How To Reduce Temperature Measurement Uncertainty

Introduction

This is Part 2 of a two part series exploring how to choose conductive type temperature sensors. Part 1 explored the construction and operation of Thermocouples, Thermistors, and RTD's and the inherent strengths and weakness of each type of sensor. In this part we explore how the choice of sensor, sensor placement and environmental factors contribute to measurement uncertainty and the calibration techniques that can be used to reduce this uncertainty. Final sensor selection is dependent upon both the inherent capabilities of the sensor and where and how is to be used.

Types and Sources of Temperature Measurement Error

• Inherent Accuracy and Calibration Error
• Location and Gradient Induced Error
• Heat Transfer and Self Heating
• Thermal transients
• Other External or Environmental Influences
  • Atmosphere & Environment: Moisture, Oxidation, Reduction
  • Magnetic, Capacitive, RF and Grounding Effects
  • Mechanical Stress, Acoustics, Vibration and Thermal Dynamics

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Inherent Accuracy and Calibration Error

The inherent accuracy of a sensor relates to how well a sensor conforms to its intended temperature profile temperature straight out-of-the box in an ideal isothermic environment. The inherent accuracy of a sensor is primarily determined by its construction which was covered in Part I. For a NIST traceable sensor such as thermocouples or RTDs, the ideal temperature profile is the NIST standard. Calibration is typically used to correct for inherent accuracy errors, but some contributors, such as thermocouple hysteresis require knowledge of the temperature dynamics the sensor will be exposed to in order to correct for it.

The common way to correct for inherent accuracy errors is to calibrate the sensor in a controlled isothermal liquid bath and compare temperature readings against a standard. For an absolute temperature measurement, the standard could be another calibrated sensor. Alternatively, point calibration such as immersion in an ice bath (0?C standard) or other standardized liquid in phase-transition (such as a gallium freezing bath at 30°C) may be a way to spot-check absolute accuracy.

If just relative accuracy is important, an array of sensors may be calibrated to each other by immersing all the sensors in a common bath at a know temperature (like 0?C for an ice bath), then slowly raising the temperature of the bath while tracking all sensor responses. To achieve the best results, the calibration bath should span the same temperature range as the intended measurement and the rate of temperature increase should slow relative to sensor responsiveness to reduce time-transient errors. The method absolutely requires all sensors to be subject to the exact same thermal conditions.

The limiting factor for minimizing the inherent sensor error is the accuracy and precision of the calibration process. Generally, thermistors and RTD's have better inherent accuracy than thermocouples, but all three types of sensors will require calibration to achieve accuracies down to 0.1?C. As described in Part I, the challenge is more difficult in calibrating thermocouples as compared to thermistors and RTDs because calibration must consider both hot and cold junction temperature errors.

Of the three types of sensors, the RTDs win on inherent accuracy with thermistors close behind (depending upon the specific choice). Thermocouples have the least inherent accuracy due to the mixture of alloys used and the extra work associated with dealing with both hot and cold junctions.

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Location and Gradient Induced Error

It is nearly impossible to sense temperature exactly where you need it. At the very least, the sensor itself has a finite size that displaces the sensing element from its attachment. So the error we are talking about here is a result of the sensor being at a different location than the desired measurement location. This error can be significant in applications where are not uniform or stable and the problem worsens if the sensors are large or inadequately attached. Sensors such as thermistors and RTD's will have more potential for location error than an equivalently placed thermocouple simply because of their size. The figure below illustrates how errors in sensor location result in temperature measurement error. Location error 'A' is a direct result of the entire sensor being displaced from its desired location typically because of interference. Location error 'B' is a direct result of the sensor element displaced from the intended surface by its encasement.

The thing to note here is that while the error resulting from location A is due to the thermal gradient along the mounting bar, the error that results from location B also involves ambient environmental conditions that can remove (or add) heat from the sensor case (which we call heat transfer error).

Location errors often just result in an offset error that be compensated for to the extent that all surrounding heat sources and sinks are known. For many systems, this can be very complex. Tracking heat flow through real three-dimensional systems involving conductive, convective and radiative energy exchange can be time consuming. However, to the extent that they are not known or are in transition, location errors will resist calibration and can strongly affect the accuracy and precision of system temperature measurements.

Thermistors and RTDs are usually larger sensors than thermocouples and they tend to produce voltages (or rather resistances) that average the temperature throughout the element. This does not mean that they will indicate temperatures half-way between hot and cold sides, rather it means that they will register a temperature that lies between the extremes but integrates the temperature distribution throughout the element. A non-linear temperature gradient across the element will bias the temperature reading towards either hot or cold sides.

Fully calibrating a system to compensate for location error can be very daunting as each heat source and heat sink (including ambient air temperature) and the dynamics of how heat is exchanged between them can form a very complex problem. If location error looks like it could be a problem in your application, the best way to avoid complex calibration is to utilize the smallest sensors possible and place them as close to the source as possible.

