Using Monte Carlo Yield Analysis
Monte Carlo yield analysis methods have traditionally been widely used and accepted as a means to estimate yield. The method simply consists of performing a series of trials. Each trial results from randomly generating yield variable values according to statistical-distribution specifications, performing a simulation and evaluating the result against stated performance specifications.
The power of the Monte Carlo method is that the accuracy of the estimate rendered is independent of the number of statistical variables and requires no simplifying assumptions about the probability distribution of either component parameter values or performance responses.
The weakness of this method is that a full network simulation is required for each trial and that a large number of trials is required to obtain high confidence and an accurate estimate of yield. Fortunately, the simulator uses state-of-the-art techniques to significantly boost the efficiency of the Monte Carlo method [1, 2, 3] while retaining its generality.
Consult the following references for details concerning state-of-the-art Monte Carlo techniques.
- M. D. Meehan and J. Purviance. Yield and Reliability Design for Microwave Circuits and Systems, Norwood, MA: Artech House, 1993.
- R. Spence and R. S. Soin. Tolerance Design of Electronic Circuits, Addison-Wesley, 1988.
- D. C. Hocevar, M. R. Lightner, and T. N. Trick. "A study of variance reduction techniques for estimating circuit yields," IEEE Trans. CAD, vol. CAD-2, pp. 180-192, July 1983.
Monte Carlo Trials and Confidence Levels
The following discusses how to calculate the number of trials necessary for a given confidence and estimate error.
Confidence level is the area under a normal (gaussian) curve over a given number of standard deviations. Common values for confidence level are shown in the following.
| Standard Deviations | Confidence Level |
|---|---|
| 1 | 68.3% |
| 2 | 95.4% |
| 3 | 99.7% |
Error is the absolute difference between the actual yield, Y, and the yield estimate, 
, given by:
where ε is the percent error. The low value limit of
is given by:
The sample or trial size, N, is then calculated from:
where
is the confidence expressed as a number of standard deviations.
Example
For a 95.4% confidence level 
, an Error = ±2% and a yield of 80%
N = 1600 trials
Refer to the section Confidence Tables for help in determining the number of trials suitable for yield analysis.
The graphs shown in the figures Yield for C = 1 (68.3% confidence) through Yield for C = 3 (99.7% confidence) may also be helpful in determining the accuracy of a yield analysis that you've already performed. These graphs plot error bounds of actual yield versus estimated yield for various values of N (number of trials).
The graph shown in the figure Yield for C = 1 (68.3% confidence) plots error bounds with a confidence interval of one standard deviation, or 68.3% confidence level.
The graph shown in the figure Yield for C = 2 (95.4% confidence) plots error bounds with a confidence interval of two standard deviations, or 95.4% confidence level.
The graph shown in the figure Yield for C = 3 (99.7% confidence) plots error bounds with a confidence interval of three standard deviations, or 99.7% confidence level.
Suppose you ran a yield analysis on your design using 100 trials and the estimated yield was 50%. Referring to the graph in the figure Yield for C = 1 (68.3% confidence), the lower bound on the actual yield is 45% and the upper bound is 55%.
From the graph shown in the figure Yield for C = 2 (95.4% confidence), for 100 trials and an estimated yield of 50%, the lower bound on the actual yield is 40% and the upper bound is 60%.
Finally, from the graph shown in the figure Yield for C = 3 (99.7% confidence), for 100 trials and an estimated yield of 50%, the lower bound on the actual yield is about 35% and the upper bound is about 65%.
Thus if you performed a yield analysis (either Monte Carlo or shadow model) using 100 trials, and the estimated yield was 50%, you have a 68.3% probability (confidence) that the actual yield is between 45% and 55%. You have a 95.4% probability that the actual yield is between 40% and 60%, and the probability is 99.7% that the actual yield is between 35% and 65%.
Yield for C = 1 (68.3% confidence)
Yield for C = 2 (95.4% confidence)
Yield for C = 3 (99.7% confidence)
Confidence Tables
The confidence tables that follow can be used to determine the number of trials suitable for yield analysis.
