Compact Behavioural Modelling Behavioural Modelling Compact of - - PowerPoint PPT Presentation

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Compact Behavioural Modelling Behavioural Modelling Compact of - - PowerPoint PPT Presentation

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PBLM 1 Compact Behavioural Modelling Behavioural Modelling Compact of Electromagnetic Effects of Electromagnetic Effects in On- -Chip Interconnect Chip Interconnect in On Invited presentation at Invited presentation at MACSI- -NET

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Compact Compact Behavioural Modelling Behavioural Modelling

  • f Electromagnetic Effects
  • f Electromagnetic Effects

in On in On-

  • Chip Interconnect

Chip Interconnect

Invited presentation at Invited presentation at MACSI MACSI-

  • NET 2003, May 2

NET 2003, May 2-

  • 3,

3, Zürich Zürich

Peter B.L. Peter B.L. Meijer Meijer

Philips Research Laboratories Philips Research Laboratories Peter.B.L.Meijer@ Peter.B.L.Meijer@philips philips.com .com

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Contents Contents

  • Introduction

Introduction

  • Modelling

Modelling “flow” “flow”

  • From Maxwell’s Equations to circuit simulation

From Maxwell’s Equations to circuit simulation – – Example 1: Example 1: one wire, load

  • ne wire, load modelling

modelling – – Example 2: Example 2: two wires with cross two wires with cross-

  • talk

talk

  • Conclusions

Conclusions

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– – Find limiting factors in Find limiting factors in high high-

  • speed digital circuits

speed digital circuits – – Reference for validation Reference for validation

  • f future design rules
  • f future design rules

– – Reference for validation Reference for validation

  • f alternative models
  • f alternative models

Why? Why?

Source: Maly

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RLC lumped models RLC lumped models Transmission line models Transmission line models ROM techniques ROM techniques ... ... Modelling Modelling assumptions? assumptions?

Other approaches Other approaches

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Maxwell Equations Circuit Simulation ``FDTD’’ gives V(t), I(t)

Fit a linear dynamic model Our generalized formalism Post-optimize & generate syntax for simulation model

Modelling Modelling flow? flow?

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FDTD method FDTD method

  • Discretizes

Discretizes and and solves Maxwell’s solves Maxwell’s Equations in both Equations in both space and time space and time

  • CPU intensive!

CPU intensive!

FDTD = Finite Difference Time Domain: a method for solving the Maxwell Equations

Also yields voltages and currents

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FDTD: CPU time = f(mesh) FDTD: CPU time = f(mesh)

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Using FDTD Using FDTD

Mesh: 250,000 cells @ 2500 time points

60 micron Al strip

5 m i c r

  • n

1u x 1u 80 micron I d e a l c

  • n

d u c t

  • r

SiO2

Voltage step: 0.05 ps delay + 0.25 ps risetime SINSQ Pstar

T = 5 ps

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Using FDTD Using FDTD

t = 0.2 picosecond t = 0.4 picosecond t = 4.6 picosecond Top view Cross section view

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t = 0.2 picosecond t = 0.4 picosecond t = 4.6 picosecond

Using FDTD Using FDTD

SiO2

80 micron Al strip Perfectly conducting ground plane

Side view

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Using FDTD Using FDTD

  • 500 GHz reflections

500 GHz reflections

– – Reflections give Reflections give

  • vershoot and
  • vershoot and

affect wave shape affect wave shape at source side at source side – – Analyze beyond Analyze beyond measurement options measurement options – – Results sanity check: Results sanity check:

~0.4 ~0.4 ps ps for 60 micron: for 60 micron: 1.5 x 10 1.5 x 108

8 m/s ~= c /

m/s ~= c /

3.9,

3.9, (3.9 (3.9 is is rel rel. . perm perm. . SiO SiO2

2 )

)

T = 5 ps

~0.4 ps

Voltages Vs(t) and Vt(t)

Current Is(t) at source side

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Maxwell Equations Circuit Simulation FDTD gives V(t), I(t)

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Linear State Space Linear State Space Modelling Modelling

  • Assume

Assume linear state space model: linear state space model: Matrix equations Matrix equations x x’(t) = ’(t) = A x A x(t) + (t) + B u B u(t) (t) y y(t) = (t) = C x C x(t) + (t) + D u D u(t) (t)

  • Determine

Determine parameter matrices parameter matrices A, B, C, D A, B, C, D for given input vectors u for given input vectors u(t) (t) and output and output vectors y vectors y(t) (t) such that a best fit is obtained such that a best fit is obtained

  • MOESP/4SID class of

MOESP/4SID class of subspace identification algorithms subspace identification algorithms

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Linear State Space Linear State Space Modelling Modelling ( (MOESP/4SID)

Behavioural modelling based on time domain data (waveforms)

DeWilde, Verhaegen, Ciggaar, Meijer, Schilders

Steps: Steps:

  • Initial order

Initial order sufficiently high, e.g., sufficiently high, e.g., 200 200 SVD plot to estimate # time constants SVD plot to estimate # time constants

  • Reduce order

Reduce order to desired value, e.g., to desired value, e.g., 10 10

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Singular value plot Mapping PDE’s (Maxwell Eqs.) to equivalent high order (200) ODE system; no apparent SV clustering for selecting order

Dynamic wire current modelling

Using MOESP/4SID

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10-th order linear dynamic model fit (green) versus

  • riginal FDTD results (red)

Dynamic wire current modelling

Using MOESP/4SID

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Maxwell Equations Circuit Simulation FDTD gives V(t), I(t)

Linear state space modelling to get linear dynamic model

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Generalized Generalized Modelling Modelling Formalism Formalism

  • Map

Map linear state space model + parameters linear state space model + parameters to our neural network to our neural network modelling modelling formalism formalism Constructive and mathematically exact! Constructive and mathematically exact!

