HBM torque transducers have an excellent linear characteristic in practical use. This is documented in the test report that is provided as standard and in the results recorded for the calibration. Nevertheless, it may be useful, in particular with measurements in the partial range, to reduce the deviation of linearity and hysteresis to a minimum through approximation of the characteristic curve.
In principle, two methods are available for determining the transducer's characteristic curve. On the one hand, the measurement results from a calibration in compliance with DAkkS (national accreditation body for the Federal Republic of Germany) regulations documented in a calibration certificate, on the other hand, the results of an on-site calibration in the drive train determined using a lever arm dead weight machine or a reference torque transducer. The resulting data points enable the two components of the measurement chain, i.e. the torque transducer and the TIM-EC EtherCAT interface module to be ideally matched to each other.
Various methods are available for approximation of a characteristic curve. Gauss's method of least squares is often used for approximation. The characteristic curve function is analyzed and determined from the data points resulting from the measured setpoint and actual values. This is the preferred method used in compliance with the German standards DIN51309 or VDI/2646 for calibration of torque measuring devices. Since the function we are looking for is a straight line, the linear regression fitting method is used.
Fig. 1: Example of a linear regression line through given data points
The characteristic curve does not run exactly through the data points and is not linear. However, with torque transducers, the characteristic curve can be described and thus linearized very well by the linear function
y = f(x) = m•x+b
The unknown slope coefficient m can then be determined applying the linear regression equation to the n measurement points (xk, yk) resulting from the calibration.
Figure 1 shows that the distance of a measurement point Pk = (xk, yk) from the best-fit line y = f(x) = m•x+b can be represented as follows.
Gauss's method of least squares is used to minimize the sum of the squared deviations sk of the measured values (xk, yk) through the origin to enable the fitting function in the calibration range under consideration to be determined .
This is achieved by calculating the partial derivative with respect to the coefficient.
The coefficient m and thus the slope of the best-fit line can then be calculated as follows: