Elmore Model for Energy Estimation in RC Trees

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Elmore Model forEnergy Estimationin RC Trees

• Quming Zhou and Kartik Mohanram
• Dept. of Electrical and Computer EngineeringRice
  University, Houston

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Purpose

• Analysis model for energy estimation
• Time-varying voltage source
• Delay-energy model
• RC tree/interconnect

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Outline

• Introduction
• Motivation
• Delay-dependent energy model
• Energy estimation in resistive elements
• Simulation results
• Model validation
• Model comparison
• Conclusions

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Self-heating

• Interconnect is becoming highly resistive
• Small geometric size
• More Joule heating
• Impacts on performance and reliability
• Thermal gradient
• Delay penalty
• Electromigration effect

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Conventional energy model

• Conventional model, 1/2CV2
• Not accurate
• No source delay
• No spatial information
• Model for distributed resistive elements
• Fast and accurate
• Energy dissipated by total distributed load
• Energy dissipated by an individual resistor

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Potential applications

• Delay sources determine the energy dissipation
• Buffer
• Time-varying power supply
• Resonant clock generator
• Reduce energy consumption less than 1/2CV2
• Adiabatic circuits, recycle the power
• Improve thermal estimation
• Spatial information

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Observation
Faster driver
Technology trend
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Delay-dependent energy model

• Self-heating of R
• Scaling factor
• Less energy dissipated at resistor R as the
  source switches slower

R
C
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Distributed delay-energy model

• Elmore model to estimate energy in RC tree
• Elmore delay
• General approach
• Calculate the current through each resistor
• Integrate I2R over time for each resistor
• Sum up energy dissipated by each resistor

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Energy estimation

• Second-order model for current
• Smooth shape in RC tree
• Self-heating as I2R
• Scaling factor

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Resistor energy
Weighted average Elmore delay of all downstream
nodes of node i
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Simple example
Individual energy ?
Total energy ?
t
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Simple example
Individual energy ?
Total energy ?
t
0.5CV2 3
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RC interconnect

• Interconnect modeled by uniform RC segments
• Otherwise, RC tree model
• Weighted average Elmore delay of each segment is
  bounded
• Bounded energy model

RICI/3
RICI/2
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Interconnect energy estimation

• Interconnect without a load CL
• Interconnect with a load CL, t 0
• CLltltCI
• CLCI

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Model validation

• General energy model for RC tree
• Random RC tree with 10000 RC elements
• SPICE takes 10s
• Proposed model takes 10ms
• 2.5 error in total energy estimation
• Bounded energy model for interconnects
• 1/3 lower bound 2/3 upper bound
• 10k simulations on random RC values
• 3.5 error in total energy estimation

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Selfheating profile
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Previous approaches

• Model-order reduction 15 (Shin, TCAD 2002)
• Model-order reduction for current of each
  resistor
• General approach for any network
• Very accurate
• High computational cost, not analytical model
• RC ladder analysis model 2 (Alioto, TVLSI 2004)
• RC ladder network
• Only total energy
• Less accurate

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AWE circuit
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Accuracy comparison ()
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Conclusions

• We present a delay-energy model
• RC tree and RC interconnect
• Time-varying voltage supply
• Energy estimation is similar to Elmore delay
  estimation
• Recursive tree traversal
• Energy of each resistor as well as total energy
• Spatially aware