Dead Band

The dead band is the insensitivity of a sensor in a specific range of input signals (Fig.2.7B). In that range, the output may remain near a certain value (often zero) over an entire dead-band zone.

Repeatability

A repeatability ( reproducibility) error is caused by the inability of a sensor to represent the same value under identical conditions. It is expressed as the maximum difference between output readings as determined by two calibrating cycles (Fig. 2.7A), unless otherwise specified. It is usually represented as % of FS:

Possible sources of the repeatability error may be thermal noise, buildup charge, material plasticity, and so forth.

Saturation

Every sensor has its operating limits. Even if it is considered linear, at some levels of the input stimuli, its output signal no longer will be responsive. A further increase in stimulus does not produce a desirable output. It is said that the sensor exhibits a span-end nonlinearity or saturation (Fig. 2.6).

 Fig. 2.6. Transfer function with saturation.

Fig. 2.7. (A) The repeatability error. The same output signal S1 corresponds to two different input signals. (B) The dead-band zone in a transfer function.

Nonlinearity

Nonlinearity error is specified for sensors whose transfer function may be approximated by a straight line [Eq. (2.1)].Anonlinearity is a maximum deviation (L) of a real transfer function from the approximation straight line. The term “linearity” actually

 Fig. 2.4. Transfer function with hysteresis.

means “nonlinearity.” When more than one calibration run is made, the worst linearity seen during any one calibration cycle should be stated. Usually, it is specified eitherin percent of span or in terms of measured value (e.g, in kPa or degree C). “Linearity,” when not accompanied by a statement explaining what sort of straight line it is referring to, is meaningless. There are several ways to specify a nonlinearity, depending how the line is superimposed on the transfer function. One way is to use terminal points (Fig. 2.5A); that is, to determine output values at the smallest and highest stimulus values and to draw a straight line through these two points (line 1). Here, near the terminal points, the nonlinearity error is the smallest and it is higher somewhere in between.

 Fig. 2.5. Linear approximations of a nonlinear transfer function (A) and independent linearity (B).

Another way to define the approximation line is to use a method of least squares (line 2 in Fig. 2.5A). This can be done in the following manner. Measure several (n) output values S at input values s over a substantially broad range, preferably over an entire full scale. Use the following formulas for linear regression to determine intercept a and slope b of the best-fit straight line: 

In some applications, a higher accuracy may be desirable in a particular narrower section of the input range. For instance, a medical thermometer should have the best accuracy in a fever definition region which is between 37 degree C and 38 degree C. It may have a somewhat lower accuracy beyond these limits. Usually, such a sensor is calibrated in the region where the highest accuracy is desirable. Then, the approximation line may be drawn through the calibration point c (line 3 in Fig. 2.5A). As a result, nonlinearity
has the smallest value near the calibration point and it increases toward the ends of the span. In this method, the line is often determined as tangent to the transfer function in point c. If the actual transfer function is known, the slope of the line can be found from Eq. (2.5).
Independent linearity is referred to as the so-called “best straight line” (Fig. 2.5B), which is a line midway between two parallel straight lines closest together and enveloping all output values on a real transfer function.

Depending on the specification method, approximation lines may have different intercepts and slopes. Therefore, nonlinearity measures may differ quite substantially from one another.Auser should be aware that manufacturers often publish the smallest possible number to specify nonlinearity, without defining what method was used.

Hysteresis

A hysteresis error is a deviation of the sensor’s output at a specified point of the input signal when it is approached from the opposite directions. For example, a displacement sensor when the object moves from left to right at a certain point produces a voltage which differs by 20 mV from that when the object moves from right to left. If the sensitivity of the sensor is 10 mV/mm, the hysteresis error in terms of displacement units is 2 mm. Typical causes for hysteresis are friction and structural changes in the materials.

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. 2.3). 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,

 and the slope will be calculated with error:

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.