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.

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.

Full-Scale Output

Full-scale output (FSO) is the algebraic difference between the electrical output signals measured with maximum input stimulus and the lowest input stimulus applied. This must include all deviations from the ideal transfer function. For instance, the FSO output in Fig. 2.2A is represented by SFS.

Fig. 2.2. Transfer function (A) and accuracy limits (B). Error is specified in terms of input value.

Span (Full-Scale Input)

A dynamic range of stimuli which may be converted by a sensor is called a span or an input full scale (FS). It represents the highest possible input value that can be applied to the sensor without causing an unacceptably large inaccuracy. For the sensors with a very broad and nonlinear response characteristic, a dynamic range of the input stimuli is often expressed in decibels, which is a logarithmic measure of ratios of either power or force (voltage). It should be emphasized that decibels do not measure absolute values, but a ratio of values only. A decibel scale represents signal magnitudes by much smaller numbers, which, in many cases, is far more convenient. Being a nonlinear scale, it may represent low-level signals with high resolution while compressing the high-level numbers. In other words, the logarithmic scale for small objects works as a microscope, and for the large objects, it works as a telescope. By

definition, decibels are equal to 10 times the log of the ratio of powers (Table 2.1):

In a similar manner, decibels are equal to 20 times the log of the force, current, or voltage:

Transfer Function

An ideal or theoretical output–stimulus relationship exists for every sensor. If the sensor is ideally designed and fabricated with ideal materials by ideal workers using ideal tools, the output of such a sensor would always represent the true value of the stimulus. The ideal function may be stated in the form of a table of values, a graph, or a mathematical equation. An ideal (theoretical) output–stimulus relationship is characterized by the so-called transfer function. This function establishes dependence between the electrical signal S produced by the sensor and the stimulus s :

 That function may be a simple linear connection or a nonlinear dependence, (e.g., logarithmic, exponential, or power function). In many cases, the relationship is unidimensional (i.e., the output versus one input stimulus). A unidimensional linear relationship is represented by the equation

  

 where a is the intercept (i.e., the output signal at zero input signal) and b is the slope, which is sometimes called sensitivity. S is one of the characteristics of the output electric signal used by the data acquisition devices as the sensor’s output. It may be amplitude, frequency, or phase, depending on the sensor properties.
Logarithmic function:

 Exponential function:

 Power function:

 where k is a constant number.
A sensor may have such a transfer function that none of the above approximations fits sufficiently well. In that case, a higher-order polynomial approximation is often employed.
For a nonlinear transfer function, the sensitivity b is not a fixed number as for the linear relationship [Eq. (2.1)]. At any particular input value, s0, it can be defined as

 In many cases, a nonlinear sensor may be considered linear over a limited range. Over the extended range, a nonlinear transfer function may be modeled by several straight lines. This is called a piecewise approximation. To determine whether a function can be represented by a linear model, the incremental variables are introduced for the input while observing the output.Adifference between the actual response and a liner model is compared with the specified accuracy limits (see 2.4).
A transfer function may have more than one dimension when the sensor’s output is influenced by more than one input stimuli. An example is the transfer function of a thermal radiation (infrared) sensor. The function connects two temperatures (Tb, the absolute temperature of an object of measurement, and Ts , the absolute temperature of the sensor’s surface) and the output voltage V :

 where G is a constant. Clearly, the relationship between the object’s temperature and the output voltage (transfer function) is not only nonlinear (the fourth-order parabola) but also depends on the sensor’s surface temperature. To determine the sensitivity of the sensor with respect to the object’s temperature, a partial derivative will be calculated as

 The graphical representation of a two-dimensional transfer function of Eq. (2.6) is shown in Fig. 2.1. It can be seen that each value of the output voltage can be uniquely

determined from two input temperatures. It should be noted that a transfer function represents the input-to-output relationship. However, when a sensor is used for measuring or detecting a stimulus, an inversed function (output-to-input) needs to be employed. When a transfer function is linear, the inversed function is very easy to compute. When it is nonlinear the task is more complex, and in many cases, the analytical solution may not lend itself to reasonably simple data processing. In these cases, an approximation technique often is the solution.

Sensor Characteristics

From the input to the output, a sensor may have several conversion steps before it produces an electrical signal. For instance, pressure inflicted on the fiber-optic sensor first results in strain in the fiber, which, in turn, causes deflection in its refractive index, which, in turn, results in an overall change in optical transmission and modulation of photon density. Finally, photon flux is detected and converted into electric current. In this chapter, we discuss the overall sensor characteristics, regardless of its physical nature or steps required to make a conversion. We regard a sensor as a “black box” where we are concerned only with relationships between its output signal and input stimulus.