1.4 System Accuracy* B. G. LIPTÁK (1982, 1995, 2003) DEFINITIONS OF TERMS

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1 1.4 System Accuracy* B. G. LIPTÁK (1982, 1995, 23) DEFINITIONS OF TERMS Accuracy (Webster). Freedom from error or the absence of error. Syn. precision, correctness, exactness. (In this sense, the term is a qualitative, not quantitative, concept.) Accuracy (ISA). In process instrumentation, degree of conformity of an indicated value to a recognized accepted standard value, or ideal value. Accuracy, measured (ISA). The maximum positive and negative deviation observed in testing a device under specified conditions and by a specified procedure. Accuracy of measurement (NIST). Closeness of the agreement between the result of a measurement and the value of the measurand. Because accuracy is a quantitative concept, one should not use it quantitatively or associate numbers with it. (NIST also advises that neither precision nor inaccuracy should be used in place of accuracy.) Error (ISA). In process instrumentation, the algebraic difference between the indication and the ideal value of the measured signal. It is the quantity that, algebraically subtracted from the indication, gives the ideal value. Range (ISA). The region between the limits within which a quantity is measured, received, or transmitted, expressed by stating the lower and upper range values. Rangeability (recommended by IEH). Rangeability of a sensor is the measurement range over which the error statement, in the units of a percentage of actual reading, is guaranteed. Repeatability (ISA). The closeness of agreement among a number of consecutive measurements of the output for the same value of the input under the same operating conditions, approaching from the same direction, for full-range traverses. Repeatability (NIST). Closeness of agreement between the results of successive measurements of the same measurand carried out under the same conditions of measurement... Repeatability may be expressed quantitatively in terms of the dispersion characteristics of the results. Reproducibility (ISA). In process instrumentation, the closeness of agreement among repeated measurements of the output for the same value of input made under the same operating conditions over a period of time, approaching from both directions. Reproducibility (NIST). Closeness of agreement between the results of measurements of the same measurand carried out under changed conditions of measurement. Uncertainty (Webster). A feeling of unsureness about something. Uncertainty (IEH Section ). Measurement uncertainty is expressed to a confidence level of 95%, and it is the limit to which an error may extend. Language, Terminology, and Reality The guide titled International Vocabulary of Basic and General Terms in Metrology (commonly referred to as VIM) was published by ISO in the name of seven organizations and contains the VIM definitions of 24 terms relevant to measurement and accuracy. So, from a theoretical point of view, we do have standards and internationally agreed upon definitions. But the reality in the average industrial plant is different, and this Instrument Engineers Handbook is written for the average instrumentation and control (I&C) engineer in those plants. Therefore, when we quantify an error herein, which one should expect when making a measurement with a particular instrument, we will not (yet) use terms such as uncertainty but will try to stay on familiar grounds. On the other hand, we will try to take a step in the right direction by improving the clarity of our language. When an instrument is specified to have ±1% accuracy, people do not expect it to have 99% error! The intended meaning of that statement is ±1% inaccuracy or a ±1% error relative to some reference standard. It is important to emphasize the role of a reference standard in all measurements, as we humans are incapable of measuring anything in the absolute. All we can do is compare an unknown quantity to a known one and determine which is larger or smaller and by how much. The presence of a reference also means that a measurement can be in error not only because the sensor is inaccurate but also because the reference has drifted or was inaccurate to start with. * Used with permission of the Instrumentation, Systems and Automation Society by Béla Lipták

