delay evaluation
DESCRIPTION
Delay Evaluation. 1. Problem Description 2. Total capacitance model 3. Interconnect delay 4. Distributed RC Model 5. Other complications. 1. Problem Description. Given a pair of pins, compute pin-to-pin delay and possibly output waveform. Delay. Interconnect. Cell. Cell. …. Cell. - PowerPoint PPT PresentationTRANSCRIPT
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04/22/23 ELEN 689 1
Delay Evaluation 1. Problem Description 2. Total capacitance model 3. Interconnect delay 4. Distributed RC Model 5. Other complications
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1. Problem Description Given a pair of pins, compute pin-
to-pin delay and possibly output waveform
Cell Cell
Delay
Interconnect
Cell…
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On-going Research Difficulty:
Non-linear behavior of device Complex interconnect parasitic
No well-accepted approach Any new idea are welcome
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Circuit Model For an inverter
Csink
Csink
…
…
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Sink Capacitance Gate capacitance, input
capacitance, pin capacitance
Given for standard cells Can be found using SPICE
Apply an AC voltage and measure current
Average over a range of frequency
I
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2. Total Capacitance Model Valid for Rd >> Rmetal All fanouts have the same delay
CtotalRdRdRC
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RC Delay
Rpd
0.35VddVdd
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Driver Resistors Pull-up and pull-down resistors are
not a constant. Which value should we choose?
Use SPICE to compute Rpd and Rpd
Vds
Ids
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RC Delay Assume constant Rpd,
)(
exp35.0poutpd
PDfdddd CCR
tVV
)(35.01ln)(
poutpd
poutpdPDf
CCR
CCRt
Zhuo Li pointed out in this case Elmore delay is 35% instead of 50%
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Linear Delay Delay is linear in Ctotal
Rd is pull-up/pull-down resistor, assumed to be linear
Interconnect R ignored Common for >0.5um technology
standard cells Delay = t0+f*Ctotal
t0: Intrinsic gate delay f: Load factor
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Non-Linear: K-factor Consider input transition time tr/f Transition time is signal rising/falling time from 20% to 80% K-factor equation
Delay td=k(tr/f, Ctotal) Output transition time t’r/f=k’(tr/f, Ctotal)
rising time
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K-Factor … Synopsis K-factor form:
Piece-wise-quadratic For each piece, a*tr+b*Ctotal+c*tr*Ctotal+d Obtained from SPICE simulation
Ignore interconnect resistance shielding
Widely used
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3. Interconnect Delay Consider the first moment of H(s):
smdth(t)tsdth(t)
)dtst(1h(t)dth(t)eH(s)
100
00
st
1
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First Moment Consider h(t) as a probability density
function, then m1 is the mean of h(t):
The name moment comes from probability theory
0
1 h(t)dttm
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Mean and Median If impulse response h(t) is symmetric
Then the mean of impulse response equals median of step response, which is 50% delay
tm1
h(t)
tm2
hstep(t)
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Elmore Delay Since m1 is easy to compute,
Elmore used m1 as the delay for the RC circuit
It can be shown for RC trees, h(t) is skewed to the left. Therefore Elmore delay is always an upper bound on the 50% delay
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Example
1
1 1
1
1
1
1
1
42 31
m1_1= –4, m1_2= –7, m1_3= –8, m1_4= –8
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Application of Elmore Delay Good
Closed form expression, easy to compute Accuracy is better the ramps Useful for routing and placement
Bad Inaccurate
For less than 0.25 um technology Unbalanced RC trees Driver ignored
Not useful for timing verification
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4. Distributed RC Model Metal resistance per unit length is
increasing, while gate output resistance is decreasing
Portion of delay associated with the interconnect is increasing
Due to resistance shielding, total capacitance is an over estimation
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Two Step Approach Cell delay + interconnect delay
Cell delay and waveform is computed using K-factor
Interconnect delay is computed using Elmore delay or transfer function
Cell CellInterconnect
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Sink Waveform Given linear input waveform,
convolution is easy
m
1i
tpi
iek(t)h~
CellCtotal
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Driving Point Waveform Ctotal is inaccurate. Use load,
driving point waveforms match better
RdRdRC
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K-factor for Load? Given C1,R,C2 of a load, search
a table for linear or piece-wise linear waveform
Rd
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How to Store Table? Use load, the k-factor table is 4-
dimensional. Too large!
m
1i
tpi
iek(t)h~
Cell
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Effective Capacitance Method Use load Use 2-dimensional K-factor table
m
1i
tpi
iek(t)h~
CellCeff
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How to Compute Ceff? Basic assumption: there exist an input ramp
and Ceff, such that the driving point waveforms are the same
Match I and Ie on averageRd Rd
Ceff
I Ie
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Iteration1. Assume Ceff=Ctotal2. Use f-factor to find transition time
trf
3. Compute current for PI model and Ceff model
4. If equal then stop, otherwise compute new Ceff and go to 2
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5. Other Complications Side input
Delay from x to out is different for different values on y
Need characterize for all input combinationsVddx
y
out
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Simultaneous Switching Too many cases to consider Big impact on delay
Vddx
y
out
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Transistor Sizing Re-sized cells are common Fast techniques to derive k-factor
for re-sized transistors
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Readings on Delay Evaluation J. Rubinstein, P. Penfield Jr., and M. A.
Horowitz, “signal delay in RC tree networks,” IEEE Trans. CAD, 1983
F. Dartu, et al., “A gate delay model for high-speed CMOS circuits,” Proc. ICCAD 1994.
L. C. Chen, et al., “A new gate delay model for simultaneous switching and its applications,”, Proc. DAC, 2001.
E. Acar, et al., “TETA: Transistor-level waveform evaluation for timing analysis,” IEEE Trans. CAD, 2002.