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 89.00 | 91.00 | 900 |
| 2.0 | 88.00 | 92.00 | 225 |
| 3.0 | 87.00 | 93.00 | 100 |
| 4.0 | 86.00 | 94.00 | 56 |
| 5.0 | 85.00 | 95.00 | 36 |
| 6.0 | 84.00 | 96.00 | 25 |
| 7.0 | 83.00 | 97.00 | 18 |
| 8.0 | 82.00 | 98.00 | 14 |
| 9.0 | 81.00 | 99.00 | 11 |
| 10.0 | 80.00 | 100.00 | 9 |
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 89.00 | 91.00 | 3457 |
| 2.0 | 88.00 | 92.00 | 864 |
| 3.0 | 87.00 | 93.00 | 384 |
| 4.0 | 86.00 | 94.00 | 216 |
| 5.0 | 85.00 | 95.00 | 138 |
| 6.0 | 84.00 | 96.00 | 96 |
| 7.0 | 83.00 | 97.00 | 70 |
| 8.0 | 82.00 | 98.00 | 54 |
| 9.0 | 81.00 | 99.00 | 42 |
| 10.0 | 80.00 | 100.00 | 34 |
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 89.00 | 91.00 | 5967 |
| 2.0 | 88.00 | 92.00 | 1491 |
| 3.0 | 87.00 | 93.00 | 663 |
| 4.0 | 86.00 | 94.00 | 372 |
| 5.0 | 85.00 | 95.00 | 238 |
| 6.0 | 84.00 | 96.00 | 165 |
| 7.0 | 83.00 | 97.00 | 121 |
| 8.0 | 82.00 | 98.00 | 93 |
| 9.0 | 81.00 | 99.00 | 73 |
| 10.0 | 80.00 | 100.00 | 59 |
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 79.00 | 81.00 | 1600 |
| 2.0 | 78.00 | 82.00 | 400 |
| 3.0 | 77.00 | 83.00 | 177 |
| 4.0 | 76.00 | 84.00 | 100 |
| 5.0 | 75.00 | 85.00 | 64 |
| 6.0 | 74.00 | 86.00 | 44 |
| 7.0 | 73.00 | 87.00 | 32 |
| 8.0 | 72.00 | 88.00 | 25 |
| 9.0 | 71.00 | 89.00 | 19 |
| 10.0 | 70.00 | 90.00 | 16 |
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 79.00 | 81.00 | 6146 |
| 2.0 | 78.00 | 82.00 | 1536 |
| 3.0 | 77.00 | 83.00 | 682 |
| 4.0 | 76.00 | 84.00 | 384 |
| 5.0 | 75.00 | 85.00 | 245 |
| 6.0 | 74.00 | 86.00 | 170 |
| 7.0 | 73.00 | 87.00 | 125 |
| 8.0 | 72.00 | 88.00 | 96 |
| 9.0 | 71.00 | 89.00 | 75 |
| 10.0 | 70.00 | 90.00 | 61 |
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 79.00 | 81.00 | 10609 |
| 2.0 | 78.00 | 82.00 | 2652 |
| 3.0 | 77.00 | 83.00 | 1178 |
| 4.0 | 76.00 | 84.00 | 663 |
| 5.0 | 75.00 | 85.00 | 424 |
| 6.0 | 74.00 | 86.00 | 294 |
| 7.0 | 73.00 | 87.00 | 216 |
| 8.0 | 72.00 | 88.00 | 165 |
| 9.0 | 71.00 | 89.00 | 130 |
| 10.0 | 70.00 | 90.00 | 106 |
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 69.00 | 71.00 | 2100 |
| 2.0 | 68.00 | 72.00 | 525 |
| 3.0 | 67.00 | 73.00 | 233 |
| 4.0 | 66.00 | 74.00 | 131 |
| 5.0 | 65.00 | 75.00 | 84 |
| 6.0 | 64.00 | 76.00 | 58 |
| 7.0 | 63.00 | 77.00 | 42 |
| 8.0 | 62.00 | 78.00 | 32 |
