To determine the performance capabilities of sensors, many performance criteria have been established by industry. Armed with these criteria a person can make direct comparisons between several sensors to determine which will perform the best.
Unfortunately, there is a lack of standardization in the industry as to what constitutes a complete set of performance data, but for the most part, most of these data can be understood if you also come armed with a relatively good knowledge of electronics and mechanics.
1-9.1 Frequency Response and Bandwidth
The frequency response of a device refers to its ability to respond adequately to a given sinusoidal input signal. The adequacy is determined by the manufacturer's design specifications. If the device is designed to respond to a range of frequencies,
as is most often the case, the bandwidth of that device is specified. Often, this specification also includes the 3-dB down points of that bandwidth.
Figure 1-12 illustrates a frequency response curve that may be found for a typical transducer device. As is frequently found in these curves, there is a relatively flat portion of the curve that exists before each end of the curve tends to drop off at a fairly rapid rate. This flat-area portion is indicated at the top of the curve. Note, however, that there is also a portion at each end which is also included with the original flat portion, bringing the total indicated bandwidth to a value of 19,985 Hz in our example.
Often what is done is to include that additional bandwidth enclosed by a 29.3%
reduction in signal output in the device being tested. When converted to the decibel scale, the same scale discussed earlier in Section 1-5.3, this reduction amount is the equivalent of a 3-dB reduction in output signal. This is also equivalent to reducing the output power of the device by one-half. As a result, the 3-dB down points are often referred to as the half-power points on a frequency response curve. The original justification for the existence of these points is based on the fact that such a reduction in signal is barely detectable by the average human ear. The reason for this lack of detection is due to the ear's logarithmic response.
Figure 1-12 Frequency response curve for a typical transducer. (From James R.
Carstens, Automatic Control Systems and Components, copyright 1990, p. 56.
Reprinted by permission of Prentice Hall, Englewood Cliffs, NJ.)
1-9.2 Sensitivity
The sensitivity of a transducer refers to its ability to generate an output response for a given change in the measurand. Mathematically, this is expressed as
change in output indicator sensitivity =
change in measurand that produced change in indicator (1-29) where a change in the output indicator is measured in output scale units such as degrees, millimetres, digits, divisions, and radians.
Example 1-21
An electronic thermometer reading changed from 55.0°F to 56.7°F, a change of 2.6 divisions on its output scale as a result of a 3.1°F actual change in the measured temperature. What is the instrument's sensitivity?
Solution: Using eq. (1-29) gives us
change in output indicator sensitivity =
change in measurand that produced change in indicator 2.6 div
3.1 F 0.829 div/ F
= °
= °
1-9.3 Static and Dynamic Error .
The static error of a sensor is defined simply as
static error = measured value -actual value (1-30) The term static error means that the measured value of the measurand does not change with time; it is relatively constant. Dynamic error, on the other hand, is calculated in precisely the same manner as static error. However, the characteristic of the measured value is such that it does change with time. Consequently, the determined error also changes with time.
Example 1-22
If a reluctive-type rpm transducer used for measuring the revolutions per minute of a rotating shaft measured a consistent rotation rate of 1190 rpm, where in fact the rpm was actually found to be 1255 rpm, find the transducer's static error.
Solution: From eq. (1-30), the static error is static error = measured value -actual value
= 1190 rpm -1255 rpm
= -65 rpm
1-9.4 Percent Accuracy
The percent accuracy of a sensor or transducer is defined as a percentage comparison of the static error to the actual value. In other words,
measured value - actual value
% accuracy = x 100
actual value (1-31)
where % accuracy is expressed as a percentage.
Example 1-23
Find the percent accuracy in the device described in Example 1-22.
Solution:
measured value - actual value
The responsiveness of a transducer is the ratio resulting from dividing the change in a measured quantity needed to produce an indicated change in the transducer's output by the measured quantity itself, expressed as a percentage. Mathematically,
change in measured quantity needed to produce indicated change responsiveness =
measured quantity (1-32)
Example 1-24
A strain gage pressure transducer reads 375 psig (pounds per square inch, gage) on a pipeline. A change of 1.5 psig in the measurand is necessary to cause a change in the output reading. Find the transducer's responsiveness. Solution: Referring to eq. (1-32) yields
change in measured quantity needed to produce indicated change responsiveness =
Reproducibility in a transducer refers to its ability to indicate identical values of the measurand at its output each time a measurement is made, assuming that all environmental conditions are the same for each measurement. Another way to state this is: the degree of agreement that a value of a measured variable has when measured at different times. All instruments, including transducers, possess a certain amount of inherent uncertainty in their ability to reproduce the same output readings time after time. Variations in output readings may be unpredictable and random. Table 1-5 shows an example of this characteristic using the readings obtained from a pressure transducer. This reproducibility may be expressed as a dynamic error.
Another way to quantify reproducibility is through graphing. A graph such as the one in Figure 1-13 may be used to show the variations of output readings for a given number of identical input measurand values.
There is no equation for determining reproducibility. Instead, it is a quality possessed by a transducer rather than a quantity.
Table 1.5 Example of Output Variations Reading
2 152.5 151.4 3 152.5 151.9 4 152.5 153.0 5 152.5 151.1 6 152.5 151.8 7 152.5 152.4 8 152.5 151.7 9 152.5 153.0 10 152.5 151.9 1-9.7 Range
A transducer's range refers to the two stated values, the lower-limit value of the measurand and the upper-limit value of the measurand, between which the transducer is designed to operate.
Figure 1-13 Reproducibility of a transducer's output for several different readings of the same measurand value.
Example 1-25
What is the range of a sensor that is designed to sense a temperature from -10 to + 250°F?
