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Breaking Down the Sources of Error

April 10, 2003

Sometimes, trying to improve a process measurement such as temperature is really about gaining a better grasp of the measurement systems already in use. Any attempt to improve a temperature control loop will be a trying experience without a full understanding of the sensor's capability to provide both an accurate and reliable signal.

Table 1. Because thermocouples are the most widely used temperature-sensing device, they are used as an example to show how error contributors are identified.

In spite of the continuing efforts of most companies' engineering and technical staff members to automatically control the process, many processes still utilize measurement devices that must be read manually. To reasonably evaluate the true accuracy of a temperature sensing system, one must assess the accuracy of all the measurement devices used in the manufacturing process.

There are a limitless number of temperature sensors, interfaces and signal processing instruments, but all decisions about the process begin with the output of a measurement device. All process systems utilize one or more of the five categories of temperature measurement instruments, which are:

Thermocouples.
Resistance temperature detectors (RTDs) and thermistors.
Bimetal thermometers.
Liquid-in-glass thermometers.
Digital and analog instrumentation.

The level of confidence placed in the output values of these devices is important. For instance, are the thermocouples accurate within one, three or 10 degrees? This type of question can be difficult to answer. However, some estimations or generalizations can be made for each device that will help in understanding the output value, whether it is a transducer signal or a manually read device.

The confidence in a reading is a reflection of the certainty or uncertainty of the reading. As stated in a NIST publication on uncertainty, "A measurement result is complete only when accompanied by a quantitative statement of its uncertainty." Although originating from a laboratory or metrology point of view, this statement has industrial meaning as well. What engineer working with temperature measurement issues has not been asked, "How accurate is the sensor? How do you know the accuracy?" Subsequent discussions of the question -- "I am not 100 percent certain, but I am 93 percent confident that the reading is within a determined value of the mean" -- usually produce more questions than answers. The goal is to determine the confidence of the measurement in terms of some deviation from the mean.

Table 2. Potential error can be introduced by the extension wire.

Accuracy Defined

The true accuracy of a measurement device depends on all of the possible measurable sources of error and not just on the fundamental accuracy of the device. For this reason, accuracy will be referred to in terms of the "true accuracy" as better defined by the overall uncertainty of the reading to a 95 percent confidence level. Properly identifying and summing all of the possible sources of error contained in a reading build the true accuracy. There are an infinite number of potential contributors, but industrial users need only be concerned with the contributors that are significant enough to impact the process. As a rule of thumb, if the sum of the positive and negative contributors is one half or less of the device's base tolerance, that error source can be ignored. The statistical validity of this statement is based upon the Root Sum Squares Method, which is expressed as

where

µ_i is the individual uncertainty components for combining sensor uncertainties of uncorrelated contributors having normal distributions

Table 3. Because it is unlikely that all measuring devices would be tested at the processing equipment, the values listed, which have been estimated based on published reference and laboratory experience, can be used instead.

Sources of Error

For each device listed, there will be a different set of error contributors developed. An added benefit to realizing the true accuracy is that the value can be used to develop the calibration requirement for the device.

Thermocouple Error. Because the thermocouple is the most widely used temperature-sensing device, I will use it as an example to show how error contributors are identified and combined to produce the true accuracy. Table 1 shows a simplified base tolerance for thermocouples from -200 to 538^oC. (The temperature range can be expanded to include higher temperatures.) This base tolerance is the starting point for the system. Potential error contributors include:

Extension wire tolerance.
Drift.
Inhomogeneity.
Cold end errors.
Voltaic errors.
Linearization error.
Electromagnetic interference/radio frequency interference (EMI/RFI) noise.
Dynamic error.

Table 4. RTDs have a different set of error contributors than thermocouples.

Each error must be quantified in terms of temperature. So, for AC noise, which is measured in terms of RMS voltage, the corresponding temperature for the AC RMS value would have to be calculated or estimated based on the instrument's AC noise rejection capability.

Each individual error contributor will be measured or estimated by its minimum and maximum value. That limit of error then can be divided by two and used as the uncertainty value for that parameter; however, it is beneficial to use the full limit of error if estimates are being made and measured values are not being used. An example of error for the extension wire contributor is shown in table 2.

Table 3 shows a list of the estimated error for each thermocouple type and temperature range. These values have been estimated based on published reference and laboratory experience. Each contributor could be measured physically at the processing equipment and adjusted to reflect actual values. It is unlikely all of the devices would be tested, so reasonable estimates such as those shown in table 3 could be used instead.

RTDs. Resistive temperature sensors will have a different set of error contributors than thermocouples or other devices. Table 4 shows the assessed error for industrial RTDs. Not all of the error contributors fit the greater-than-one-half-of-the-base-tolerance rule of thumb because these tables were developed for completeness. After all, the error contributor must be included until it can be proven it is not adversely affecting the measurement. It may be that a particular RTD will exhibit greater error than estimated due to high temperature shunting errors. The tables may be useful as a sort of checklist of potential sources of error when measuring temperature.