| 9.0 | 61.00 | 79.00 | 25 |
| 10.0 | 60.00 | 80.00 | 21 |
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 69.00 | 71.00 | 8067 |
| 2.0 | 68.00 | 72.00 | 2016 |
| 3.0 | 67.00 | 73.00 | 896 |
| 4.0 | 66.00 | 74.00 | 504 |
| 5.0 | 65.00 | 75.00 | 322 |
| 6.0 | 64.00 | 76.00 | 224 |
| 7.0 | 63.00 | 77.00 | 164 |
| 8.0 | 62.00 | 78.00 | 126 |
| 9.0 | 61.00 | 79.00 | 99 |
| 10.0 | 60.00 | 80.00 | 80 |
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 69.00 | 71.00 | 13924 |
| 2.0 | 68.00 | 72.00 | 3481 |
| 3.0 | 67.00 | 73.00 | 1547 |
| 4.0 | 66.00 | 74.00 | 870 |
| 5.0 | 65.00 | 75.00 | 556 |
| 6.0 | 64.00 | 76.00 | 386 |
| 7.0 | 63.00 | 77.00 | 284 |
| 8.0 | 62.00 | 78.00 | 217 |
| 9.0 | 61.00 | 79.00 | 171 |
| 10.0 | 60.00 | 80.00 | 139 |
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 59.00 | 61.00 | 2400 |
| 2.0 | 58.00 | 62.00 | 600 |
| 3.0 | 57.00 | 63.00 | 266 |
| 4.0 | 56.00 | 64.00 | 150 |
| 5.0 | 55.00 | 65.00 | 96 |
| 6.0 | 54.00 | 66.00 | 66 |
| 7.0 | 53.00 | 67.00 | 48 |
| 8.0 | 52.00 | 68.00 | 37 |
| 9.0 | 51.00 | 69.00 | 29 |
| 10.0 | 50.00 | 70.00 | 24 |
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 59.00 | 61.00 | 9219 |
| 2.0 | 58.00 | 62.00 | 2304 |
| 3.0 | 57.00 | 63.00 | 1024 |
| 4.0 | 56.00 | 64.00 | 576 |
| 5.0 | 55.00 | 65.00 | 368 |
| 6.0 | 54.00 | 66.00 | 256 |
| 7.0 | 53.00 | 67.00 | 188 |
| 8.0 | 52.00 | 68.00 | 144 |
| 9.0 | 51.00 | 69.00 | 113 |
| 10.0 | 50.00 | 70.00 | 92 |
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 59.00 | 61.00 | 15913 |
| 2.0 | 58.00 | 62.00 | 3978 |
| 3.0 | 57.00 | 63.00 | 1768 |
| 4.0 | 56.00 | 64.00 | 994 |
| 5.0 | 55.00 | 65.00 | 636 |
| 6.0 | 54.00 | 66.00 | 442 |
| 7.0 | 53.00 | 67.00 | 324 |
| 8.0 | 52.00 | 68.00 | 248 |
| 9.0 | 51.00 | 69.00 | 196 |
| 10.0 | 50.00 | 70.00 | 159 |
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 49.00 | 51.00 | 2500 |
| 2.0 | 48.00 | 52.00 | 625 |
| 3.0 | 47.00 | 53.00 | 277 |
| 4.0 | 46.00 | 54.00 | 156 |
| 5.0 | 45.00 | 55.00 | 100 |
| 6.0 | 44.00 | 56.00 | 69 |
| 7.0 | 43.00 | 57.00 | 51 |
| 8.0 | 42.00 | 58.00 | 39 |
| 9.0 | 41.00 | 59.00 | 30 |
| 10.0 | 40.00 | 60.00 | 25 |
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 49.00 | 51.00 | 9604 |
| 2.0 | 48.00 | 52.00 | 2401 |
| 3.0 | 47.00 | 53.00 | 1067 |
| 4.0 | 46.00 | 54.00 | 600 |
| 5.0 | 45.00 | 55.00 | 384 |
| 6.0 | 44.00 | 56.00 | 266 |
