4. 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 x - 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 1 shows 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 ±Δ. The real functions deviate from the ideal by ±δ, where δ < Δ.
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 shows that ±Δ 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 specified 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:
1. Directly in terms of measured value (Δ)
2. In percent of input span (full scale)
3. In terms of output signal
For example, a piezoresistive pressure sensor has a 100-kPa input full scale and a 10Ω full-scale output. Its inaccuracy may be specified as ±0.5%,±500 Pa, or ±0.05Ω.
In modern sensors, specification of accuracy often is replaced by a more comprehensive value of uncertainty because uncertainty is comprised of all distorting effects both systematic and random and is not limited to the inaccuracy of a transfer function.
5. 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°C; however, an available sensor is rated as having an accuracy of ±1°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. (1)], then the calibration should determine variables a and b; if it is exponential [Eq. (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 calibraction 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:
v = a + bt. (10)
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. (10), we arrive at:
v1 = a + bt1 (11)
v2 = a + bt2
v2 = a + bt2
and the constants are computed as:
b = (v1 - v2) / (t1 - t2) and a = v1 - bt1 (12)
To compute the temperature from the output voltage, a measured voltage is inserted into an inversed equation:
t = (v - a) / b (13)
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/°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. (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. (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.
6 Calibration Error
The calibration error is inaccuracy permitted by a manufacturer when a sensor is calibrated in the factory. This error is of a systematic nature, meaning that it is added to all possible real transfer functions. It shifts the accuracy of transduction for each stimulus point by a constant. This error is not necessarily uniform over the range and may change depending on the type of error in the calibration. For example, let us consider a two-point calibration of a real linear transfer function (thick line in Fig. 3).
To determine the slope and the intercept of the function, two stimuli, s1 and s2, are applied to the sensor. The sensor responds with two corresponding output signals A1 and A2. The first response was measured absolutely accurately, however, the higher signal was measured with error - Δ. This results in errors in the slope and intercept calculation. A new intercept, a1, will differ from the real intercept, a, by:
δa = a1 - a = Δ / (s2 - s1) (14)
and the slope will be calculated with error:
δb = -Δ / (s2 - s1) (15)
Reference Books About Sensor
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