One additional note for thermocouples is offered here. Location error can be a problem for both the hot and cold junctions. In the case of a thermocouple measurement instrument, the cold junction temperature is usually determined by another sensor located nearby. As is often the case, the cold junction measurement is performed inside an electronics enclosure that contains multiple heat sources and sinks. If ultra-high accuracy and precision is required, location error of the cold junction within the electronics enclosure must also be considered.

Because the thermocouple is the smallest, most durable, and most 'point-sensing' of the three types of sensors, they can typically be placed better than RTD's or thermistors to minimize location error at the hot junction. With care taken towards the cold-junction measurement, which is often not as critical for placement, these sensors usually offer the best solution for minimizing location error. Overall, to minimize location error, choose the smallest sensor.

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Assessing Heat Transfer and Self Heating Error

Even if the errors associated with inherent accuracy and location have been fully assessed and compensated for, a temperature sensor still may not be able to accurately determine the desired temperature. The sensor receives other conductive, convective and/or radiative inputs besides the desired location temperature. In the previous figure, it can be represented by ambient conditions that heatup or cool down the sensor. It can also result from close proximity of the thermally conductivity electrical wires used in thermistors, RTD's and some thermocouples to a nearby heating element. In this instance, heat from a local source travels up the copper wire to the sensing element and distorts the measurement. Many thermocouple types uses alloys that are not very conductive, so this kind of error is less critical for them. For instance, a Type T thermocouple has one copper wire that could be affected by this kind of error, while the nickel alloy wires of a Type E thermocouple are far less sensitive.

The third form of this kind of error applies to Thermistors and RTDs and results from heat dissipated inside the sensing element itself (thermocouples do not require external power). Thermistors and RTDs require an external source to power them. Because these two types of sensors are resistive elements, power is dissipated inside them as per Ohm's law: current squared times resistance. To minimize this error, current to these sensors should be kept low. The milliwatt (or fractional milliwatt) of power that these sensors normally dissipate is typically quite small in comparison to the heat in which the sensor is exposed. However, if the thermally conductive path from the sensor element to the environment is not very conductive relative to the power dissipated inside the sensor, then the heat inside the sensor may continue to build and the temperature inside the sensor will continue to rise. The measured temperature may then not be indicative of the environment.

The right choice of thermocouple often offers the best choice for minimizing heat transfer errors in critical applications. But as previously stated, care must be taken with both the hot and cold junctions so that accuracy error is not introduced inadvertently by cold-junction measurements.

Transient Errors

Errors due to thermal transients are typically the most difficult to compensate for. While they are a result of temperature changing with time, they would not exist without previously mentioned errors conditions such as those arising from location and heat transfer. They are usually the most difficult to address because every material within the thermal system has its own unique thermal conductivity and capacity. In electrically equivalent lumped-mass circuits this is like saying that every material has its own thermal time constant, so that trying to determine the actual time-varying temperature external to a sensing element requires all the same knowledge needed to resolve static errors mentioned in previous sections, plus knowledge of the time constants involved and what each of the sources and sinks are doing in time. Because of the complexity of solving transient temperature problems, the majority of temperature measurements are performed at near steady-state conditions. Of the three types of sensors, once again, the smallest wins (thermocouple) because it corresponds to the sensor with the smallest time constant.

Thermal resistances associated with different conduction paths or materials are calculated by dividing the path length by the product of the thermal conductivity and cross-sectional area. For example the thermal resistance associated with heat propagating down the length of a 0.1 meter 316 stainless steel rod of 0.01 meter cross section is:

The thermal capacitance associated with the rod is the product of the mass (density times volume) times the thermal capacity of steel:

So the thermal time constant associated with heat propagating down the length of the rod is the product of these two or:

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Other External Influences

Atmosphere & Environment: Moisture, Oxidation, Reduction

For all three sensor types, operating or cycling them near their temperature limits can cause deterioration that results in drift from the initial profile. Thermistors and RTDs are usually well sealed from the environment and so are less susceptible to internal corrosion; however, these sensors are usually connected by means of copper wires so deterioration of wiring leading to the sensor is also of concern. For RTDs the lead-wire corrosion problem is mitigated by the use of 3-wire or 4-wire units that do an excellent job of measuring the resistance of the sensing element and not the connection wire. For the sensing element itself, RTDs exhibit the greatest stability. Thermistors usually exhibit some initial drift but are generally stable after initial aging. For thermocouples, addressing this problem is more complex. The voltages produced are a direct result of the dissimilar metals used and alloy formulation which changes as the metal ages and deteriorates. Furthermore, thermocouples are often used bare (without encasements) in order to improve responsiveness and/or operate at extreme temperatures. Without encasements, the thermocouples least susceptible to oxidation are the platinum and tungsten based types (B, C, D, G, R, and S). Of the base-metal type thermocouples (E, J, K, N, T), types K and N have the best oxidation resistance but are not suitable for reducing, vacuum, or sulfurous atmospheres without encasement. Types E, J, and T are generally suitable for all atmospheres as long as their temperature limits are not exceeded. Type E is less able to handle sulfurous atmospheres, while the type J rusts in elevated or subzero temperatures and become brittle at subzero temperatures as well. Special formulations of type E, K, N, and T are capable of being used down to cryogenic temperatures of -250?C.

