Optimal two-point static calibration of measurement systems with quadratic response

Measurement devices and instruments must be calibrated after manufacture to correct for component and assembly tolerances, and periodically to correct for drift and aging effects. The number of reference inputs needed for calibration depends on the actual transfer characteristic and the desired accuracy. Often, a linear characteristic is assumed for simplicity, either for the overall input range (global linearization) or for successive input subranges (piecewise linearization). Thus, only two reference inputs are needed for each straight line. This two-point static calibration can be easily implemented in any system having some basic computation capability and allows for the correction of zero and gain errors, and of their drifts if the system is periodically calibrated. Often, the reference inputs for that calibration are the end values of the measurement range (or subrange). However, this is not always the optimal selection because the calibration error is minimal for those reference inputs only, which are not necessarily the most relevant inputs for the system being considered. This article proposes three optimization criteria for the selection of calibration points: limiting the maximal error (LME), minimizing the integral square error (ISE), and minimizing the integral absolute error (IAE). Each of these criteria needs reference inputs whose values are symmetrical with respect to the midrange input (x c ) , have the form x c ±Δx/(2√n) when the measurand has a uniform probability distribution function, Δx being the measurement span, and do not depend on the nonlinearity of the actual response, provided this is quadratic. The factor n depends on the particular criterion selected: n=2 for LME, n=3 for ISE, and n=4 for IAE. These three criteria give parallel calibration lines and can also be applied to other nonlinear responses by dividing the measurement span into convenient intervals. The application of those criteria to the linearization of a type-J thermocouple illustrate their performance and advantages with respect to the customary end-point linearization (n=1) even for nonquadratic responses. For quadratic responses, n=1 yields the maximal error at the center of the input measurement range.