Calibration

If the sensor’s manufacturer’s tolerances and tolerances of the interface (signal conditioning) circuit are broader than the required system accuracy, a calibration is required. For example, we need to measure temperature with an accuracy ±0.5 degree C; however, an available sensor is rated as having an accuracy of ±1 degree C. Does it mean that the sensor can not be used? No, it can, but that particular sensor needs to be calibrated; that is, its individual transfer function needs to be found during calibration. Calibration means the determination of specific variables that describe the overall transfer function. Overall means of the entire circuit, including the sensor, the interface circuit, and the A/D converter. The mathematical model of the transfer function should be known before calibration. If the model is linear [Eq. (2.1)], then the calibration should determine variables a and b; if it is exponential [Eq. (2.3)], variables a and k should be determined; and so on. Let us consider a simple linear transfer function. Because a minimum of two points are required to define a straight line, at least a two-point calibration is required. For example, if one uses a forward-biased semiconductor p-n junction for temperature measurement, with a high degree of accuracy its transfer function (temperature is the input and voltage is the output) can be considered linear:

 To determine constants a and b, such a sensor should be subjected to two temperatures (t1 and t2) and two corresponding output voltages (v1 and v2) will be registered. Then, after substituting these values into Eq. (2.10), we arrive at

 and the constants are computed as

 To compute the temperature from the output voltage, a measured voltage is inserted into an inversed equation

In some fortunate cases, one of the constants may be specified with a sufficient accuracy so that no calibration of that particular constant may be needed. In the same p-n-junction temperature sensor, the slope b is usually a very consistent value for a given lot and type of semiconductor. For example, a value of b=−0.002268 V/ degree C was determined to be consistent for a selected type of the diode, then a single-point calibration is needed to find out a as a =v1 +0.002268t1. 

For nonlinear functions, more than two points may be required, depending on a mathematical model of the transfer function. Any transfer function may be modeled by a polynomial, and depending on required accuracy, the number of the calibration points should be selected. Because calibration may be a slow process, to reduce production cost in manufacturing, it is very important to minimize the number of calibration points.
Another way to calibrate a nonlinear transfer function is to use a piecewise approximation. As was mentioned earlier, any section of a curvature, when sufficiently small, can be considered linear and modeled by Eq. (2.1). Then, a curvature will be described by a family of linear lines where each has its own constants a and b. During the measurement, one should determine where on the curve a particular output voltage S is situated and select the appropriate set of constants a and b to compute the value of a corresponding stimulus s from an equation identical to Eq. (2.13). 

To calibrate sensors, it is essential to have and properly maintain precision and accurate physical standards of the appropriate stimuli. For example, to calibrate contacttemperature sensors, either a temperature-controlled water bath or a “dry-well” cavity is required. To calibrate the infrared sensors, a blackbody cavity would be needed. To calibrate a hygrometer, a series of saturated salt solutions are required to sustain a constant relative humidity in a closed container, and so on. It should be clearly understood that the sensing system accuracy is directly attached to the accuracy of the calibrator.An uncertainty of the calibrating standard must be included in the statement on the overall uncertainty, as explained in 2.20.