One further note on atmospheric environments, if the intention is to measure the temperature on a surface, forced air flow on and around a sensor can cause a false reading because convective currents can add or remove heat from the sensor and/or measurement surface. If the atmosphere is a different temperature than the surface or the measurement environment is moist, then the heat flow associated with convection must be considered just as if it were another heat source or sink.

Mechanics, Acoustics, Vibration and Triboelectric Effects

In choosing the right sensor, the mechanical environment must be considered as well. If the sensor will be subject to a great deal of mechanical motion, vibration or high intensity acoustics, small wire gages and fragile sensors should be avoided. The most common wire failures occur at connection points where there is the greatest amount of flexure. Mechanical motion or vibration can stimulate internal resonances inside the sensor that can cause early failure. While thermocouples are generally the most durable of the three sensor types, some of the alloys used in thermocouple wire are more ductile than others and capable of handling more motion. The iron lead of a Type J thermocouple is among the worst for resisting fatigue.

Besides fatigue, cables in motion also generate low voltage triboelectric effects that are normally too small to be of concern. However, for microvolt sensors such as thermocouples or RTD's they could become a contributor to measurement uncertainty if the motion stimulating the effect is of the same order as the thermal responsiveness you intend to measure.

Magnetic, Capacitive, RF, and Grounding Effects

Because of the relatively small signals involved, thermocouples and RTDs generally have the lowest noise immunity of the three sensor types. Shielding and properly grounding these sensors go along way to improving noise immunity from potential offsets due to capacitive, RF, and offset currents. However, immunity from magnetic sources is not so easily conquered.

The environments in which these sensors must operate may contain large motors and solenoids or high current devices like welders that cause large transient currents or magnetic surges. For those sensor types that require stimulating electronics (thermistors and RTDs), these transients or power droops could potentially affect the power supplies and sensing circuits inside the sensor electronics and consequently affect temperature readings. Alternatively, large inductive spikes can cause circulating currents that alter ground potentials near the sensors that, in-turn, bias the voltage read from the sensor and create a false reading.

When thermistors are used to measure temperatures near their lower extremes, the resistance may approach 100K or more. When this happens, long runs of thermistor wire can create an antenna that can add a great deal of noise to the measurement system. While most of this can be filtered out, the potential for biasing the measurement up or down becomes greater because DC charge can collect as well (electret effect). Normally, thermistors would not used near their temperature extremes unless the loss of accuracy was already acceptable.

The best method to protect from outside electrical and magnetic sources is to keep the sensor and lead wires away from them. It also helps to shield them and pay close attention to electronics isolation and grounding. In general, keep sensor lead wires short. If possible, convert the signals to digital form as close to the measurement point as possible to minimize noise.

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System Calibration and Analysis - Putting it all Together

It may be clear by now that good measurement accuracy repeatability requires as much knowledge of the environment as it does of the sensors. We can however, still look at system calibration as consisting of two parts. First, there is the calibration that demonstrates how a sensor will perform in an ideal environment, or its inherent accuracy. Next, through thermal analysis, we can apply the predicted environment to the sensor to determine how well we can achieve our required accuracies and so estimate repeatability.

The first level of calibration is required to extract the inherent accuracy of the sensor through a batch calibration of all sensors in a controlled bath. This data is then stored in a sensor calibration table for use in separating out heat transfer and location errors.

The second level of analysis is to characterize the thermal environment by identifying and quantifying all the heat sources and sinks as well as the expected heat transfer mechanisms in the system relative to the anticipated sensor locations. For this step, conductive, convective, and radiative heat transfer should be considered as well as whatever thermal transient conditions may be present. For a first-order analysis of transient conditions, it is often helpful to calculate lumped-mass thermal time constants to determine the expected thermal responsiveness of the system. Later, if necessary, a more complete thermal analysis can be performed by finite-element methods. Above all, don't ignore the recommended temperature limits of each sensor type. It is usually better to use a sensor that operates well within the operational temperature limits than to pick one that barely fits.

The final level of analysis is to assess how the sensor will behave in the thermal environment. What is the mass of the sensor and how does it compare to the system time constants? What are the expected location errors due to any inabilities to place the sensor exactly where it needs to be? What are the atmospheric influences that could affect the ability of the sensor to correctly determine the temperature at the desired location? And finally, what electrical, magnetic, or mechanical disturbances could mask the desired measurements? The answers to these questions can help lead you to the optimal sensor choice.