| 7.0 | 43.00 | 57.00 | 196 |
| 8.0 | 42.00 | 58.00 | 150 |
| 9.0 | 41.00 | 59.00 | 118 |
| 10.0 | 40.00 | 60.00 | 96 |
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 49.00 | 51.00 | 16576 |
| 2.0 | 48.00 | 52.00 | 4144 |
| 3.0 | 47.00 | 53.00 | 1841 |
| 4.0 | 46.00 | 54.00 | 1036 |
| 5.0 | 45.00 | 55.00 | 663 |
| 6.0 | 44.00 | 56.00 | 460 |
| 7.0 | 43.00 | 57.00 | 338 |
| 8.0 | 42.00 | 58.00 | 259 |
| 9.0 | 41.00 | 59.00 | 204 |
| 10.0 | 40.00 | 60.00 | 165 |
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 39.00 | 41.00 | 2400 |
| 2.0 | 38.00 | 42.00 | 600 |
| 3.0 | 37.00 | 43.00 | 266 |
| 4.0 | 36.00 | 44.00 | 150 |
| 5.0 | 35.00 | 45.00 | 96 |
| 6.0 | 34.00 | 46.00 | 66 |
| 7.0 | 33.00 | 47.00 | 48 |
| 8.0 | 32.00 | 48.00 | 37 |
| 9.0 | 31.00 | 49.00 | 29 |
| 10.0 | 30.00 | 50.00 | 24 |
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 39.00 | 41.00 | 9219 |
| 2.0 | 38.00 | 42.00 | 2304 |
| 3.0 | 37.00 | 43.00 | 1024 |
| 4.0 | 36.00 | 44.00 | 576 |
| 5.0 | 35.00 | 45.00 | 368 |
| 6.0 | 34.00 | 46.00 | 256 |
| 7.0 | 33.00 | 47.00 | 188 |
| 8.0 | 32.00 | 48.00 | 144 |
| 9.0 | 31.00 | 49.00 | 113 |
| 10.0 | 30.00 | 50.00 | 92 |
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 39.00 | 41.00 | 15913 |
| 2.0 | 38.00 | 42.00 | 3978 |
| 3.0 | 37.00 | 43.00 | 1768 |
| 4.0 | 36.00 | 44.00 | 994 |
| 5.0 | 35.00 | 45.00 | 636 |
| 6.0 | 34.00 | 46.00 | 442 |
| 7.0 | 33.00 | 47.00 | 324 |
| 8.0 | 32.00 | 48.00 | 248 |
| 9.0 | 31.00 | 49.00 | 196 |
| 10.0 | 30.00 | 50.00 | 159 |
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 29.00 | 31.00 | 2100 |
| 2.0 | 28.00 | 32.00 | 525 |
| 3.0 | 27.00 | 33.00 | 233 |
| 4.0 | 26.00 | 34.00 | 131 |
| 5.0 | 25.00 | 35.00 | 84 |
| 6.0 | 24.00 | 36.00 | 58 |
| 7.0 | 23.00 | 37.00 | 42 |
| 8.0 | 22.00 | 38.00 | 32 |
| 9.0 | 21.00 | 39.00 | 25 |
| 10.0 | 20.00 | 40.00 | 21 |
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 29.00 | 31.00 | 8067 |
| 2.0 | 28.00 | 32.00 | 2016 |
| 3.0 | 27.00 | 33.00 | 896 |
| 4.0 | 26.00 | 34.00 | 504 |
| 5.0 | 25.00 | 35.00 | 322 |
| 6.0 | 24.00 | 36.00 | 224 |
| 7.0 | 23.00 | 37.00 | 164 |
| 8.0 | 22.00 | 38.00 | 126 |
| 9.0 | 21.00 | 39.00 | 99 |
| 10.0 | 20.00 | 40.00 | 80 |
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 29.00 | 31.00 | 13924 |
| 2.0 | 28.00 | 32.00 | 3481 |
| 3.0 | 27.00 | 33.00 | 1547 |
| 4.0 | 26.00 | 34.00 | 870 |