2 1.4 System Accuracy 79 CLARIFYING THE ACCURACY STATEMENT ± Bias In a volume dealing with process measurement, no subject is more deserving of in-depth evaluation than the error that is inherent in all measurement. Good control is possible only if the controlled variable is precisely measured. Yet the term accuracy (or, more precisely, inaccuracy or uncertainty) itself is poorly defined, frequently misunderstood, and often used as a sales gimmick. Consequently, use of this term cries out for international standardization and, as was noted above, ISO has already prepared such standards. The need for clarity of language and standardization exists for the following reasons: 3 2 5% of Area ±.67δ 68.3% of Area ±1δ Precision 95% of Area ±2δ Total Uncertainty True Value When the error or inaccuracy of an instrument is stated to be ±1%, one would assume that this statement refers to the actual measurement the actual reading. One would assume that, if this particular instrument happens to read 1, the true value of that measurement must fall between 99 and 11, but this frequently is not the case. Some manufacturers express their error statements (inaccuracy percentages) on the basis of percent of actual span, while others might base it on percent of full scale, percent of range, or percent of upper range value, and so on. This inconsistency is undesirable, because it is confusing. It would be better if all measurement error statements always referred to the actual measurement. 2. To make error statements expressed as percentages of the actual measurement truly meaningful, the statement should also specify the measurement range over which the statement holds true. This would be a simple matter if all manufacturers agreed to define rangeability as the measurement range over which their error statement (as a percentage of actual reading units) is guaranteed. This approach would allow all sensor inaccuracies to be stated on the same basis and therefore would eliminate the confusion. If all detector inaccuracies were stated as x% of actual reading throughout the range of y, users could be comparing apples with apples when comparing bids, and the room for creative specmanship would at least be reduced. 3. Further confusion occurs because different manufacturers include different factors in their error statements. Most suppliers include only linearity, rangeability, and hysteresis errors in their total error statement; they list the error contributions caused by drift, temperature effects, overrange, power supply, humidity, RFI, and vibration separately. ly, some manufacturers claim an apparent increase in accuracy not by improving precision but by considering fewer and fewer effects in the total error statement. Naturally, to reverse this trend, international agreement is needed with regard to the amount of variation (in ambient temperature, power supply, and others) that the manufacturer s error statements must include. FIG. 1.4a In any measurement, the total uncertainty (total error) is the sum of the sensor s random error (precision) and its systematic error (bias). 4. Yet another source of confusion is the fact that, when the error of 1 sensors is tested, the results fall onto a bell curve (Figure 1.4a). It would be desirable to reach international agreement so that all error statements would always be based on the performance of at least 95% of the units tested. In addition, an error statement should always state if it is based on self-evaluation performed by the supplier or on an evaluation by an independent testing laboratory and, in the latter case, if the test report is available for review. If the above four recommendations were universally accepted, the subject of sensor error and inaccuracy would be much less confusing. While this is not likely to occur soon, a better understanding of the factors that cause the present state of confusion should be helpful, because it can speed the development of universal standards for sensor error and performance. TERMINOLOGY OF INACCURACY AND REPEATABILITY The purpose of all measurement is to obtain the true value of the quantity being measured, and error is thought of as the difference between the measured and the true quantity. Because it is impossible to measure a value without some uncertainty, it is equally impossible to know the exact size of the error. What is possible is to state the limits within which the true value of a measurement will fall. The accuracy-related terminology used in the process control industry can be illustrated by an example of target shooting (Figure 1.4b). The spread of the nine shots fired into the upper right-hand corner of the target in a tight pattern represents the random error of the shooter. Looking at the penetration of the bullets, one can say that his shooting is repeatable and precise, but precision alone does not guarantee accuracy; it is only the measure of the ability of the shooter, which is called random error. 23 by Béla Lipták

