July Tips column explained how to use a sum of errors (S) or the
square-root-of-sum-of-squares method (RSS) to create an error budget. But how
do you know which formula to use? This column looks at those methods in more
detail. To start, I created an Excel spreadsheet with four columns of 500 uniform
random errors, each spread equally between 0 and 10 ppm, and then calculated
the 500 S and RSS values plotted in Figure
Note the S error values average about 20 ppm and the RSS values average about
11. Figure 1b plots
the 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 superimposed
dark-red curve plots a 10-point moving average of the values for the S errors
and the dark-blue line represents the moving average of the RSS values. The
moving average shifts results to slightly higher error values, but it makes
trends easier to see.
Click here to see all spreadsheets referred to in this column.
in mind that error budgets use magnitude rather than signed data. Thus for
budgets, a Â±10 ppm error simply becomes 10 ppm. And the squaring operations in
the RSS method create positive values.
do the graphs in Figure 1 present realistic error conditions and values?
Click here to view all charts referred to in this column.
find this out, rather than using four errors, all between 0 and 10 ppm, I
created 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 the
respective distributions shown in Figure
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 seen
one of my analog-electronics colleagues noted, use the
square-root-of-sum-of-squares formula when you have errors that follow a
Gaussian distribution. (If you want to know if an error source has a Gaussian
distribution, ask the device manufacturer.)
to compare Gaussian and uniform errors, I created another spreadsheet with
random error values that fit normal, or
Gaussian curves. The four data sets have standard-deviation values (Ïƒ) of 5,
7.5, 10, and 12.5 and a peak-center value (Î¼) of zero to closely approximate
the ranges of uniform random error values created earlier. I used absolute
values of Gaussian data and plotted S and RSS values shown in Figure 3. I
expected a significant difference between plots of the uniform- and
Gaussian-type errors, but saw only a slight difference. In my opinion, when you
plan for a worst-case error budget, use the S value, even though a circuit will
rarely experience that magnitude of error. On the other hand, an RSS value -
Gaussian or otherwise - provides information about typical error values you
will see. And it's always best to have
real information to illustrate rules of thumb.
generate the random numbers for Excel, I used EasyFitXL, a software package
that can produce sets of random numbers and perform curve fitting for many