More about Error Budget Math

TheJuly Tips column explained how to use a sum of errors (S) or thesquare-root-of-sum-of-squares method (RSS) to create an error budget. But howdo you know which formula to use? This column looks at those methods in moredetail. To start, I created an Excel spreadsheet with four columns of 500 uniformrandom errors, each spread equally between 0 and 10 ppm, and then calculatedthe 500 S and RSS values plotted in Figure1a.Note the S error values average about 20 ppm and the RSS values average about11. Figure 1b plotsthe frequency distribution of these error values. The S values peak at 20 ppm(37 counts) and the RSS values peak at 12 ppm (74 counts). The superimposeddark-red curve plots a 10-point moving average of the values for the S errorsand the dark-blue line represents the moving average of the RSS values. Themoving average shifts results to slightly higher error values, but it makestrends easier to see.

Click here to see all spreadsheets referred to in this column.

Keepin mind that error budgets use magnitude rather than signed data. Thus forbudgets, a A plus or minus 10 ppm error simply becomes 10 ppm. And the squaring operations inthe RSS method create positive values.

Butdo the graphs in Figure 1 present realistic error conditions and values?

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Tofind this out, rather than using four errors, all between 0 and 10 ppm, Icreated four columns of uniform random error values with ranges of 0-10, 0-15,0-20 and 0-25 ppm. Excel calculated the 500 S and RSS values and plotted therespective distributions shown in Figure2.In this case, the S errors peak at 35 ppm (26 counts) and the RSS errors at 26(38 counts). The moving average shows the same peak relationship as seenearlier.

Asone of my analog-electronics colleagues noted, use thesquare-root-of-sum-of-squares formula when you have errors that follow aGaussian distribution. (If you want to know if an error source has a Gaussiandistribution, ask the device manufacturer.)

So,to compare Gaussian and uniform errors, I created another spreadsheet withrandom error values that fit normal, orGaussian curves. The four data sets have standard-deviation values (I florin ) of 5,7.5, 10, and 12.5 and a peak-center value (I.25) of zero to closely approximatethe ranges of uniform random error values created earlier. I used absolutevalues of Gaussian data and plotted S and RSS values shown in Figure 3. Iexpected a significant difference between plots of the uniform- andGaussian-type errors, but saw only a slight difference. In my opinion, when youplan for a worst-case error budget, use the S value, even though a circuit willrarely experience that magnitude of error. On the other hand, an RSS value -Gaussian or otherwise - provides information about typical error values youwill see. And it's always best to havereal information to illustrate rules of thumb.

Togenerate the random numbers for Excel, I used EasyFitXL, a software packagethat can produce sets of random numbers and perform curve fitting for manydistribution types.