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SPICEDiego
A Transistor Level Full System Simulator
Chung-Kuan Cheng
May 27, 2004
Computer Science & Engineering Department
University of California, San Diego
Outline
 Motivation
 Status of Commercial Simulators
 Solver Engine: Multigrid Review
 Activity Driven Analysis
 Nonlinear Transistor Devices
 Experimental Results
 Conclusion
Motivation
 Moore’s Law
 # transistors and clock frequency double /2 years
1984: 100K transistors, 10M Hz
2003: 100M transistors, 5G Hz
 More Challenges for Circuit Simulator
 Electrical Coupling (C&L):
interconnect delay, crosstalk, voltage drop, ground
bounce
 Short
Channel Devices
 SPICE
 Cannot perform large chip analysis, capacity limit to
50,000 transistors ( O(n2) complexity )
Status of Commercial Simulators
Partition Based Simulation
 Most commercial fast spices (HSIM, Power Mill / Time Mill / Rail
Mill, NaroSim, RedHawk, Ultrasim)

Advantage
• Smaller Matrix Size
• Easy to apply varies time step to different subcircuits
 Disadvantage
• Hard to catch coupling effect between subcircuits
• Device Model ignoring Miller’s effect
• Potential convergence problem
• Accuracy not guaranteed.
Motivation
High
Direct Method
Complexity
Basic Iterative
Slow
Convergence
Conjugate Gradient
Method
Multigrid Method
Multigrid Review
 Error Components

High frequency error (More oscillatory between neighboring
nodes)

Low frequency error (Smooth between neighboring nodes)

Basic iterative methods only efficiently reduce high
frequency error
 Basic Idea of Multigrid

Convert hard-to-damp low frequency error to easy-to-damp
high frequency error
Multigrid : A Hierarchy of Problems
A2• X2=b2
A1 • X1=b1
2
1
Gauss Elimination
4
2
1
Restriction
3
Smoothing
A0 • X0=b0
4
2
1
Interpolation
3
6
5
Restriction
Smoothing
Smoothing
Interpolation
Smoothing
Hierarchically, all error components smoothed efficiently
Geometric vs Algebraic
 Geometric multigrid method
 Require
Regular Grid Structure
 Algebraic Multigrid
 Coarsening


Relied on Matrix,
No requirement of regular grid structure
Coloring scheme
 Error
Smoothing Operator: Gauss-Seidel
 Interpolation
Ae  0


aiiei    aije j
j i
Small residue but the error decreases very slowly.
In practice, we use only coarse node at the RHS of above
formula to approximate error correction of fine node.
Convergence of Multigrid Method
 The matrix needs to be symmetric positive definite
 Key
to the convergence of iterative method
SOR, PCG, Multigrid
 RC network
 The
system matrix is S.P.D(symmetric positive definite)
System Equation:
CX (t )  GX (t )  U (t )
Apply Trapezoidal Rule:
2
2
(G  C ) X (t  h)  (G  C ) X (t )  U (t )  U (t  h)
h
h
LHS matrix is S.P.D, it is also valid for B.E. and F.E formulae
Convergence of Multigrid Method
 RLKC network
System Equation:
C
0

0  V   G
 
L   I   Al
T
 Al  V  U 
    
0  I   0 
Apply Trapezoidal Rule:
T
 2C
 Al  V (t  h)  2C
G
G

 h
2L  
h

 A
  I (t  h)    A
l
l
h 


Al   Vt )  U (t  h)  U (t )
2L     

  I (t ) 
0

h 
T
The LHS matrix is not S.P.D, but can be converted to S.P.D matrix
 2C
h T 1

G

A L A )V (t  h ) 

 h
l
l
2


h
 2C

 G  A T L 1A )V (t )  [U (t )  U (t  h )]  2 A T I (T )

l
l
2 l

 h

h
I (t  h )  I (t )  L 1A [V (t  h )  V (t )]

l

2
The LHS matrix of first equation is now S.P.D. Similar for B.E and F.E
L-1 is called K / Susceptance / Reluctance Matrix
Why Algebraic Method
 No Requirement of Regular Grid
 Works
for general circuits.
 Circuit with Mutual Inductance
 Adjacency
graph of the converted system matrix
is different from circuit topology.
Converted System Matrix:
2C
h
 G  A T L 1A
l
h
2 l
Activity Driven Analysis
 Circuit Latency & Multi-rate Behavior

Spatial Latency

Only portions of the circuit is active at any given time
80%-90% of total gates are non-switching

Temporal Latency
A given portion of circuit is not always active.

Multi-rate Behavior

Varies time constant  multi-rate behavior
 How to utilize ?

Circuit Partitioning: common technique used in timing
simulators.
Adaptive Smoothing
 HOW?
 Only active regions get error smoothed
 Varies “time step size”
 inactive subcircuits may only get chance to have error
smoothed at finest level once every several time points
 WHY?
 Error smoothing operation at finer levels takes most of the
iteration time
 Smoothing at coarser level is sufficient for inactive portions
of circuit
Adaptive smoothing at finest grid level
Incorporating Transistor Devices (1)
•Direct Simulation of Transistor Devices Makes Linear Solver Diverge
•Conventional Method: Abstract Device as Current Waveform, Ignore the
Interaction with VDD/VSS.
• How to include Transistor Devices?
Inside the inner most NewtonRaphson linearization iteration,
decouple the linear and
nonlinear interface, replaced by
Norton Equivalent Circuit.
Incorporating Transistor Devices (2)
 Advantage
 Possible
to use fast linear matrix solver (require
symmetric positive definite matrix properties , which is
not hold for nonlinear transistors)
 Less Memory Requirement: Matrix for nonlinear
components can be generated on the fly. Possible to
run large case with millions of transistors.
 Decouple linear-nonlinear only at the inner most
Newton-Raphson iteration of transient analysis.
Accuracy guaranteed via linear-nonlinear iteration
(typically 4 ~ 10 iterations)
Experimental Results (1)
 Test Case #1
 Board / Packaging / Chip Power Network
 Fully coupled packaging inductance
 60k elements, 5000 nodes.
 Spice failed
 Our tool
 Less than 2 minutes
chip
board
Power Supply
Experimental Results (2)
 Power/Clock network case.
 30k
nodes, 1000 transistor devices
 Spice run time 41323s
 Our Run time: 1859s 22x speedup
Experimental Results (3)
 1K cell design
 10,286
nodes
 751 Gates
 Spice run time: 2121s
 Our run time: 26.1s
8x Speedup
 10K cell design
 123,590
nodes
 7,481 Gates
 Spice Run time: 44293s
 Our run time: 3572s, 12.4x Speedup
Why SPICEDiego is better?
 SPICEDiego: fast accurate transistor level circuit simulator
Powerful Matrix Solver Engine
 Transistor devices.
 Capable of capturing coupling effects.
 Device Model including Miller’s effect
 Less Memory Requirement (no LU factorization, dose not
save matrix for transistors)

 Application
 interconnect delay
 Crosstalk
 voltage drop, ground bounce
 simultaneous switching noise
Conclusion
 Moore’s Law demands an extraordinary
fast circuit simulator with guaranteed
accuracy.
 Current tools cannot cover Miller’s
effect, mutual inductance. There is no
bound on the error either.
 SPICEDiego offers a solution for circuit
designers