Bayesian Sensor Calibration of a CMOS-Integrated Hall Sensor Against Thermomechanical Cross-Sensitivities

For the first time, Bayesian sensor calibration is used to identify efficient calibration procedures for a sensor cross-sensitive to two parasitic influences. The object under study is a thermomechanically cross-sensitive sensor system for determining the magnetic induction ${B}$ . The packaged system comprises a Hall sensor, a stress sensor, and a temperature sensor. The three sensor signals are combined in a polynomial sensor response model with 11 parameters to determine ${B}$ compensated for offset and cross-sensitivities. For the calibration, sensors are exposed to mechanical stress values between 0 and −68 MPa, temperatures between −40 and $100 ^{\circ} \text{C}$ , and ${B}$ values between −25 and 25 mT. A sample of 35 sensors serves to extract the prior model parameter distribution of their fabrication run. The Bayesian experimental design is applied to identify sets of 2–8 optimal calibration conditions under I-optimality and G-optimality. The Bayesian inference then allows to obtain the posterior model parameter distribution of any uncalibrated sensor from the same run. Any such sensor is thereby turned into a ${B}$ measuring device with individually quantified accuracy. The method was successfully applied to 15 validation sensors. In the case of I-optimality, the median root-mean-square (rms) textsigma values of the ±1 σ confidence intervals for the extracted ${B}$ values were found to be 113–71 $\mu \text{T}$ after near-I-optimal calibrations based on 2–8 measurements. Over the entire range of temperature and mechanical stress and for applied $| {B} | \leq $ 25 mT, corresponding experimentally determined medians of the rms deviations between predicted and applied ${B}$ values were found to be 89–71 $\mu \text{T}$ . Analogous observations apply to G-optimality. In short, Bayesian calibration made it possible to obtain functional ${B}$ sensors of known accuracy with significantly fewer calibration measurements than model parameters. This was enabled by prior knowledge collected by the thorough characterization of 35 prior-generating specimens.