Calibrating Slow, Exponential Sensors
The Problem
Many sensors used in science and industry are slow to respond to changes in their environment. In order to calibrate these kind of sensors against a standard reference, many calibration programs set arbitrarily long equilibrium periods during which data is logged. Each sensor is deemed to have approached equilibrium to a sufficient degree to utilize an average of an arbitrary number of the "last" readings in an appropriate curve fitting procedure that maps what the sensor measures against the standard reference. The standard reference could be temperature, or pressure, or humidity, or any primary quantity.

An examination of plotted results for these typical calibrations often reveals significant residual slopes in the data indicating that sensors are not all that close to equilibrium for the last data set used. An alternative method is presented here that solves for the equilibrium condition based on the entire equilibrium logging period. The equilibrium conditions for each sensor, instead, are then fed to the curve fitting procedure in the normal manner. The results are sensor coefficients that produce engineering units in the field that better indicate equilibrium conditions.
The Solution
For this method to work, sensors are presumed to be "exponential" - that is they obey the relation:
Y = Ye - [ (Ye - Yo) e-ct]
where
Y is the measured quantity
Ye is the final, equilibrium (asymtotic) value of the measured quantity
Yo is the beginning value of the measured quantity
c is the sensor's equilibrium time constant
t is time This behavior is depicted above noting that the actual data is often noisy but closely follows the curve indicated.

The (Ye - Yo) term can be regarded as the local perturbation of each sensor at each standard reference set point. Each sensor, then has its own unique values of Ye, Yo and c.
Solving for Ye (the equilibrium sensor value of interest):
Y' = c (Ye - Yo) e-ct
Y'(0) = c (Ye - Yo)
Y'' = - c² (Ye - Yo) e-ct
Y''(0) = - c² (Ye - Yo)

therefore
- (Y')² +Yo
Ye = ————————————
Y''

with

c = - Y''(0) / Y'(0)
and
Yo = Y(0)

Fitting an exponential function by the method of least squares to the data, solves for the three characterizing values for each sensor at each standard reference set point. Finally, the results for Ye are used in whatever curve fitting procedure is normally used to produce sensor coefficients.