| 5.0 | 25.00 | 35.00 | 556 |
| 6.0 | 24.00 | 36.00 | 386 |
| 7.0 | 23.00 | 37.00 | 284 |
| 8.0 | 22.00 | 38.00 | 217 |
| 9.0 | 21.00 | 39.00 | 171 |
| 10.0 | 20.00 | 40.00 | 139 |
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 19.00 | 21.00 | 1600 |
| 2.0 | 18.00 | 22.00 | 400 |
| 3.0 | 17.00 | 23.00 | 177 |
| 4.0 | 16.00 | 24.00 | 100 |
| 5.0 | 15.00 | 25.00 | 64 |
| 6.0 | 14.00 | 26.00 | 44 |
| 7.0 | 13.00 | 27.00 | 32 |
| 8.0 | 12.00 | 28.00 | 25 |
| 9.0 | 11.00 | 29.00 | 19 |
| 10.0 | 10.00 | 30.00 | 16 |
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 19.00 | 21.00 | 6146 |
| 2.0 | 18.00 | 22.00 | 1536 |
| 3.0 | 17.00 | 23.00 | 682 |
| 4.0 | 16.00 | 24.00 | 384 |
| 5.0 | 15.00 | 25.00 | 245 |
| 6.0 | 14.00 | 26.00 | 170 |
| 7.0 | 13.00 | 27.00 | 125 |
| 8.0 | 12.00 | 28.00 | 96 |
| 9.0 | 11.00 | 29.00 | 75 |
| 10.0 | 10.00 | 30.00 | 61 |
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 19.00 | 21.00 | 10609 |
| 2.0 | 18.00 | 22.00 | 2652 |
| 3.0 | 17.00 | 23.00 | 1178 |
| 4.0 | 16.00 | 24.00 | 663 |
| 5.0 | 15.00 | 25.00 | 424 |
| 6.0 | 14.00 | 26.00 | 294 |
| 7.0 | 13.00 | 27.00 | 216 |
| 8.0 | 12.00 | 28.00 | 165 |
| 9.0 | 11.00 | 29.00 | 130 |
| 10.0 | 10.00 | 30.00 | 106 |
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 19.00 | 21.00 | 899 |
| 2.0 | 18.00 | 22.00 | 224 |
| 3.0 | 17.00 | 23.00 | 100 |
| 4.0 | 16.00 | 24.00 | 56 |
| 5.0 | 15.00 | 25.00 | 36 |
| 6.0 | 14.00 | 26.00 | 25 |
| 7.0 | 13.00 | 27.00 | 18 |
| 8.0 | 12.00 | 28.00 | 14 |
| 9.0 | 11.00 | 29.00 | 11 |
| 10.0 | 10.00 | 30.00 | 9 |
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 19.00 | 21.00 | 3457 |
| 2.0 | 18.00 | 22.00 | 864 |
| 3.0 | 17.00 | 23.00 | 384 |
| 4.0 | 16.00 | 24.00 | 216 |
| 5.0 | 15.00 | 25.00 | 138 |
| 6.0 | 14.00 | 26.00 | 96 |
| 7.0 | 13.00 | 27.00 | 70 |
| 8.0 | 12.00 | 28.00 | 54 |
| 9.0 | 11.00 | 29.00 | 42 |
| 10.0 | 10.00 | 30.00 | 34 |
| Error +/- % | Estimated % Yield | Number of Trials | |
|---|---|---|---|
| Low | High | ||
| 1.0 | 19.00 | 21.00 | 5967 |
| 2.0 | 18.00 | 22.00 | 1491 |
| 3.0 | 17.00 | 23.00 | 663 |
| 4.0 | 16.00 | 24.00 | 372 |
| 5.0 | 15.00 | 25.00 | 238 |
| 6.0 | 14.00 | 26.00 | 165 |
| 7.0 | 13.00 | 27.00 | 121 |
| 8.0 | 12.00 | 28.00 | 93 |
| 9.0 | 11.00 | 29.00 | 73 |
| 10.0 | 10.00 | 30.00 | 59 |
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