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2.1 The Sensor Model

As depicted in Figure 2.1, a generic process sensor determines the current state o theprocess rom one or more physical observables, which by de nition are physical quanti-ties that can be directly measured. Physical observables include temperature, requency,light intensity, speed, and so orth. Te sensor uses these inputs to produce a measure-ment, which may be a simple number (e.g., the local temperature) or a more complexdata set such as an image.

I the measurement is to have any validity it must be true that a given set o physicalobservable values must always produce the same measurement value; otherwise, themeasurement would be ambiguous. By taking a process measurement, we are assumingthat the quantity to be measured can be measured and that it depends uniquely on thephysical observables. Tis requirement is similar to the so-called zeroth law o thermody-namics (which states that temperature exists as a measurable quantity), and we could evencall it the zeroth law o measurement. Tis general concept is described mathematically by a unctionS, whose domain is the set o all possible values o the physical observablesand range is the set o all possible measurement values:

measurement S observables= ( ) (2.1)

Te act that a sensors output can be described by a unction (at least in theory) ollowsdirectly rom the causal relationships linking the quantity o interest in the process, thephysical observable, and the resulting measurement.

Another requirement or making process measurements is that the unction(equation 2.1) must be continuous, which suggests that in nitesimal changes in thephysical observable can only cause in nitesimal changes in the measurement. Tisrequirement ensures that the measurement data is di erentiable and that the rate o change in the measurement remains nite.

A nal requirement is that the measurement reported by the sensor must be linearly dependent on the process quantity to be measured. Ideally, the output o the sensor

2Measurement

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Industrial Process Sensors

(denoted as s) gives the exact value o the quantity (q) to be measured, but in practicethere is o en a small discrepancy given by

s Aq b= + (2.2)

where A is a gain actor andb is a constant o set. An important unction o calibrationis to adjust the sensors electronics so that A =1 and b =0, thus ensuring that s = q. Inmost process control applications it is not necessary to get the calibration exact, becausethe controller usually bases its corrective actions on changes in the measurement ratherthan on the absolute value o the measurement.

Te sensor model S represented by equation 2.1 is a set o equations that describe thephysical mechanism o the interaction between the physical observable and the process. Itis possible to ormulate a mathematical model or any process sensor; whether or not themodel captures the physics o the measurement is another matter. For instance, a proxim-

ity sensor that is based on electrical capacitance may be sensitive to changes in humidity or barometric pressure due to its design. I the model used by the sensor ails to includethe environmental e ects on the capacitance, then it will ail to describe the output o thesensor under all operating conditions. Te sensor will always generate a signal, but thesignal could be wrong. I however the model adequately describes the sensor design, thenthe sensitivity to these extraneous e ects can be determined or even corrected.

2.2 Units of Measure

Te sensor, together with whatever support circuitry is required, produces a measure-ment value that is usually represented by a voltage or current level, serial port signal, orother type o computer inter ace signal. Te signal may be displayed on a panel i there isno direct computer connection. When measurements are represented by a voltage or cur-rent level (as in the case o 420 milliamp [mA] loop control systems), the pre-establishedrange (minimum and maximum) o possible readings is used to interpret the sensoroutput. Tus a current output o 5 mA rom one sensor might mean 1 meter per second(1 m/s), whereas 5 mA rom a di erent sensor might mean 30 C. Tere ore, not only therange but also the unit o measure is assumed. Sensors that communicate directly witha computer system generally do not have this ambiguity, since the units can be includedin the string o characters that convey the measurement result.

Te International System o Units include undamental quantities such as the meter,kilogram, and second, and derived units such as the newton (the unit o orce equal to1 kgms2). Since it is o en necessary to express a reading that is either very large or very

Figure 2.1 A generic process sensor.

PhysicalObservable(s) Measurement

ProcessSensor


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Measurement

close to zero, the International System o Units also de nes pre xes that multiply thebase unit by a power o 10 (see able 2.1).

