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3/6/98 1
CarnegieMellon
Delay Metrics for the Next 50Years
Larry PileggiCarnegie Mellon University
Department of Electrical and ComputerEngineering
3/6/98 2
CarnegieMellon Outline
◆ Introduction– Interconnect delay dominance– Back-end models and analyses– Front-end metrics
◆ Elmore delay– Introduced in 1948– Applied to digital IC problems in early 1980’s– Somewhat ineffective for deep submicron (DSM)
◆ Probability Interpretation of Moments (PRIMO)◆ Stable n-Pole Models (SnP)◆ Conclusions
3/6/98 3
CarnegieMellon
◆ Metal resistance per unit length is increasing, while gate outputresistance is decreasing, with scaling
◆ Average wire lengths are not scaling, so portion of delay associatedwith the interconnect is increasing
◆ Gate delay is further decreasing with increasing metal resistancedue to shielding effects
Interconnect Dominance
3/6/98 4
CarnegieMellon
Back-end Analyses
Extraction
Physical Design
Test Generation
Design Verification Timing Verification
Simulation Floorplanning
Logic PartitioningDie Planning
LogicSynthesis
Logic Design andSimulation
Behavioral Level Design
◆ DSM interconnectdominance impacts allaspects of the top-downdesign flow
Back-end
3/6/98 5
CarnegieMellon
◆ Model order reduction via moment matching can be usedeffectively for interconnect verification
◆ Orthonormalized moments, or Krylov subspace methods wererecently proposed for increased numerical accuracy
Back-end Verification
CalculateMoments ∑
==
n
i
tipeiktY1
)(
...)( 2210 +++= smsmmsY
3/6/98 6
CarnegieMellon Krylov Reduction Methods
◆ Same as momentmatching if we haveinfinite precision
◆ Can capture dozensof dominant poles
◆ Approximations tothe 10’s of gigahertzis straightforward
◆ Some issues remainto be solved withregard to passivity
exact
PRIMA
MPVL
Frequency (GHz)10 20 300.00
0.05
0.10
0.15
0.20
Y11(s)60 poles
1 7
2 8
3 9
4 10
5
6 12
Coupled RLC Lines
11
3/6/98 7
CarnegieMellon Reduction Methods
◆ But there are very fewapplications which requirethis level of detail
◆ There is a greater need forimproved interconnectmodeling at the front-endand physical design levels
3/6/98 8
CarnegieMellon
Back-end Analyses
Extraction
Physical Design
Test Generation
Design Verification Timing Verification
Simulation Floorplanning
Logic PartitioningDie Planning
LogicSynthesis
Logic Design andSimulation
Behavioral Level Design
◆ Catching all of theinterconnect problems atback-end is too late!
3/6/98 9
CarnegieMellon
◆ Even with an approximate interconnect topology and values,moment matching and Krylov subspace methods areinappropriate for the front-end of design
◆ Higher order moments can be calculated at a fraction of the cost[RICE] required to calculate the first one
◆ But calculating the delays requires nonlinear iterations
Front-end Metrics
CalculateMoments
∑=
==n
i
tipeikVtv d
DD1
5.0)(
...)( 2210 +++= smsmmsV
3/6/98 10
CarnegieMellon The Elmore Delay
◆ Metric of choice for front-end applications and performance-driven physical design
◆ Explicit delay metric, yet can still capture interconnect resistanceeffects
◆ Primarily applied to RC tree circuits [Penfield & Rubenstein]R4
C4( )
44)432(2
432114
CRCCCR
CCCCRTD
+++++++=R1 R2 R3
C1 C2 C3
◆ The first moment of the impulse response
...)( 2210 +++= smsmmsH
3/6/98 11
CarnegieMellon The Elmore Delay
◆ Elmore (1948) proposed to treat the derivative of a monotonicstep response as a PDF, and estimate the median (50% delaypoint) by the mean
100
50f
80 100 100
50f 50f 50f
100 100
50f 150f
100
150f
0.0 200.0 400.0 600.0 800.0 1000.00.0
0.2
0.4
0.6
0.8
1.0
time (ps)
impulse response, h(t)
3/6/98 12
CarnegieMellon The Elmore Delay