3 8 General Considerations Repeatability Random Error (Precision) Systematic Error (Bias) THE ACCURACY STATEMENT The accuracy of a product category (sensors, transmitters, and so on) is established on the basis of testing large numbers of these products. For the more important sensors, the accuracy statements 1 should also include information on testing. An example of such a statement is quoted below from a National Bureau of Standards Calibration Certificate for a turbine flowmeter: Illegitimate Error FIG. 1.4b Accuracy terminology. Total Error (Inaccuracy) The distance between the mean impact of the nine bullets and the bull s-eye of the target is the systematic error. This error (caused by the wind or by the faulty adjustments of the sights) is repeatable and can be eliminated (in case of sensors, by calibration or by rezeroing; in case of the shooter, by waiting until the wind stops or by readjusting the sights). This error is not related to the shooter s inability to duplicate his shots. Systematic error is also referred to as bias, which is the systematic displacement of the measured value from the true one. It can be reduced by recalibrating the sensor against a reference standard, such as a calibrated (standard) thermal element, a known composition analytical sample, or dead weights. The shot in the lower left-hand corner of the target is an illegitimate error, which is caused by blunders and can be totally eliminated. The total error in a measurement can thus be defined as the sum of the random error and the systematic error or bias. If the purpose of an installation is to maintain the process conditions at previously experienced levels, and there is no interest in their true values, then the goal is to reduce the random error, without paying much attention to the remaining bias. In many industrial installations, such a repeatable (but inaccurate) measurement is sufficient. Conversely, if the interest is in determining the true value of the measurement, because the installation serves such absolute purposes as accounting or quality control, the repeatable measurement is insufficient, and attention must be concentrated on absolute (total) accuracy. This can be obtained only through the reduction of both the random and the systematic errors, which is usually achieved by recalibration. The results given are the arithmetic mean of ten separate observations, taken in groups of five successive runs on two different days. The reported values have an estimated overall uncertainty of ±.13%, based on a standard error of ±.1% and an allowance of ±.1% for possible systematic error. Figure 1.4a illustrates the results of such a test. In that test, the precision (half of repeatability) of 68% of the devices tested has been found to be ±1% of the true value, while, for 95% of the devices, it fell within ±2%, and, for all 1% of the devices, it amounted to ±3%. The total error (total uncertainty) is the sum of precision plus bias, which is the systematic error of the bell curve itself. Because the bias can be reduced by calibration and rezeroing, but the precision (or repeatability) cannot, it would be desirable if manufacturers identified both of these values. Manufacturers should also state if the basis of this data is 68, 95, or 1% of the devices tested. To allow the I&C profession to mature, manufacturers should eliminate specmanship from their sales literature, so that users can compare apples with apples when making a selection. FLOW MEASUREMENT EXAMPLE Figure 1.4c illustrates three flow sensors installed in series in the same process pipe, with each measurement signal being totalized. All three flow sensors are sized for the same full range of 1 GPM (38 l/min) flow rate. The goal of this example is to illustrate how the total system error is determined at the flow rates of 2 GPM (76 l/min) and 8 GPM (34 l/min) in two different cases. In Case 1, the basic assumption is that the component errors are additive. In Case 2, the assumption is that the total system error will be the error of the least precise component in the system. Errors introduced by countertotalizers, which is usually one count, will be neglected. For the purposes of the example of Figure 1.4c, the magnetic flowmeter, transmitters, and integrators will all have ±.5% full-scale (FS) error. The orifice plate will be assumed to have an inaccuracy of ±.5% of rate and the error of the turbine flowmeter will be assumed to have an inaccuracy of ±.25%. (The orientation table in Chapter 2 [Table 2.1b] provides complete data for all flow sensors, including their performance characteristics.) 23 by Béla Lipták

