SYSTEM ANALYSIS USING RESPONSE SURFACE MODELING - Advanced Signal Integrity for High-Speed Digital Design - page 622

14.8 DEFECT RATE PREDICTION USING MONTE CARLO
SIMULATION

At this point we have identified the worst-case corners and the optimum equalizer
settings. We have also seen that our worst-case corner does not meet the eye
height spec, which means that we do not yet have a working design. However,
we can use our response surface model to evaluate whether or not our design
can meet the 1000-ppm defect rate target through the Monte Carlo simulation
method. In Monte Carlo simulation, we generate random samples that adhere to
the input sample distributions for a large number of cases of each input variable.
To do so, we require knowledge of the probability distributions of the input
variables. We then use the fitted RSM equations to calculate the predicted output
responses. By doing so for a large number of samples (we use 500,000 for our
examples), we generate distributions for the responses from which we can then
estimate the defect rates.
For our example system, the circuits that provide the on-die termination for
both transmitter and receiver are identical designs with built-in impedance control
features that limit the variation to 10 . Characterization of the impedance

variability due to process shows that it is normally distributed with a 50- mean
and a 4- standard deviation. The way that the impedance control circuit works
is to “clamp” the impedance of any circuit that would exceed the limits to the
corresponding limiting value. For example, if the termination impedance that
occurs for a given part due to natural variation were 38 , the circuit will set the
actual impedance to 40 . Without this control, approximately 1.2% of the parts
would fall outside

±

±

10 . This type of distribution is called a censored normal

distribution .
In addition, characterization of the printed circuit board differential impedance
reveals that it is normally distributed with a mean of 100 and a standard
deviation of 6 . 78 . This distribution results in approximately 1.3% of the product
failing to meet the 84- to 117.9- design window that we specify. However, the
PCB vendor can screen all product and remove any parts that fall outside our
specs. This type of distribution is called a truncated normal distribution . The
screening process requires that the PCB vendor add an impedance test to the
process flow and throw away parts that don't meet the specs, which increases
the cost of the boards by approximately $1 each. We may wish to simulate our
system with and without truncation to see whether or not the benefit justifies the
cost. We will set the trace length to the maximum value (0.508m), since it is the
design case of interest. We will estimate the defect rates at multiple equalization
settings in order to obtain an understanding of the sensitivity of defect rate to the
amount of equalization.
Histograms showing the distribution of the input variables for 500,000 cases
are shown in Figure 14-7. The distributions for R TT and R Tx appear normal, with
deviations at the edges that are creating by the impedance control. The “censor-
ing” has the effect of taking the tails of the distribution that would fall beyond
50 ±

10 and stacking them at the extremes. The differential impedance