Accuracy

Avery important characteristic of a sensor is accuracy which really means inaccuracy. Inaccuracy is measured as a highest deviation of a value represented by the sensor from the ideal or true value at its input. The true value is attributed to the object of measurement and accepted as having a specified uncertainty.
The deviation can be described as a difference between the value which is computed from the output voltage and the actual input value. For example, a linear displacement sensor ideally should generate 1 mV per 1-mm displacement; that is, its transfer function is linear with a slope (sensitivity) b=1 mV/mm. However, in the experiment, a displacement of s =10 mm produced an output of S =10.5 mV. Converting this number into the displacement value by using the inversed transfer function (1/b=1 mm/mV), we would calculate that the displacement was sx =S/b=10.5 mm; that is sx −s =0.5 mm more than the actual. This extra 0.5 mm is an erroneous deviation in the measurement, or error. Therefore, in a 10-mm range, the sensor’s absolute inaccuracy is 0.5 mm, or in the relative terms, inaccuracy is (0.5mm/10mm)×100%=5%. If we repeat this experiment over and over again without any random error and every time we observe an error of 0.5 mm, we may say that the sensor has a systematic inaccuracy of 0.5 mm over a 10-mm span. Naturally, a random component is always present, so the systematic error may be represented as an average or mean value of multiple errors.

Figure 2.2Ashows an ideal or theoretical transfer function. In the real world, any sensor performs with some kind of imperfection. A possible real transfer function is represented by a thick line, which generally may be neither linear nor monotonic. A real function rarely coincides with the ideal. Because of material variations, workmanship, design errors, manufacturing tolerances, and other limitations, it is possible to have a large family of real transfer functions, even when sensors are tested under identical conditions. However, all runs of the real transfer functions must fall within the limits of a specified accuracy. These permissive limits differ from the ideal transfer function line by 

 

The real functions deviate from the ideal by ±δ, where 

For example, let us consider a stimulus having value x. Ideally, we would expect this value to correspond to point z on the transfer function, resulting in the output value Y . Instead, the real function will respond at point Z, producing output value Y'. This output value corresponds to point z' on the ideal transfer function, which, in turn, relates to a “would-be” input stimulus x' whose value is smaller than x'. Thus, in this example, imperfection in the sensor’s transfer function leads to a measurement error of −δ.

The accuracy rating includes a combined effect of part-to-part variations, a hysteresis, a dead band, calibration, and repeatability errors (see later subsections). The specified accuracy limits generally are used in the worst-case analysis to determine the worst possible performance of the system. Figure 2.2B shows that

 

may more closely follow the real transfer function, meaning better tolerances of the sensor’s accuracy. This can be accomplished by a multiple-point calibration. Thus, the specifiedm accuracy limits are established not around the theoretical (ideal) transfer function, but around the calibration curve, which is determined during the actual calibration procedure. Then, the permissive limits become narrower, as they do not embrace part-to-part variations between the sensors and are geared specifically to the calibrated unit. Clearly, this method allows more accurate sensing; however, in some applications, it may be prohibitive because of a higher cost. The inaccuracy rating may be represented in a number of forms:

In modern sensors, specification of accuracy often is replaced by a more comprehensive value of uncertainty (see Section 2.20) because uncertainty is comprised of all distorting effects both systematic and random and is not limited to the inaccuracy of a transfer function.