  • Post

Post-

  • optimize
  • ptimize to deal with numerical

to deal with numerical artefacts artefacts

  • f MOESP/4SID (instability & no implicit DC)
  • f MOESP/4SID (instability & no implicit DC)
  • Automatically

Automatically generate generate lumped linear circuit lumped linear circuit models for models for Pstar Pstar, , Spectre Spectre, VHDL , VHDL-

  • AMS, …

AMS, …

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Weighted sum Weighted sum s sik

ik

Differential equation Differential equation for neuron output for neuron output y yik

ik Basic Basic multilayer perceptron multilayer perceptron theory theory + extensions ( + extensions (Meijer Meijer 1996) 1996) Inputs Outputs

Feedforward Feedforward neural network neural network

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Learning Learning ( (= =Optimization) Optimization)

  • Define cost function, e.g.,

Define cost function, e.g.,

(model

(model -

  • data)

data)2

2

  • Discretize

Discretize and apply optimization algorithm and apply optimization algorithm* *, , involving combinations of involving combinations of

  • DC, TR and AC small signal analysis

DC, TR and AC small signal analysis

  • DC, TR and AC sensitivity (for gradients)

DC, TR and AC sensitivity (for gradients)

* *Conjugate gradient, BFGS, … Conjugate gradient, BFGS, …

  • Risks: slow convergence, local minima

Risks: slow convergence, local minima Neural Networks f Neural Networks for

  • r Device and Circuit Modelling

Device and Circuit Modelling

http://server506.hypermart.net/meijerpb/thesis/thesis_meijer.zip (11.5 MB)

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Applying Generalized Formalism Applying Generalized Formalism

Post-optimize

Fix MOESP/4SID artefacts: ensure stable model & fit with DC initial state

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Maxwell Equations Circuit Simulation FDTD gives V(t), I(t)

Linear state space modelling to get linear dynamic model Our generalized formalism Post-optimize & generate syntax for simulation model (lumped linear circuit model)

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Circuit simulation results Circuit simulation results vs vs FDTD FDTD

FDTD results and FDTD results and Neureka Neureka/ /Pstar Pstar NN model results NN model results

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Verify model generalization Verify model generalization

  • Linear model:

Linear model: modelling modelling for one signal with all for one signal with all (relevant) frequencies should suffice (relevant) frequencies should suffice -

  • in theory!

in theory!

  • Verify:

Verify: define a define a different stimulus different stimulus and check if and check if the FDTD simulation still matches results for the the FDTD simulation still matches results for the unchanged unchanged circuit model simulation circuit model simulation If so, that will confirm that the circuit model If so, that will confirm that the circuit model indeed applies to indeed applies to all stimuli all stimuli -

  • and not just the

and not just the

  • ne(s) used during
  • ne(s) used during modelling

modelling

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  • New

New stimulus stimulus STEP with STEP with slope /= 10, slope /= 10, applied to applied to

– – FDTD FDTD simulation simulation – – Unchanged Unchanged circuit circuit simulation simulation model model

Verify model generalization [1] Verify model generalization [1]

Circuit simulation model versus FDTD

Excellent fit!

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Verify model generalization [2] Verify model generalization [2]

  • New

New stimulus stimulus GAUSSIAN GAUSSIAN applied to applied to

– – FDTD FDTD simulation simulation – – Unchanged Unchanged circuit circuit simulation simulation model model

Excellent fit!

Circuit simulation model versus FDTD

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Single Wire Single Wire Modelling Modelling

FDTD simulation of FDTD simulation of Maxwell Equations Maxwell Equations in space and time in space and time Lumped linear dynamic Lumped linear dynamic circuit simulation model circuit simulation model

Complex wire load modelling

Orders of magnitude gain in simulation speed while preserving detailed (parasitic) effects

# Model Eqs. 10 5 - 10 6 # Model Eqs. 10 1 - 10 2

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Twoport modelling: dynamic load effects including cross-talk Example 2: Two wires with cross-talk

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Circuit simulation results Circuit simulation results vs vs FDTD FDTD

FDTD results and FDTD results and Neureka Neureka/ /Pstar Pstar NN model results NN model results

Circuit simulation model versus FDTD

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Verify Verify twoport twoport model generalization model generalization

  • New

New stimulus stimulus GAUSSIAN GAUSSIAN applied to applied to

– – FDTD FDTD simulation simulation – – Unchanged Unchanged circuit circuit simulation simulation model model

Excellent fit!

Circuit simulation model versus FDTD

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  • New numerical

New numerical interconnect analysis interconnect analysis options

  • ptions

going well going well beyond 100 GHz beyond 100 GHz

  • Many/All RF4D

Many/All RF4D parasitics parasitics can be included: can be included: capacitive capacitive and and inductive inductive effects, effects, skin skin effect effect

  • Reference for validation of new design rules

Reference for validation of new design rules

  • Reference for validation of alternative models

Reference for validation of alternative models

Conclusions Conclusions