4 1.4 System Accuracy 81 The Inaccuracy of These Expressed As ±.5% of Full Scale Range FQI FQ FT Totalizer Integrators FQI FQ FY FT Scaler and Totalizer FQI The Inaccuracy of These Devices is ± I Count Inaccuracy is Εxpresssed as ± % of Measurement Flow Sensor Output FEM Linear Analog System (Magnetic Flowmeter) Nonlinear Analog System (Orifice Plate) Linear Digital System (Turbine Flowmeter) Flow ( 1 GPM) Ideal 8 1 FIG. 1.4c Errors of the components of three different flow totalization loops. In the more detailed discussion that follows, it will be shown that the overall system error can be much greater than the component errors. It will be shown that, at 2% of full-scale flow, the error of a turbine flowmeter will be around.25% of actual flow, the error of a magnetic flowmeter might range from 3% to 9% of actual flow, and the measurement error of an orificebased measurement error will range from 5 to 12% of actual rate. The ±.5% maximum inaccuracy (some based on actual readings, others on full-scale readings) was selected to reflect the typical installations in the existing plants. Today, when smart transmitters and improved sensors are available, one can select more accurate system components, some with maximum errors of ±.1% of actual span. Here, we will assume that the maximum error of any of the system components is ±.5% and, based on that assumption, we will determine the resulting total system error. The performance of analog and digital, linear and nonlinear devices will be discussed separately. Analog and Linear Devices Traditional Magnetic Flowmeters The performance of a linear analog flow sensor, such as a magnetic flowmeter, is shown in Figure 1.4d. The line marked actual represents the relationship between the actual flow and the output signal generated by the flow sensor. Figure 1.4e illustrates this error as a percentage of full scale (FS), with the error limits being ±.5% FS. In Figure 1.4f, the same ±.5% FS sensor performance is illustrated, but against a vertical coordinate that is a percentage of actual flow units (instead of full scale). The specific detector performance is likely to be better at most points of its range than what these error limits would imply. The main message is that, for sensor with percent-fs performance, the measurement error increases as the flow rate drops, as shown in Figure 1.4g. FIG. 1.4d Performance of a linear analog flow sensor, such as a magnetic flowmeter. Flow Sensor Error Based on Full Scale Ideal FIG. 1.4e Error plot for a percentage of full-scale flow sensor. It should be noted that the performance described here is representative of the older designs of magnetic flowmeters, which continuously maintained their magnetic fields. In the newer designs, the field is cycled on and off, and the sensor can be automatically rezeroed, so the measurement error can be reduced. Therefore, the inaccuracy of these newer magnetic flowmeters can approach ±.5% of actual flow. Analog, Nonlinear Orifice Plates The orifice plate itself is rather accurate and is a percent-ofactual-flow sensor, having an error limit of ±.5% of actual flow rate, as shown in Figure 1.4h. 23 by Béla Lipták

5 82 General Considerations Flow Sensor Error Based on % of Flow Rate Reading (%) + 1. Inaccuracy Based on Reading Ideal FIG. 1.4f The error of a linear flow sensor shown in units of percentage of full-scale flow. Inaccuracy Based on Flow Rate Reading FIG. 1.4h The error contribution of the orifice plate alone. Desired Measurement, Flow (%) 1 8 ±.5 % of Measurement Ideal Measurement, Orifice Pressure Drop (%) FIG. 1.4i Performance of an orifice type nonlinear analog flow sensor. FIG. 1.4g The error of a linear flow sensor shown in units of percentage of actual flow. The pressure drop through an orifice relates to the square of the flowing velocity or volumetric flow rate through the orifice plate. Figure 1.4i illustrates both, i.e., this ideal nonlinear (square root) relationship and the actual performance of a specific differential-pressure (d/p) cell used in an orifice type, nonlinear flow sensor. To the error contribution of the orifice plate shown in Figure 1.4h (±.5% of actual flow rate), one must add the error of the differential-pressure transmitter shown in Figure 1.4i (±.5% FS). In addition, when the square root must be extracted before the signal can be integrated (Figure 1.4c), the error contribution (the gain effect) of this extraction must also be recognized. Figure 1.4j illustrates that this extraction of the square root improves the accuracy at the higher flow rates but degrades it as the flow rate is reduced. Digital Linear Turbine Flowmeter The calibration of a turbine meter in terms of the K factor, given in units of pulses per gallon, is rather similar to the calibration curve of an orifice plate (Figure 1.4k). The inaccuracy of a turbine meter is also in units of percentage of the actual flow and is rather constant over a fairly wide range of flows. Turbine flowmeter inaccuracy can be improved by reducing the rangeability requirement of the unit (Figure 1.4l). 23 by Béla Lipták

