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In the comments to blog #4, Frank Wiedmann asked about the correlation between the results of mismatch from Monte Carlo analysis and DC mismatch analysis. It is a fair question and here is a short blog to explore the topic. The example may not be realistic, but it is a useful for exploring the effects of mismatch on a circuit.
Let’s start with a simple circuit—A resistively loaded differential amplifier with cascodes, shown in Figure 1. The process is the Cadence® 45nm GPDK. This test circuit gives us a good platform for exploring the effect of mismatch on circuit performance. The GPDK includes models for Monte Carlo analysis and the results are easy to share.
Figure 1: Differential Amplifier
As background, DC mismatch is an analysis that estimates the effect of mismatch on circuit performance from a single simulation. It is considerably faster than using Monte Carlo analysis. The drawback is that only the DC operating point effect of mismatch is considered, so we could not use it for the dynamic comparator, see part 5. Originally, the model card needed to be modified for DC mismatch analysis. DC mismatch used different mismatch parameters than Monte Carlo analysis. Since about 2012, DC mismatch analysis reads either the stats block or Monte Carlo process variations. The original DC mismatch parameters are still supported for backwards compatibility. For Virtuoso® ADE Explorer users, look in the Analyses tab for dcmatch. One point to keep in mind is that DC match simulates the offset voltage at a specified output due to process variation, however, it can’t be used for derived measurements.
For the Monte Carlo analysis, Low-Discrepancy Sampling was used. To generate a "good" distribution, 1400 iterations were used. From experience, this number of iterations should give a reasonable approximation to the expected distribution. The standard deviation of the output offset voltage is 10.56mV. The Monte Carlo analysis results are compared with the dc mismatch analysis results in Table 1.
Figure 2: Output Offset Voltage Distribution
The DC Mismatch analysis was run using the stats option, that is, the statistical information in the stats block is used for the DC mismatch analysis.
Table 1: Comparison of Monte Carlo and DC Mismatch Results
The comparison results show that the offset voltages are close but are not quite identical. The difference in the results comes down to the approximation that is used when performing DC mismatch analysis. DC mismatch analysis assumes that the output distribution is Gaussian. The assumption allows us to estimate the variation without requiring the many iterations required by Monte Carlo to calculate the actual distribution. This is an example where the assumption breaks down because the tails of the distribution are not Gaussian. The output referred offset voltage is plotted using the normal quantile-quantile plot, shown in Figure 3. The results show that the tails of the distribution are not Gaussian, see the areas in the green boxes.
Figure 3: Quantile-quantile plot of Output Offset Voltage
One other item to notice is that DC mismatch and Monte Carlo mismatch analysis report the same contributors. The contributors are the random variables that result in the largest variation in the output offset voltage.
The summary is that DC mismatch provides a reasonable approximation to Monte Carlo mismatch results and can be used for predicting trends and worst-case corners. The limitation is that DC mismatch relies on the assumption that the distribution is Gaussian. As a result for signoff, Monte Carlo analysis is the appropriate choice.