◆ Exact only if h(t) is symmetrical◆ We’ve proven that RC tree impulse responses have positive
skew
0.0 200.0 400.0 600.0 800.0 1000.00.0
0.2
0.4
0.6
0.8
1.0
time (ps)
impulse response, h(t)
mean=134ps
median=100ps
3/6/98 13
CarnegieMellon Central Moments
◆ The circuit response moments
are related to the Central Moments of the h(t) Distribution by:
◆ Roughly speaking:
266 var2
)(
312133
2122
10n11
mmmmiancemm
mmk
nmeanm kn
knk
−+−=≡−=
−
=≡= −
=∑
µµ
µµ
25.1
2
3
µµµ MedianMean
Skew−==
∫∞−=→+++=
++++++=
0
2210
1
1 )(!)1(
1
1)( dttht
qmsmsmm
sbsb
sasasH q
q
qmm
nn K
K
K
3/6/98 14
CarnegieMellon
◆ Skew is a measure of the asymmetry
◆ We proved that all RC interconnect trees:– have unimodal impulse responses, h(t)– and that the h(t) distributions have positive skew
◆ It is then easily shown for such a distribution that
◆ The Elmore delay is an upper bound on the 50% step response delay
Elmore Delay Bound
negative zero positive
MeanMedianMode ≤≤
3/6/98 15
CarnegieMellon Elmore Bound
◆ Bounds get tighter toward the interconnect loads◆ Repeated convolutions make the distributions more “normal” ---
positive skew decreases toward a constant value
impulse responses100
50f
80 100 100
50f 50f 50f
100 100
50f 150f
100
150f
0.0 100.0 200.0 300.0-1.0e+10
0.0e+00
1.0e+10
2.0e+10
3.0e+10
4.0e+10
time (ps)
3/6/98 16
CarnegieMellon Finite Rise Times
◆ Any input voltage with a unimodal derivative will also make theresponse more normal (finite tin) --- and the first moment boundstill holds
◆ For finite rise times, the pulse response distribution becomes moresymmetrical as the rise time increases
◆ In the limit, the mean of the pulse response equals the median andthe Elmore delay becomes exact
◆ A large percentage of responses will fall into this category
)( tV in
derivative)( tV in′
response )( tv out′
3/6/98 17
CarnegieMellon Elmore Errors
-20 0 20 40 60 80 100 120 140 1600
100
200
300
400
500
600
700Elmore
Percentage Error
◆ 1200 response nodes for 700 nets from a 0.35 micron CMOS µP
max
100ps rise times
3/6/98 18
CarnegieMellon Dominant Time Constant
◆ The Elmore delay as a dominant time constant
◆ If one time constant dominates all others, and there are no lowfrequency zeros, we can approximate the dominant pole by m1
11 τ≈m
∑∑==
−=→−−−−−−=
n
i i
m
i im
n
zpm
pspsps
zszszssH
111
21
21 11
)())((
)())(()(
K
K
7.0)5.0ln( 11 mmtdelay ⋅≅=
◆ This approximation only scales the step response delay by aconstant factor
3/6/98 19
CarnegieMellon Dominant Time Constant
-40 -20 0 20 40 60 800
100
200
300
400
500
600
7000.7 Elmore
Percentage Error
◆ Max error is reduced, but ramp follower responses are optimistic
max
3/6/98 20
CarnegieMellon Using The Elmore Delay
◆ Not a good approximation forgeneral DSM trees
◆ Worst case error for busseswith near- and far-end loads
bus
◆ Works well when the rise timeis slow
◆ Or for balanced interconnectssuch as clock trees
clktree
3/6/98 21
CarnegieMellon µP Clock Tree
◆ Given the following floorplanfor a µP clock tree, optimizethe metal widths in terms ofthe Elmore delays to balancethe skew
◆ R and C per unit lengthvalues are pre-layoutestimates
1.98pF
3.01pF
1.30pF
2.06pF2.18pF
2.75pF1.97pF
1.37pF 0.85pF
287 287
287 287
287 287
287 287
575
575
575
575718 1202
610
610
610
1077
All lengths arein microns
3/6/98 22
CarnegieMellon µP Clock Tree
◆ Widths for zero Elmoreskew produced 8 ps ofskew with moment-matching models
◆ But correlations foroptimization of signalpaths are not as good
◆ Signal paths requiresmall absolute errors,whereas clock treesrequire only smallrelative errors
4.00
3.01pF
1.30pF
2.06pF2.18pF
2.75pF1.97pF
1.37pF 0.85pF
5.90 5.90
5.95
6.70
12.05.00
5.005.00
5.00
5.00
1.98pF
11.6
9.65
10.7
7.30 5.10
7.30 6.50
3/6/98 23
CarnegieMellon Higher Order Metrics
