Proceedings of the Fifteenth Symposium on Explosives and Pyrotechnics, Essington, PA, April 1994.
Barry T. Neyer
EG&G Mound Applied Technologies
Miamisburg, OH 45343-3000
Barry T. Neyer
1100 Vanguard Blvd
Miamisburg, OH 45342
(937) 865-5170 (Fax)
We measured the symmetry of the distribution of slapper thresholds to determine if the distribution was symmetric about the mean. An asymmetric distribution would imply that the usual assumption of a normal distribution of thresholds would be in error, while a symmetric distribution would tend to confirm that assumption. We found no asymmetry of the distribution between the 10\% and 90\% points. Thus, we found no significant deviation from the assumption that the distribution of thresholds were distributed normally.
Technical Papers of Dr. Barry T. Neyer
Sensitivity tests are often performed on slapper detonators (also called Electric Foil Initiators or EFIs) to estimate various parameters of the distribution of slapper thresholds. (Two such levels are the "no fire" level, usually the level at which 0.1% or less of the parts fire and the "all fire" level at which 99.9% of the parts will fire.) One of the assumptions generally made when performing the test and analysis is that the thresholds, or a simple known transformation of the thresholds, are distributed normally. Given this assumption, it is a simple matter to determine all the parameters of the distribution once estimates of the mean, m, and standard deviation, s, have been determined.
If the thresholds are distributed normally, then all of the parameters can be estimated accurately. However, if the distribution differs significantly from a normal distribution, then the estimates will be in error. Thus, knowledge of the distribution of thresholds is vital for determining these extreme percentiles. However, determination of the distribution of the thresholds is difficult because the experimenter does not have any threshold information about the individual slappers in the sample. Thus, it is not possible to determine the distribution by computing a histogram of the thresholds.
A determination of the distribution of the population between the 0.1% and 99.9% points could be determined by measuring the probability of response at a number of stress levels. However, such a determination would require testing tens of thousands of components. An alternative to a complete determination of the distribution function is a test to determine the distribution at several points. The engineer could analyze the data from the test to determine if the distribution was consistent with the assumption of normality.
Because the normal distribution, as most other distribution functions commonly used in statistics, has two parameters, the experimenter must measure the distribution at three or more distinct points. If one of the three points is the mean, and the other two are located symmetrically around the mean, then it would be possible to determine if the distribution is symmetrical, and thus compatible with the hypothesis of being a normal distribution. If a significant asymmetry were found the hypothesis of normality of the distribution would have to be abandoned. Normality would not be proved by finding a symmetric distribution, but such a finding would rule out many other types of distributions.
The next section gives a description of the test, the third section describes various methods used to analyze the test, the fourth section describes work to expand this study, and the final section presents various consequences of the test.
We tested one hundred MAD1158 "mini blue light" slappers. These slappers were connected to a fireset with a 0.264 mF capacitor through a fast switch. The circuit had a nominal impedance of 20nH. The voltage on the capacitor was varied to apply different stresses to the slappers.
The experimenter must ensure that he tests a large number of components to ensure that enough successes and failures are recorded at each level to determine the probability of success at the level with sufficient precision. Because of economic constraints, we were limited to approximately 100 components in our test series. Since we needed to test at three levels, and wanted to ensure that there were a number of successes and failures at all three test levels, we chose to test 80 of these detonators in the extremes of the distribution, half at the lower end, and half at the upper end. The remaining 20 would be tested near the mean.
The first eighty slappers were tested according to the Neyer test (Neyer 1994), procedure. This test concentrates the test levels at the two points corresponding to m ± s . Although the test is designed to measure the parameters of a normal distribution, it also accurately determines the 16% and 84% points of an arbitrary distribution. Because the experimenter does not know the 16% and 84% points before beginning the test, the first part of the test concentrates on estimating these levels. As a result, components were tested throughout the 10--20% and 80--90% region of the distribution.
Analysis of the data indicated that the 16% level was at 1470.1 volts, and the 84% level was at 1543.4 volts. If the distribution is normal (or any distribution symmetric about the mean) then the mean (and median) should be half way between these two levels. The last twenty slappers were fired at the average of these two levels, 1506.7 volts. If significantly more or less than half of the parts fired, then the distribution would not be symmetric, and thus could not be normal. However, if approximately half of the parts fired at this level, then the assumption of normality and symmetry would be strengthened.
Figure 1 shows the results of the test. The "X"s represent parts that fired, while the "O"s represents parts that did not fire. As can be seen from the figure, exactly half of the last twenty parts fired halfway between the m ± s points responded, indicating that the distribution is approximately symmetric about the mean between the m ± s (16% to 84%) points. Thus no deviation was found from a normal or symmetric distribution.
Figure 1: The graph shows the results of the test to measure symmetry of the response. The Xs represent fires, and the 0s represent parts that did not fire
The data were also analyzed with the program MuSig with various transformations and probability functions to determine if any of these distributions had a significantly higher likelihood of producing the test results. The analysis showed that both the normal and logit distribution had essentially similar likelihood for producing the test results recorded. This result is not surprising, because both distributions are very similar in the 10% to 90% part of the distribution.
The linear (untransformed) distribution had a larger likelihood than the log and various power transformations, but the results were not statistically significant. Again, this result was expected because the m to s ratio was small. As a result all the transformation functions used were essentially linear throughout the test region.
This work shows that the distribution of these slapper thresholds was symmetric over the limited range of 10% to 90% of the distribution. However, most researchers are concerned with more extreme levels, such as the 99% or 99.9% level. A number of distributions commonly used in statistical analysis are similar to the normal distribution in the 10% to 90% range, but depart from normality outside of this range. Thus, the results presented here are not able to distinguish between these distributions.
To measure the distribution at an extreme level requires a much larger sample size. For example, suppose the experimenter wishes to determine if the distribution was normal versus logit. Testing would have to occur near the 5% and 95% points of the distribution. In order for the test to be statistically significant, the experiment would have to yield a number of successes at the 5% point and a number of failures at the 95% point.
If N components are tested at a level with probability of success p, then pN components would be expected to function, and qN would be expected to fail, where q = 1 - p. Suppose the experimenter tested a number
of components at the level p, where p is a small number. The experimenter wants to ensure that at least one part responds at this extreme level. The probability that all fail to respond is given by
This function attains the minimum value of P = 0.0625a when p = q = 0.25 and the maximum value of P = e-a when p 0. If a = 10 then the probability that all components fail is on the order of 2 parts in a million when p = 0.05. Because the number of specimens expected to respond is a, with a standard deviation of a, setting a = 10 would also ensure that the probability of success at that level could be measured to ±30%.
Thus, to measure the distribution near the 5% and 95% points would require a sample size of approximately 500. One hundred of these would be tested as they were in this test. The first part of the test established the parameters and . The remainder of the test would be conducted at the 5% and 95% levels, 200 at each level. Subsequent analysis will then reveal if the distribution is symmetric about the mean, and if it is consistent with a normal, logit, or other distribution.
This work shows that the distribution of thresholds for slapper detonators is symmetric, and thus consistent with the assumption of normality most often assumed. While this work does not prove that the distribution of slapper thresholds follows a normal distribution, it does help to establish that such an assumption is not unreasonable.
I wish to thank James J. Heinrichs for technical assistance. EG&G Mound Applied Technologies is operated for the U.S. Department of Energy under Contract DE-AC04-88DP43495.
Barry T. Neyer (1994) "A D-Optimality-Based Sensitivity Test," Technometrics, 36, 61-70.