 2 1 0

0 0 0 1 0 2

0 0 1 0 2 0

0 1 11 2 1

1 0 11 1 2

0

0

0

0

0

0

,

, ,

1. Start with an initial polynomial coefficient

2. For each iteration

3. Perturb a coefficient ai’ = ai +

4. Compute mean square gate delay mismatch

5. If reduces, commit perturbation ai = ai +

6. Else, go other direction ai = ai –

7. Stop if no improvement

8. Reduce step for another iteration

9. Compute I(Vi, Vo) and C

Our Constructive Method Our Constructive Method

ApplicationsApplications

More accuracy, arbitrary waveform

Efficiency advantage over SPICE simulation

Gate delay calculation for Long interconnects

Cross-coupling interconnects

Supply voltage drop effect

Supply current calculation

Noise calculation

Supply Voltage Variation Effect on Gate Delay CalculationSupply Voltage Variation Effect on Gate Delay Calculation

There exists an equivalent inverter macro-model for each input combination for any (inverting) complex gate

Adjust input and output voltages for I(Vi, Vo) table lookup for a falling input signal, but not for a rising input signal

VVV

VVV

VVII

oo

ii

oi

'

'

)','('

ExperimentsExperiments BPTM 70nm technology cell library

Compare (our) constructed and (SPICE simulation based) pre-characterization models

Our Constructed

Quad. Pre -Characterization

Cubic Pre -Characterization

invx4 1.25 5.7 9.73 45.2 11.12 37.8

invx8 1.85 16.9 12.93 27.2 14.67 27.8

nor2x4 0.46 4.8 7.06 152.7 5.91 56.5

ExperimentsExperiments Gate delays by (1) our model and (2) pre-characterized

model normalized by SPICE simulation resultsFor Ideal (1.0V) Supply Voltage

(1) 93.0 97.4 99.0 101.0 97.8 94.9 95.7 98.0

(2) 100.0 98.7 99.3 100.0 102.5 101.2 100.6 100.0

(1) 98.4 102.6 99.0 95.5 103.9 100.8 100.2 99.5

(2) 104.3 101.8 100.0 100.0 97.6 98.5 99.2 100.1

For Degraded (0.9V) Supply Voltage

(1) 102.0 105.4 106.7 108.6 101.2 106.8 108.6 106.9

(2) 102.1 99.8 98.7 98.8 100.8 99.6 98.7 97.9

(1) 107.8 106.4 106.6 106.2 107.8 106.4 106.6 108.3

(2) 100.0 99.6 99.1 97.8 95.6 97.9 98.7 100.2

SummarySummary Utilize existing timing libraries for application of

novel current-based gate modeling

Wide range of applications: supply current calculation, delay calculation for complex waveforms, e.g., resistive shielding, crosstalk coupling, supply voltage variation, etc.

Slightly less accurate than pre-characterized current-based gate models, e.g., within 8.6% vs. 4.4% for gate delay calculation with varied supply voltage

Reasonable runtime for model construction, 28.3 seconds in average on a 2.8GHz P4 system

Thank you !Thank you !