More about Error Budget Math

The 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 1a. 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.

Keep in mind that error budgets use magnitude rather than signed data. Thus for budgets, a ±10 ppm error simply becomes 10 ppm. And the squaring operations in the RSS method create positive values.

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

Click here to view all charts referred to in this column.

To 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 2. 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 earlier. 

As 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.)

So, 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 (σ) of 5, 7.5, 10, and 12.5 and a peak-center value (μ) 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.

To 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 distribution types.