2.3 Simple Statistics

Consider a new thermocouple that has been installed in the process and is now gener-ating temperature data. Suppose the rst reading is 30.0 C and the second reading is30.2C. wo possibilities can be considered: either the temperature is increasing or thethermocouple reading is uncertain by at least 0.2 C. All sensor readings are subject toa certain amount o variability, so in order to know whether or not the temperature isactually increasing, we must know whether or not a change o 0.2 C is signi cant in thiscase. In other words, i the temperature is truly constant, how much variability can beexpected rom the thermocouple readout?

Simple statistical descriptors can be used to assess whether or not a given variation

in a reading is noteworthy. Te rst o these is the mean, or numerical average. Given asequence o N readings {r1, r2, , rN}, then the mean M is de ned as

M N r i

i

N

==1

1 (2.3)

Te standard deviation s o the readings is de ned to be

=

=12

1N

r M ii

N

( ) (2.4)

Te signi cance o the standard deviation is that it provides an indication o the vari-ability o the data. Small values o s result rom readings that are uni ormly near the

Table 2.1 Pre xes Used with the International System o Units

Symbol Pre x Multiplier

P peta 1015

tera 1012

G giga 109

M mega 106

k kilo 103

c centi 102

m milli 103

m micro 106n nano 109

p pico10

12

emto 1015


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10 Industrial Process Sensors

mean, and large values o s are caused by large variations rom one reading to the next.Te ratio ( s / M ) is the coe cient o variation, which can be used to compare the vari-ability between two sets o readings with di erent mean values.

As an example, consider the hypothetical thermocouple mentioned earlier, and sup-pose that its output is recorded once a minute or a period o 50 minutes, as shown inFigure 2.2a. Te mean o these 50 measurements is 30.04 C, and the standard deviationis 0.30C; there ore, the coe cient o variation is only 1%. Although there are randomfuctuations rom one reading to the next, there does not appear to be either an upwardor a downward trend in the data. Since the standard deviation is 0.30 C and the di -

erence between the rst two readings is only 0.20C, it seems likely that the initialincrease in the reading was not signi cant (i.e., no action should be taken by the processcontroller).

Te measurement values in this hypothetical example are normally distributed,which is to say that the probability P that the reading equals r is given by

P r

r M ( ) exp

( )= 12 2

2

2

(2.5)

29.0

29.5

30.0

30.5

31.0

31.5

0 10 20 30 40 50Time (min)

Temperature(C)

29.0

29.5

30.0

30.531.0

31.5

Temperature(C)

0 10 20 30 40 50

Time (min)

(a)

(b)

Figure 2.2 Te output rom a hypothetical thermocouple as a unction o time when (a) thetemperature is steady and (b) the temperature is increasing. Te dashed lines indicate the plus andminus 2 s levels or the baseline reading.


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Measurement 11

where the mean M and standard deviation s o the measurement are de ned above. Tedi erence (r M) is usually interpreted as instrumental error. It is known rom statisti-cal analysis that in the special case o normally distributed data, about 68% o the values

are within one standard deviation o the mean, and about 95% are within two standarddeviations.1 Only one in a million readings will deviate arther than 5 s rom the mean(assuming that the physical quantity being measured actually remains constant). In thisexample s =0.3C, so 95% o the readings are expected to lie in the range o 29.4C to30.6C as indicated by the dashed lines in Figure 2.2a. In act, the gure shows that 49o the 50 readings (i.e., 95%) do lie within this region, which means that the temperatureis stable according to this hypothetical thermocouple. I it is known that the standarddeviation is 0.3C, then it becomes clear that the apparent increase o 0.2 C in the sec-ond temperature reading is not a signi cant change.

Figure 2.2b shows the output o the same hypothetical thermocouple during a heat-ing cycle, with the plus and minus 2 s limits indicated by dashed lines as be ore. Te

rst two readings just happen to be 30.0 C and 30.2C, as be ore. As argued above, theincrease in the second reading is not large enough to indicate that the temperature isincreasing. However, a er about 20 minutes the temperature readings requently exceedthe limit indicated by the upper dashed line. At that point it is clear that the temperaturehas increased by an amount that is statistically signi cant.

