**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 s_{x}= 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**error**s.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**error**s, manufacturing tolerances, and other limitations, it is possible to have a large family of real transfer functions, even when**sensor**s 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**error**s (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**sensor**s 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

**sensor**s, 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 (t_{1}and t_{2}) and two corresponding output voltages (v_{1}and v_{2}) will be registered. Then, after substituting these values into Eq. (10), we arrive at: v

v

_{1}= a + bt_{1}(11)v

_{2}= a + bt_{2}and the constants are computed as:

b = (v

_{1}- v

_{2}) / (t

_{1}- t

_{2}) and a = v

_{1}- bt

_{1}(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 =v

_{1}+0.002268t

_{1}.

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

**sensor**s, it is essential to have and properly maintain precision and accurate physical standards of the appropriate stimuli. For example, to calibrate contacttemperature**sensor**s, 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, s

_{1}and s_{2}, 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**error**s in the slope and intercept calculation. A new intercept, a1, will differ from the real intercept, a, by:δa = a

_{1}- a = Δ / (s

_{2}- s

_{1}) (14)

and the slope will be calculated with

**error**:

δb = -Δ / (s

_{2}- s

_{1}) (15)

## 2 comments:

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February 24, 2011 at 1:01 AM## Post a Comment