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Libraries with Scalable Polynomial Delay Model Improve Modeling Accuracy
By Mehmet A.Cirit
When it comes to the problems of deep-submicron modeling,the interconnectand its effect on cell and chip timing have taken on paramount importance.
There have also been changes in design style and methodology to enable
large designs,control power dissipation and maintain performance in the face
of limited power budgets.
Although it is clear that timing and analysis tools should be prepared to address the problems of variable supply and shifting temperature,the existing library methodology and modeling tools are not adequate to satisfy such
requirements.
The Synopsys Liberty library format has been the standard depository of timing and power information.It is widely available for almost any process,and quite
a number of tools can read and write it.Throughout the years,it has consistently
grown and expanded to accommodate new requirements and new analysis
types.Meanwhile,the Nonlinear Delay Model (NLDM),also introduced by Synopsys,is essentially the standard delay-modeling format in use today.
Synopsys recently introduced libraries based on the Scalable Polynomial
Delay Model (SPDM)to address voltage and temperature variation and to remedy some well-known
shortcomings of NLDM.The new model not only can address temperature and voltage variations,
but it is also highly accurate,which has not been the case with other modeling techniques.And it
can be used to model almost all electrical parameters as well as delays.
We now know that there are intrinsic limits to the accuracy that can be achieved with NLDM -lim-
its that hamper its usefulness and accuracy.Delay parameters,especially,are very smooth functions
of the input slope and load.Except for some minor nonlinearity at specific regions,the delay surface
hardly presents a challenge to the modeling enthusiast.
Yet despite the simplicity of the delay as a function of input slope and load,modeling has been a
problem,for a number of reasons:
That makes it necessary to use special circuits to suppress the dependency and interaction.This,of course,
compromises the accuracy.
The NLDM delay equation is first order in load and
input slope.SPDM generalizes and extends that model
into higher-order polynomials of input slope,load,voltage and temperature variables.In addition,instead of insisting that the delay equation go through a fixed set of
points,it requires that the deviation be minimized.Thus,
a single equation could span the usage domain of the
cell.In principle,the higher the polynomial orders,the
better the models they produce.To calculate the polynomial coefficients accurately,one needs to have enough
data points.
Much as NLDM 's capability to reduce errors can be
improved by employing finer grids at nonlinear regions,
SPDM can be improved by using higher-order polynomials and fine grid-input data.Both interpolation and ex-
trapolation capabilities can be thus improved.
In order to illustrate the capability of the model,we
have generated delay data using a 7 x 7 table over the
full voltage and temperature range of a typical process
corner for an inverter.An SPDM model,second order in
load and input slope and first order in temperature and
voltage,has been generated using five pieces of temperature-voltage corner data.
Delay data obtained using simulation and delays calculated using SPDM are shown in Fig.1 (page 57).In
order to determine the interpolation accuracy,we have
generated oversampled 13 x 13 tables and used NLDM
and SPDM to predict the additional data values.Since
this additional data falls on the center points of the initial table points,it is expected to have the maximum deviation.The accompanying table summarizes the mean,
rms and maximum deviation between the model and
simulated data:
Rise
T ... V ... Mean ... RMS ... Max
0 ... 1.98 ... -0.00071 ... 0.00289 ... 0.009704
25 ... 1.80 ... -0.00024 ... 0.00295 ... 0.012418
50 ... 1.89 ... -0.00042 ... 0.00285 ... 0.008511
75 ... 1.71 ... -0.00033 ... 0.00293 ... 0.010466
100 ... 1.62 ... -0.00007 ... 0.00289 ... 0.010890
Fall
T ... V ... Mean ... RMS ... Max
0 ... 1.98 ... -0.00047 ... 0.00323 ... 0.011897
25 ... 1.80 ... -0.00045 ... 0.00337 ... 0.013309
50 ... 1.89 ... -0.00027 ... 0.00302 ... 0.010047
75 ... 1.71 ... -0.00052 ... 0.00337 ... 0.012592
100 ... 1.62 ... -0.00011 ... 0.00362 ... 0.014218
Here are similar parameters for NLDM:
Rise
T ... V ... Mean ... RMS ... Max
0 ... 1.98 ... -0.00024 ... 0.00181 ... 0.008154
50 ... 1.89 ... -0.00003 ... 0.00165 ... 0.007000
75 ... 1.71 ... 0.00058 ... 0.00130 ... 0.004925
100 ... 1.62 ... 0.00048 ... 0.00109 ... 0.003563
25 ... 1.80 ... 0.00075 ... 0.00113 ... 0.002637
Fall
T ... V ... Mean ... RMS ... Max
0 ... 1.98 ... 0.00050... 0.00119 ... 0.002770
50 ... 1.89 ... 0.00046 ... 0.00126 ... 0.002984
75 ... 1.71 ... 0.00072 ... 0.00113 ... 0.003235
100 ... 1.62 ... 0.00054 ... 0.00112 ... 0.002558
25 ... 1.80 ... 0.00042 ... 0.00119 ... 0.002592
We used a single SPDM model to cover the wide temperature and voltage range.At first glance,it may seem
that NLDM is superior to SPDM,but one gains insight
into the behavior of the models by looking at the actual
error distributions.Fig.2 shows the interpolation errors,
or the difference between expected delays and the calculated values.NLDM delay distribution is expected to be
nonuniform,since each grid element uses a different
equation.Since the delay equations adapt themselves to
local conditions,it is not possible to determine if there is
noise in the delay data reported by the circuit simulator.
But SPDM uses a single equation.The presence of sudden changes in error distribution indicates errors in the
input data obtained from the simulators.In the absence
of noise,interpolation errors should vary gracefully.
Sudden jumps and dips indicate noise.Fig.2 shows
the presence of noise in the simulation results.This effect
can be amplified by changing the time-step algorithm and
integration method used by the circuit simulator.One set
of options is not necessarily accurate for all operating
conditions of a cell.NLDM repeats those errors;SPDM filters them out.If the changes in delays are comparable to
the resolution of the circuit simulator,one cannot improve
the accuracy by taking more samples;maximum error
does not necessarily improve with bigger tables.This
shows itself as larger rms errors for SPDM.Thus,SPDM
may be the best approximation of the actual cell behavior.
SPDM in its current form does not allow process variations.Such variations can be modeled using various sigma parameters,but there is no standard.Along with
temperature and voltage,one can introduce a “sigma ”
variable to account for process variations.
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