Saurabh K Tiwary

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Faster, Parametric Trajectory-based Macromodels
Via Localized Linear Reductions
Rob A Rutenbar Carnegie Mellon University rutenbar
_at_ece.cmu.edu

• Saurabh K Tiwary
• Cadence Berkeley Labs
• stiwary_at_cadence.com

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The Holy Grail for Circuit Macromodeling

• From a device-level circuit, to a lightweight
  model, automatically
• ..that does the following
• Captures important behaviors for verifying the
  circuit in system
• Jettisons the unimportant behaviors
• Simulates as fast as possible

auto
F(x)
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Background Trajectory Based Macromodels

• Original idea from Rewienski, Vasilyev, White at
  MIT
• Several recent extensions by Dong, Roychowdhury
  at Minnesota
• We model the circuit using a classical state
  space (SS) formulation

m inputs
k outputs
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Background Trajectory Based Macromodels

• Problem n huge. Solution Reduced
  approximation for x(t)

m inputs
k outputs
k
q

q
Cr
Br

m
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Background Trajectory Based Macromodels

• How do we get the reduced xr(t)? Projection
  methods

V obtained through Krylov methods andTruncated
Balanced Realization (TBR)
V (n?q) matrix
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Background Trajectory Based Macromodels

• Problem f (?) g(?) are still very nonlinear
• Solution Linearize the equation at different
  points in state space

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How Trajectory-Based Methods Really Work

• Simulate training inputsin state space
• Choose center pts xion this trajectory
• Linearize near each pt,projecting down toa
  smaller state vec
• Vote all linearizations(weighted by distance)to
  predict dynamics

State space
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Closer Look at Reduction

• Linear Circuits
• Reduce order of state space eqn.
• Small state-var ? Fast matrix solve
• Methods Krylov, PVL etc.

• Non-linear circuits
• Multiple linear ckts in state-space
• Generate reductions at each lin. pt.
• Different red. matrix (Vi) at each pt.

V
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Common Reduction Matrix (Vcommon)

• Interpolation for predicting dynamics
• Interpolate in reduced SS
• Need common reduction matrix (Vcommon) for all
  linearization pts
• Biorthogonalization for Vcommon
• Vcommon SVD(V1, V2, , Vs)
• Vcommon of size Nxq
• q lt N
• q ltlt N

Interpolation in common reduced state-space
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Our Prior Work Scalable Trajectory Models
Stored trajectory sample
xr2

• Early trajectory methods
• Broke down if we had lots of training data
• Could not handle data from many simulations to
  capture all behavior
• Scalable models
• Only use a small set of the nearby visited
  trajectory training points to create the
  linearization at new point x

You are here current point instate space for
reduced model
x
xr1
This point is ignored too far
xr2
Interpolation only uses K5 nearest sample pts,
combining thesematrices with someappropriate
weighting
x
xr1
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(1) Local Reduction

• Big Idea Only the k nearest pts need to be in
  reduced space
• Generate local reductions for k pts (Vlocal)
• Vlocal SVD(V1, V2, , Vk) -- k5 and s10,000
• Vlocal of size Nxq
• q ltlt N
• Mechanics
• Create groups s.t. for any point, there
• are k nearest pts belonging to same group
• For each group, create local reductions
• Interpolate locally

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Creating Local Groups

• Create clusters using GMMs
• Fringe pts with low membership prob.
• Create new cluster with nearest neighbors
• Create Vlocal for each group
• Store reduced lin. for each group member
• Memory Implication
• Multiple copies of same linearization
• One for each group membership
• q ltlt N -- not a big penalty

fringe points
trajectory linearizations
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(2) Loading Effects

• Obtained through circuit simulation
• KCL at the in/out node of the form
• C and G are the capacitance and conductance
  associated with the node
• C and G values are state dependent
• Interpolate them during model simulation

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(3) Model for AC Simulation

• SS eqn in s domain at DC operating pt
• Can use MOR for fast AC evaluation
• Goal Fast AC sim across large range of device
  process params
• Useful for circuit sythesis

Vin
Vout
Parameters W,L,tox,vth, etc.
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Earlier Approach Variational Trajectory Models

• Nonlinear transmission line
• Nonlinear element diode
• N 799
• q (nominal) 31
• a varied over 4 orders of mag
• q (parameterized) 34

Bond Daniel, DAC 2005
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Support for Variational AC Model

• Scalable MOR formulation
• DC linearizations in param space
• Simulate ckt for diff process design parameter
  values
• Generate local Models
• Interpolate

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Putting It All Together CMFB Opamp Model

• Circuit being macromodeled
• Folded cascode opamp with common mode feedback
  (CMFB) block
• 40 devices, 24 dimensions in original state
  space
• We train with 70 waveforms, mixed sinusoids and
  triangles, of widely varying amplitude and
  frequency content

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Local Vs Common Reductions

• Model Training
• N24, s8000, k5
• Common reduction
• q 19
• Simulation speed-up 9X
• Local reductions
• q 7-11
• Simulation speed-up 18.4X

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Ex Bi-quad Filter

• A bi-quad filter
• Uses the 40-transistor opamp
• Modeled as single circuit
• N70, q12-20
• Speed-up 30X

Vout
Vin
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Loading Model Works Too

• VCO in a simple 11 PLL
• New feature Loading
• Extended VCVS model to include state dependent
  input/output impedance
• Interpolated same as other dynamics
• Lets us handle loading-sensitive circuits like
  PLL

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Variational Models

• Preliminary results
• 40 transistor opamp ckt
• 5 parameters 4 device 1 process
• 800 linearization points
• Computation Gain Bandwidth
• Parameters

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Variational Models Results

• Initial training
• No. of samples 800
• Avg. error 3
• After pruning
• No. of samples 300
• Avg. error 5.5
• Nice control
• More error inc. sampling
• Speedup 200-400X
• No DC sim required

Gain (dB)
Model
Circuit
Frequency (Hz)
Frequency response at an untrained point in
parameter space
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Summary

• Trajectory methods very promising for on demand
  models.
• New contributions make methodology attractive for
  practical apps
• Local models provide additional speed-ups
• Loading models make system level simulation
  accurate
• Extensions for AC sim offers design space
  exploration opportunities
• Integration with SPICE 3f5 provides credibility
  to the approach.
• Support for BSIM3 models and hierarchical
  simulation.

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Thank You!
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Practical Issues

• Modeling infrastructure handles gt 10k pts
• Memory requirements are bottleneck
• Model potential candidate regions in design space
• Brute force complete design space modeling
  infeasible
• Eg. 10 param, 10pts each 1010 linearization pts
• Dimensionality not a problem
• Can easily handle gt30 dimensions
• Can use local functional fitting to increase
  coverage