Te preceding example demonstrates that a discussion o the reproducibility o ameasurement is closely tied to a consideration o its statistics. In particular, the expected variability in a measurement can be quanti ed by the standard deviation. Although wehave not considered the root cause o the variability in the measurement, it is clear thatthis variability must be reduced in order to improve the precision o the measurement.

2.4 Sources of Error

Te variability observed in multiple sensor readings is a combination o the real vari-ability in the quantity being measured and the variability due to the sensor itsel . Telatter is usually called the instrumental error . Te goal o the measurement is to deter-mine the process parameters (temperature, fow rate, etc.), and any variability in theprocess should become evident rom a statistical review o the process sensor data. Asdemonstrated above, the interpretation o this data depends on the standard deviationo the sensor output under quiescent conditions. Tere ore, the instrumental error mustbe assessed in order to determine the signi cance o changes in the sensor output.

Te sources o instrumental error can be classi ed into systematic error and stochastic(random ) error . Systematic error is usually due to a faw in the design, installation, cali-bration, or use o the process sensor. Tis type o error is mani ested as a constant, orslowly changing, o set or multiplicative actor in the measurement. Common causes o systematic error are sliding mechanical seals in orce or position sensors, dark current inin rared detectors and video cameras, and incorrect values o physical constants (suchas viscosity or index o re raction) in calculations. Te accumulation o process mate-rial on the windows o optical sensors ( ouling) will also lead to systematic errors as aresult o the decrease in light intensity.


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Systematic errors o en lead to instrumental dri , wherein the amount o errorchanges relatively slowly over time. Since electronic circuits are sometimes sensitive tochanges in temperature, a potential source o instrumental dri is the ambient tem-perature in the process area. Depending on the operational design o the sensor, otherenvironmental conditions (such as barometric pressure) can contribute to dri .

Stochastic errors are rapid but usually small fuctuations in the sensor output, andthis random component o the signal is o en called noise. Sensor noise comes rom a variety o sources including vibrat ions rom the process equipment, ambient sound,electrical ground loops, corroded electrical connections, and electronic circuitry.Mechanical vibrations can inter ere with the precise optical or mechanical alignmentin a sensor and cause an error in measurement. Sensors that are designed to measure very small electrical signals contain ampli er circuits with a high gain; such devicesare o en sensitive to vibration. Sensors that are very sensitive to vibration are micro-phonic and there ore respond to ambient sound in the process area (o en a very noisy environment).

Ground loops are ormed by connecting electrical equipment to ground via mul-tiple paths or ground circuits. Tese connections create a continuous circuit or loopin which alternating currents are induced by neighboring electrical equipment. Teinduced current leads to a fuctuating voltage o set as a result o the resistance o thewiring, and this o set appears as a spurious signal in the sensors electronics. 2 In aprocess environment, the current fowing through ground loops can be quite strong,so the process sensors must be adequately shielded and properly grounded in order toprovide reliable results.

Mechanical vibrations and electrical ground loops contribute stochastic noise to theprocess measurement, but it o en happens that such noise is dominated by a ew compo-nent requencies. ruly random electrical noise is caused by loose or corroded electricalconnections and electronic ampli cation. When an electrical connection becomes looseor corroded, a slight voltage di erence may develop across the junction. Tis voltage isnot well de ned because the junction resistance is not well de ned, and the fuctuating voltage a ects the sensor output in an unpredictable manner. Other sources o ran-dom electrical noise include shot noise and thermal noise (see or example Fraden 1996,pp. 212 ). Shot noise is caused by the act that electrical current is conducted by elec-trons; the current appears to be continuous, but it is in reality a rapid and randomsuccession o charge trans er events. Termal noise is due to the random motion o elec-trons in a conductor; this motion is dependent on the temperature o the conductor, soan increase in temperature produces an increase in thermal noise.