6 1.4 System Accuracy 83 Inaccuracy Based on Reading 5 Inaccuracy Based on Reading Nonlinear 1. 1 Linear FIG. 1.4j Comparing the inaccuracies of linear and a nonlinear flowmeter ±.5% at 3:1 Rangeability ±.25% at 1:1 Rangeability 2 4 FIG. 1.4l Turbine flowmeter inaccuracy as a function of rangeability Turbine Meter K Factor (Pulses/Gallon) TABLE 1.4m System Inaccuracy (Total Loop Error) in Units of Percentage of Readings Mean K Assumption Used to Estimate Accumulated Basis 1 Basis 2 System Inaccuracy Operating Flow Rate (GPM) Type of flow detection loop Analog, linear ±9.% ±% ±3.% ±.5% (magnetic flowmeter) Analog, nonlinear ±12.% ±2.% ±5.% ±.5% (orifice flowmeter) Digital, linear (turbine flowmeter) ±.25% ±.25% ±.25% ±.25% FIG. 1.4k Turbine flowmeter calibration curve. Combined System Accuracy Having reviewed the inaccuracies of the three flow sensors and the various loop components shown in Figure 1.4c, the next step is to evaluate the resulting total loop errors. There is no proven basis for determining the accumulative effect of component inaccuracies, and only an actual system calibration can reliably establish the total loop inaccuracy. Still, we have learned the following from experience. We know that, the fewer the number of components in an analog measurement loop, the better the loop s performance. In digital systems, no additional error seems to be introduced by the addition of functional modules. It has also been reported that the averaging of the outputs of several sensors that are detecting the same process variable will reduce the measurement error. These reports suggest that the error is reduced by the square root of the number of sensors in parallel. So, if two sensor outputs (each having a 1% error) are averaged, the error will be reduced to 1/ 2 =.7% (and with three outputs, to.58%, with four outputs, to.5%, and so on). Without actual system calibration, the evaluation of the overall loop accuracy must be based on some assumptions. Table 1.4m summarizes the system inaccuracies for the three loops in Figure 1.4c at 2 and 8% of flow rate and by evaluating the accumulated effect of component inaccuracies on the basis of one of two assumptions: Basis 1 Here, it is assumed that the inaccuracy of each component is additive, and therefore the total loop inaccuracy is the sum of component inaccuracies (a very conservative basis). Basis 2 Here, the assumption is that the system inaccuracy is the same as the inaccuracy of the least accurate component 23 by Béla Lipták

7 84 General Considerations and therefore other inaccuracies can be neglected (a very optimistic assumption). If Basis 1 is accepted for evaluating the total system error, an orifice-type installation operating at 2% of full-scale flow will have an error of ±12% of the reading, although the inaccuracy of any component in the loop does not exceed ±.5% FS. The data in Table 1.4m is based on the performance of conventional d/p transmitters and on conventional magnetic flowmeters. With the newer, pulsed DC magnetic flowmeters, the error can be reduced to ±.5% of actual flow over a 1:1 range. Similarly, if the intelligent, multiple-range d/p cells are used, orifice measurement error can be reduced to ±1% of actual flow over a 1:1 range. 2 To achieve this level of performance, it is necessary to automatically switch the d/p cell span from its high to its low setting, based on the actual flow measurement. If the conventional magnetic flowmeters and d/p cells are considered, and if they are evaluated on a basis that is slightly more conservative than Basis 2 but less conservative than Basis 1, the resulting loop errors are as shown in Figure 1.4n. From the data in Table 1.4m and Figure 1.4n, it can be concluded that neither error nor inaccuracy is by any means a clearly defined single number and that the required rangeability of the measurement has a substantial impact on performance. Therefore, a meaningful accuracy statement should answer the following questions: (1) What portion of the total error is the precision (random error) of the sensor? (2) Is the sensor error based on full scale (FS) or on actual reading (AR)? (3) Over what range of measurement values is the error statement applicable? System Inaccuracy Based on Reading Linear 4 Digital FIG. 1.4n Total loop inaccuracies as a function of sensor type and flow rate, calculated on the basis of equation 1.4(1), where the total loop error is obtained by taking the square root of the sum of the component errors squared. (Accuracy in simple flow measurement. TI-1-3a. The Foxboro Company.) 