◆ For signal nets is wouldappear that we should match3 moments minimally
◆ Capture shapes for goodrelative errors
◆ But we can’t afford nonlineariterations for most delaymetric applications
◆ Two potential approaches:– PRIMO– SnP
impulse responses
0.0 100.0 200.0 300.0-1.0e+10
0.0e+00
1.0e+10
2.0e+10
3.0e+10
4.0e+10
time (ps)
3/6/98 24
CarnegieMellon PRIMO - Gamma Functions
◆ Extend Elmore’s idea to matching other distribution properties◆ Requires selection of some representative distribution◆ Incomplete gamma is similar to RC impulse responses
◆ Moment matching m1, µ2, and µ3 for time-shifted incompletegamma is provably stable
t
gλ,n(t)
Γ x( ) yx 1– e y– yd
0
∞
∫=gλ n, t( ) λ ntn 1– e λ t–
Γ n( )-------------------------- =
n4 µ2( )3
µ3( )2----------------= λ
2µ2
µ3---------=
3/6/98 25
CarnegieMellon PRIMO - Gamma Fitting
◆ Since provably stable, agamma integral table canbe used for delays
◆ With rise time a 2D tableis required
◆ For this example the step-delay error is < 1%
0.00e+00 1.00e-10 2.00e-10 3.00e-100.0e+00
2.0e+09
4.0e+09
6.0e+09
8.0e+09
100
50f
80 100 100
50f 50f 50f
100 100
50f 150f
100
150f
time (ps)
exact
m1-shifted gamma
3/6/98 26
CarnegieMellon Gamma Fitting
◆ Gamma approximationstruggles for some cases
◆ DSM interconnects can havecomplex low frequency zeroeffects
◆ Step delay error isunderestimated by 8% forthis example
100
50f
80 100 100
50f 50f 50f
100 100
50f 150f
100
150f
time (ps)
0.0e+00 2.0e-11 4.0e-11 6.0e-11 8.0e-11 1.0e-100.0e+00
1.0e+11
2.0e+11
3.0e+11
exact
m1-shifted gamma
3/6/98 27
CarnegieMellon SnP
◆ We can build provablystable n-poleapproximations
◆ Driving point poleapproximations areprovably stable
◆ k’s are fitted bymatching moments atthe response nodes ofinterest
◆ Generates stable n-exponential distributionmodel which permitstable lookup evaluation
RC Tree
...)( 2210 +++= smsmmsI
MomentMatching moments
real, stable,time constants
∑=
−=
n
i
iteikth
1
/)(
τ
3/6/98 28
CarnegieMellon S2P
◆ A two-pole model, or doubleexponential distributionfunction, can be used with a3D table to evaluate finiterise time response delays
◆ Step delay error is less than1.5% in this example
time (ps)
100
50f
80 100 100
50f 50f 50f
100 100
50f 150f
100
150f
exact
S2P
0.0 100.0 200.0 300.00.0e+00
2.0e+09
4.0e+09
6.0e+09
8.0e+09
3/6/98 29
CarnegieMellon S2P: Double Exponential
-10 -5 0 5 10 15 20 250
100
200
300
400
500
600
700
800S2P
Percentage Error
◆ CMOS µP example
max
3/6/98 30
CarnegieMellon S3P
◆ It can be shown thatthree exponentials areminimally required to fitsome unimodal impulseresponses
◆ S2P step delay error is14% in this example
◆ But is a 4D tablepractical?
100
50f
80 100 100
50f 50f 50f
100 100
50f 150f
100
150f
0.0 100.0 200.0 300.0-1.0e+10
-5.0e+09
0.0e+00
5.0e+09
1.0e+10
1.5e+10
2.0e+10
time (ps)
exact and S3P
S2P
3/6/98 31
CarnegieMellon Inductance
◆ On-chip inductance is becoming a reality for long lines◆ Impulse responses are no longer unimodal◆ Skew measure (µ3) can be used to control damping
R
3pf 3pf 3pfR = 1 ohm/cmL = 0.335 nH/cmC = 0.134 pF/cmZ0 = 50 ohms
5 cm 2.5 cm
0.0 10.0 20.0 30.0 40.0 50.0
-150.0
0.0
150.0
R
µ3 (scaled)Packaging Example
3/6/98 32
CarnegieMellon RCL Interconnects
2 4 60.0
0.5
1.0
1.5
Time (ns)0
1.34ns
1.01ns
0.75ns
R
3pf 3pf 3pfR = 1 ohm/cmL = 0.335 nH/cmC = 0.134 pF/cmZ0 = 50 ohms
5 cm 2.5 cm
◆ The delays areaccurately predictedby the moment metricsonce the damping iscontrolled
3/6/98 33
CarnegieMellon Conclusions
◆ Some progress has been made on more accurate delaymetrics
◆ But more work remains to be done for the mostdifficult DSM problems
◆ Similar metrics for coupling are necessary◆ But coupled line responses are provably not unimodal
for the general case