12.2. Uncertainty for quadratic calibration of a loadcell

12.2. Type A uncertainty for quadratic calibration of a loadcell

Background on loadcell calibration. The purpose of this case study is to illustrate the calculation of uncertainties for calibrated values of a loadcell. The calibration of the loadcell depends on a calibration experiment where responses of the loadcell are observed for known loads (psi). The following are three repetitions of the response of loadcell #32066 at eleven known loads as measured at the National Institute of Standards and Technology.

                          X         Y

                         2.     0.20024    
                         2.     0.20016    
                         2.     0.20024
                         4.     0.40056    
                         4.     0.40045    
                         4.     0.40054
                         6.     0.60087    
                         6.     0.60075    
                         6.     0.60086
                         8.     0.80130    
                         8.     0.80122    
                         8.     0.80127
                        10.     1.00173    
                        10.     1.00164    
                        10.     1.00173
                        12.     1.20227    
                        12.     1.20218    
                        12.     1.20227
                        14.     1.40282    
                        14.     1.40278    
                        14.     1.40279
                        16.     1.60344    
                        16.     1.60339    
                        16.     1.60341
                        18.     1.80412    
                        18.     1.80409    
                        18.     1.80411
                        20.     2.00485    
                        20.     2.00481    
                        20.     2.00483
                        21.     2.10526    
                        21.     2.10524    
                        21.     2.10524

The loadcell in this example exhibits a quadratic response relative to load, and the calibration curve is the estimated from the calibration data by least-squares as the function:

where the estimates of a, b , c, and their associated standard deviations from the least-squares analysis as shown below.

              COEFFICIENT ESTIMATES      ST. DEV.      T VALUE

                a  -0.183980E-04         (0.2450E-04)    -0.75
                b   0.100102             (0.4838E-05)     0.21E+05
                c   0.703186E-05         (0.2013E-06)     35.

              RESIDUAL STANDARD DEVIATION = 0.0000376353

              RESIDUAL DEGREES OF FREEDOM = 30

Given a future response by the loadcell, say, Y', the problem is to correct the response to the proper load, X', and to calculate its associated uncertainty. A graph showing the calibration curve and a future response, Y' corrected to X', the proper load (psi), is shown below. The calibration data which consist of three repetitions at each of eleven loads are listed at the end of the example.

Uncertainty of the calibrated value X'. The calibrated value is usually calculated from

and one method of assessing its uncertainty is via propagation of error with the variance appproximated by

where f is the function that defines X' and denotes a partial derivative. The first step in the analysis, deriving the partial derivatives, can be simplified with software capable of algebraic reprsentation; in this case, Mathematica was used to derive the partial derivatives. The partial derivative of f with respect to Y is:

The partial derivatives with respect to the coefficients of the quadratic calibration curve are respectively:

The standard deviation of the calibrated value. The next step is to substitute the values of the coefficients and their respective standard deviations into the equations above for the range of interest for Y' where

is the residual standard deviation from the fit to the calibration data. The graph showing standard deviations of calibrated values, X', as a function of instrument response, Y', is displayed below.

Problem with propagation of error. The propagation of error shown above is not correct because it ignores the covariances among the coefficients, a, b , c. Unfortunately, some statistical software packages do not display the covariance terms with the output from the least-squares analysis. The information should be accessible from the "variance-covariance matrix".

The variance-covariance terms for the loadcell data set are shown below.

a 6.0049021-10 b -1.0759599-10 2.3408589-11 c 4.0191106-12 -9.5051441-13 4.0538705-14

The diagonal elements are the variances of the coefficients, a, b, c respectively, and the off-diagonal elements are the covariance terms.

Recomputation of the standard deviation of X'. The standard deviation of X' is redefined to account for covariance terms so that

Appropriate substitutions are made and the standard deviations are recomputed and graphed as a function of instrument response. In this case, the uncertainty is reduced by including covariance terms, some of which are negative.