6 Nonlinear 8 1 TEMPERATURE AND PRESSURE EFFECTS If a sensor such as a d/p cell has been tested at temperatures and pressures that differ from the operating temperature and pressure, this will affect the total error. The total error includes the d/p cell error (E), which is determined under atmospheric ambient conditions. Therefore, E reflects the linearity, repeatability, and hysteresis errors of the sensor. For the purposes of this example, it is assumed that E = ±.2% of actual span. Other factors that affect the total error include the zero (T z ) and span shifts (T s ) that might occur as a result of temperature variations. For a temperature variation of 1 F (55 C), T z is assumed to be ±.5% of maximum range, while T s is assumed to be ±.5% of actual reading. The effect of changes in static pressure on the zero and span are noted by P z and P s. They are evaluated as the consequence of the physical distortion caused by 2 psig (138 bars) of operating pressure. For the purposes of this example, it will be assumed that P z = ±.25% of maximum range, and P s = ±.5% of actual reading. For the purposes of this example, assume a d/p cell with a maximum range of to 75 in ( to 19 m) H 2 O and an actual span of to 1 in ( to 2.54 m) H 2 O. It is further assumed that the actual operating temperature of the d/p cell is within 5 F (18 C) of the temperature at which the unit was calibrated and that the actual operating pressure is 1 psig (69 bars). When the process measurement is 1 in. (2.54 m) H 2 O, the above assumptions will result in the following error components: E =±.2% T z =.5 (75 in./1 in.) (5 F/1 F) = ±1.875% T s =.5 (5 F/1 F) = ±.25% P z =.25 (1 psig/2 psig) (75 in./1 in.) = ±.9375% P s =.5 (1 psig/2 psig) = ±.25% If we calculate the total error (E t ) as being the square root of the sum of the square of the individual errors, the result is: E t = ( ) = ± 2. 13% 1.4(1) From the above example, one might note that the largest contributions to the total error are the zero shifts caused by the pressure and temperature differences between the calibration and the operating conditions. These errors can be reduced by selecting a d/p cell with a maximum range that is closer to the actual reading. One might also note that the total error (E t ) would have been even higher if the actual measurement did not correspond to 1% of the actual span (1 in H 2 O), but only some fraction of it. It should also be noted that the above E t value is not the total measurement error of the loop but only the error 23 by Béla Lipták

8 1.4 System Accuracy 85 contribution of the d/p cell. Finally, one should note that one advantage of the smart transmitters is their ability to reduce the pressure and temperature effects on the span and zero. Therefore, if E is ±.1% in an intelligent transmitter, the total error E t can be kept within about ±.3%. REPEATABILITY VS. TOTAL ERROR Based on the information presented above, the following qualitative conclusions can be drawn: 1. Inaccuracy is likely to be improved by reducing the number of components in a measurement loop. 2. Inaccuracy statements are meaningful only when given in combination with rangeability. The wider the rangeability required (expected load variations), the more inaccurate the measurement is likely to be. Furthermore, the rangeability effect on digital systems is the least; it increases when linear analog system are used, and it is the highest in case of nonlinear analog systems. 3. On nonaccounting systems, the interest is focused on repeatability (random error) and not on total inaccuracy. The repeatability of most measurement loops is several-fold better than their total error. 4. Instrumentation worth installing is usually also worth calibrating. In this regard, several points should be made: (a) The accuracy of a multicomponent system is unknown unless it is calibrated as a system. (b) The calibration equipment used must be at least three times more accurate than the system being calibrated. (c) Periodic recalibration is a prerequisite to good control. 5. Instrumentation worth installing should also be worth keeping in good condition. The performance of all sensors is affected by corrosion, plugging, coating, and process property variations. Therefore, scheduled maintenance is required to guarantee reliable operation. In summary, (a) inaccuracy should be stated as a function of rangeability, (b) multicomponent systems require system calibration, and (c) maintaining good performance requires periodic recalibration and scheduled maintenance. References 1. Kemp, R. E., Accuracy for Engineers, Instrumentation Technology, Inc., Painesville, Ohio. 2. Rudbäck, S., Optimization of orifice plates, venturies and nozzles, Meas. Control, June Bibliography Applicable standards: DIN/IEC Standard #77 and ASME PTC19.1. Englund, D. R., Loading Effects in Measurement Systems, Instrument and Control Syst., February 197, Shinskey, F. G., Estimating System Accuracy, Foxboro Publication #413 5, Invensys Systems, Inc., Foxboro, MA. Taylor, B. N. and Kuyatt, C. E., Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, NIST Technical Note 1297, NIST, Gaithersburg, MD, Vom Berg, H., What is accuracy? Meas. Control, April by Béla Lipták

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