Solution: The range of this sensor is the two stated limiting temperature values, -10°F and part + 250°F.
1-9.8 Span
The span of a transducer or sensor refers to the arithmetic difference between the two stated range values.
Example 1-26
What is the span of the sensor described in Example 1-24?
Solution: The span of this sensor is simply the arithmetic difference between the two range values: 250°F -(-10°F), or 260°F.
1-9.9 Deviation from Linearity
A sensor's deviation from linearity is determined by plotting its output against a linear output and then stating the amounts of departure from the linear curve. These statements are usually given for certain spans of operation, as shown in the following example.
Example 1-27
Figure 1-14 shows the output of a flow-metering reluctance-type sensor. Express the deviation from linearity for this device based on the resultant curve shown.
Solution: The following linearity deviation information can be obtained from Figure 1-14 (all figures are maximum figures within each span):
0-2 L/min = -2 mV 2-4 L/min = -3 mV 4-6 L/min = +8 mV 6-8 L/min = +8 mV 8-10 L/min = -10mV
Figure 1-14 Deviation from linearity for a transducer.
Figure 1-15 Example of hysteresis.
1-9.10 Hysteresis
Stated simply, hysteresis refers to the characteristic that a transducer has in being unable to repeat faithfully, in the opposite direction of operation, the data that have been recorded in one direction (i.e., ascending or descending in value). The data used for ascertaining the hysteresis characteristic are derived from static data points, that is, data points that do not change in value with time. Consider Figure 1-15. A pressure gage was used to record the information given in the graph.
Two sets of data were recorded. One set was recorded with the pressure increasing with time, starting at time = 0, pressure = 0. The other set of data was recorded beginning at maximum time and maximum pressure and decreased over the same time increments. Theoretically, the pressure readings should coincide within each set of data. But in reality, a different set of pressure readings resulted for the same time increments. The resultant curves form what is often referred to as the hysteresis loop for that particular instrument. In the case of our pressure gage, there is an apparent
"stretching" or perhaps "slop" in the mechanical linkages coupling the mechanical sensor to the indicating needle. In the case of solid-state components, where there are no moving mechanical parts, this hysteresis can be caused by temporary polarization of materials within the component itself that directly affects the build-up or decrease of electrical charges inside the crystalline makeup of the solid-state materials.
1-9.11 Resolution
The term resolution is often used in conjunction with digital sensing devices denoting the ability of the device to distinguish between discrete, individual signal levels. As an example, let's say that we wanted to convert an analog voltage level to a digital signal.
To do this, an analog-to-digital (A/D) converter circuit must be used. Let's assume that the output of this converter has the capability of handling 8 bits of binary data. This means that for a given input analog voltage span of, say, 40 V (representing a range of,
say, 10-50 V dc), we would be able to convert this voltage into 28 parts at the A/D's output. The resolution of this circuit would then be
input signal span
In other words, we would have the capability of reading the input voltage with our A/D converter to the nearest 0.156 V.
Because of the several methods used in expressing the performance characteristics of a transducer, and because of the apparent similarities of some of the definitions of these expressions, another example problem is presented here to help separate one characteristic from another.
Example 1-28
A particular photocell transducer produces an analog output voltage that is proportional to the light intensity to which it is exposed. A light flow rate of 0.02 lumen (1m) is incident on the transducer to produce an output voltage of 0.376 V dc according to its attached digital display. (A calibration check made earlier shows that for the same light intensity the actual output voltage should have been 0.381 V dc.) The voltage range of the digital display on the transducer is from 0 to 1.5 V dc. It was found through experimentation that a change of 0.001 lm produced a change in output voltage of 0.052 V. This reading was recorded when the input light flow rate was 0.019 lm. It was further noted at this illumination value that the smallest change in voltage that could be detected with the existing digital display was 0.001 V.
Determine the following characteristics for the transducer: (a) the sensitivity, (b) static error, (c) percent accuracy, (d) responsiveness, (e) reproducibility, (f) range, and (g) span.
Solution: (a) To determine the sensitivity [refer to eq. (1-29)]:
0.052 lm sensitivity =
0.001 V 52 lm/V
=
Note that in this case since the output indication was given in volts rather than in divisions, degrees, or some other mechanical measuring unit as dictated by eq. (1-29), the smallest unit of measurement available in this instance was the digital display's smallest display unit capability, 0.001 V.
(b) To determine the static error [refer to eq. (1-30)]:
static error = 0.376 V - 0.381 V
= -0.005 V (c) To determine percent accuracy [refer to eq. (1-31)]:
0.376 - 0.381
% accuracy = x 100
0.381
= -1.31%
(d) To determine responsiveness [refer to eq. (1-32)]: .0.001 0.001
responsiveness = x 100 0.019
= 5.26%
(e) To determine reproducibility: Because reproducibility is a quality possessed by a transducer or instrument, there is no method by which it can be calculated. However, one attribute of reproducibility is found in the percent accuracy figure that was calculated above. This figure may be used to obtain some idea of the transducer's reproducibility.
(f) To determine range (see Section 1-9.6): The range was already stated as being 0 to 1.5 V dc.
(g) To determine span (see Section 1-9.7):
span = 1.5 V - 0 V
= 1.5 V dc
1-10 - Summary
To understand how electrical transducers function, it is necessary to have a firm foundation established in physics. We have attempted to make this establishment in Chapter 1 along with defining transducers and sensors, together with how they are categorized for reference purposes. In the chapters that follow, it may be necessary to refer back to the appropriate sections in this chapter to refresh yourself with the necessary physics to understand the characteristics of the measurands in question.