Noise can o en be mitigated by averaging the sensors output signal over a numbero readings. Stochastic error is as likely to cause an increase in signal as it is a decrease,so the positive and negative fuctuations can be expected to cancel each other i asu ciently large number o measurements are summed. An example is provided inFigure 2.3, which shows a 10-point moving average o the data shown in Figure 2.2.3 Teconstant temperature case (Figure 2.3a) shows a airly constant output, and the heatingcycle case (Figure 2.3b) clearly indicates an increase in temperature. Process sensorso en per orm some internal signal averaging in order to produce a more stable signal.


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Measurement 13

2.5 Analysis of Error

Te individual systematic and stochastic errors are involved in making a measurementpropagate through the process sensor and become compounded in the resulting outputsignal. I the sensor model is known, it is possible to estimate the relative e ect o eacho these contributions through an analysis o error. A er one has identi ed the relativecontributions o each source o error, it is possible to ocus on reducing the most egre-gious o them. A thorough example o error analysis is given in chapter 10 , where thee ects o environmental conditions on a lm thickness sensor are considered.

Error analysis is essentially a calculation o the cumulative e ect caused by eithersmall changes or systematic error; it is there ore a calculation o di erentials. A simpleexample is the measurement o resistance, which is the basis o many simple sensors. I aconstant current I is passed through a resistance R, the voltage di erence V between itstwo terminals (see chapter 5) is given by

V IR= (2.6)

29.0

29.5

30.0

30.5

31.0

31.5

0 10 20 30 40 50

Time (min)

Temperature(C)

29.0

29.5

30.0

30.531.0

31.5

Temperature(C)

0 10 20 30 40 50

Time (min)

(a)

(b)

Figure 2.3 A 10-point moving average o the data shown in Figure 2.2 or a hypothetical ther-mocouple: (a) the moving average o data shown in Figure 2.2a; (b) the moving average o datashown in Figure 2.2b.

http://44168_ch10.pdf/http://44168_ch05.pdf/http://44168_ch05.pdf/http://44168_ch05.pdf/http://44168_ch10.pdf/

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Tere ore, knowing the amount o current, one could measure the voltage and calculatethe resistance according to

RV I = (2.7)

Tus, two distinct quantities are involved in this measurement, and the total error inmeasured resistance depends on the error in measured voltage and the error in theassumed current.

Te absolute error ( d R) in the measured resistance due to the error ( d V ) in the mea-sured voltage is

RR

V V

I V I = =

1(2.8)

where denotes partial di erentiation and the vertical bar signi es that the current isheld constant. Te relative or ractional error ( d R/R) can be calculated by dividing bothsides o equation 2.8 by equation 2.7:

RR

I I

V V

V V I

= = (2.9)

Similarly, it can be shown that the ractional error due to an error in the assumed valueo the current (which may have changed over time) is given by

RR

I I V

= (2.10)

Te negative sign indicates that the resistance is underestimated i the current isoverestimated.

Since the measurement o voltage is independent o the estimation (or prior mea-

surement) o the current, the contributions to the ractional error are also independentand must be added in quadrature. Tere ore, the total ractional error in the measuredresistance is

RR

RR

RR

V V I V

=

+

=

2 2 2

++

I I

2

(2.11)

Equation 2.11 shows that the relative error o the resistance measurement is greater than

the larger o the two relative errors in voltage and current. I those two error terms arecomparable to each other, then

RR

V V

1 414. (2.12)


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Measurement 15

Results similar to equations 2.11 and 2.12 can be derived or any process sensor i theunction S rom equation 2.1 is de ned. Te value o such calculations is that they pro-

vide a rigorous ramework or understanding sensor error in terms o the error in eacho the physical observables involved in the measurement. Once the sensor variability isde ned, it is possible to address the variability o the process itsel .

Suggested Reading

Bevington, P.R. and Robinson, D.K. (1992). Data Reduction and Error Analysis for thePhysical Sciences,Second Edition. New York: McGraw-Hill.

riola, M.F. (2005). Elementary Statistics, Ninth Edition. Boston: Pearson/Addison-Wesley.

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