By definition, dynamic error attempts to reasonably estimate the degree of thermal-property effects that both the sensor and its environment will have on the measurement. How well is the sensor coupled to the environment? Are thermal lags affecting the measurement? Are fluid dynamics affecting the readings? Dynamic error attempts to take these effects into account.

The following example by Michalski allows an estimation of true fluid temperature to be made based on the time response of the sensor and the rate of temperature change of the fluid. A medium's sinusoidally varying temperature was measured by a temperature sensor having a time response constant, NT, of 30 sec. The period of the temperature medium's oscillations, ?0, was 100 sec, while the amplitude of the sensor temperature, ?nT, is 5^oC.

To determine the true amplitude of the temperature of the medium we can use the following equation:

where

NT is the sensor time constant and

w is the angular frequency.

Tables 5 and 6. A parallax error only is present for instruments that must be read visually -- as is the case with bimetal and liquid-in-glass thermometers. The reading angle is relative to the devices themselves and not the surrounding mounting system.

Solving this equation produces a result of 10.65^oC. Michalski gives the derivation for the equation shown. The measurement and sensor conditions account for a significant amount of error in this scenario. The degree of error may be extreme, but it serves well to illustrate the need in some applications to properly assess the time constant of the RTD. RTDs often respond more slowly than thermocouples and frequently cannot be located in close proximity to the desired measurement point. These two limitations are what can cause the kind of error shown in the example.

Bimetallic and Liquid-In-Glass Thermometers. Tables 5 and 6 represent the sources of error for bimetal and liquid-in-glass thermometers, including parallax error. Parallax error is present for instruments that must be read visually, and it is the error associated with reading the device at an angle other than normal incidence. Remember that the reading angle is relative to the device itself and not the surrounding mounting system.

Bimetallic and liquid-in-glass thermometers generally are less susceptible to dynamic error as they are used in very specific applications. Because the base accuracy is larger for these devices, dynamic error generally does not impose any significant error contribution. A more important source of error for liquid-in-glass thermometers is immersion error. Dynamic error and immersion are very similar, and for table 6 only, immersion error is used.

Both bimetallic and liquid-in-glass thermometers are used in applications where it is either economically advantageous or safer to use devices that can be read manually. Based on these measurements, process adjustments are made manually by the operator. Manual readings and adjustments do not preclude the assessment of measurement uncertainty.

Table 6. These values summarize all of the predominant error contributors for a particular device over a specific temperature range. Cal U Req. is the required uncertainty of the calibration equipment used to calibrate the process measurement devices to the stated uncertainty.

Digital Instruments. There are many devices that fall under this heading, but issues regarding accuracy are a common factor for all. Controllers, transmitters and many others must be evaluated as to their ability to properly filter and process an incoming signal. It was noted previously that the VRMS AC noise level should be assessed or estimated for a thermocouple. Although it is true that the thermocouple acts as the source, the degree of error realized will be dependent on the digital instrumentation's ability to reject the unwanted AC signal. Other sources of error in these instruments include:

Ground loops.
Linearization error.
AC noise.
Amplifier error.
Amplifier drift.
Cold end compensation errors.
Base accuracy limitations.

Producing an accuracy table for instrumentation would be counterproductive; the specifications are supplied with the instrument and can be used to assess the influence these error sources may have on a reading. For most of them, the expected effect would be small in comparison to the sensor error, so they could be essentially negated. The primary focus is to determine if the reading instrument is at least as accurate as the sensor itself. Identifying the instrument or sensor as the predominant source of error gives immediate direction as to the best course of action for improvement.

To properly assess the sources of error in temperature measurement requires a broad analysis to be undertaken. A wide range of technical disciplines will be involved in the process, but the increased understanding of the measurement will be worth the effort.

Table 7 shows the estimated "true accuracy" for the main devices used in temperature measurement. The values summarize all of the predominant error contributors for a particular device over a specific temperature range. Total values include the base device accuracy plus all of the extraneous influences that may adversely affect the measurement. Each error contributor's limit of error has been used as an uncertainty value. Each value has been combined using the Root Sum Squares Method to form the final value expressed. The device calibration uncertainty requirement also is shown using a three 3:1 test accuracy ratio (TAR), which was developed by Nicholas and White. Calibrating the individual device beyond this ratio affords little benefit if the prevailing error component does not originate with the device itself.

Ultimately, each measurement situation and manufacturing process should be evaluated on an individual basis. But, when that is not a practical solution, the tables presented give reasonable estimates of temperature measurement devices' true accuracy.