Negative Temperature Coefficient Thermistors Part II: Determining the Level of Uncertainty

An understanding of the terms and the underlying concepts used to express measurement uncertainty will provide the basis for choosing the appropriate test equipment for an application.

Part I of this article, in the May 1997 issue of Sensors, addressed NTC thermistor characteristics, materials, and configurations. The reference numbers here are continuous from Part I.

Determining the level of uncertainty is an important part of the process used for setting up a thermistor testing system. The National Institute of Standards and Technology (NIST) [12], other national standards laboratories, and the International Organization for Standardization [13] have formed an international consensus to adopt the guidelines recommended by the International Committee for Weights and Measures (CIPM) to provide a uniform approach to expressing uncertainty in measurement. In these guidelines, terms such as accuracy, repeatability, and reproducibility have definitions that may differ from those used by some equipment manufacturers. In the NIST guidelines, for example, accuracy is defined as a qualitative concept and should not be used quantitatively. The current approach is to report a measurement result accompanied by a quantitative statement of its uncertainty [12]. Some equipment manufacturers use accuracy and others use uncertainty in their specifications, so it will probably take some time before all companies adopt the current guidelines.

Determining the Level of Uncertainty
The first step in setting up a thermistor test system is to determine the level of uncertainty allowable for the application. Because the cost of equipment increases as the level of uncertainty decreases, it is important not to overspecify the equipment. Generally speaking, test system uncertainty should be 4 to 10 times better than that of the device to be tested. A 4:1 ratio is adequate for most applications; for more stringent requirements, a 10:1 ratio may be necessary and will probably result in a more costly system [14,15]. For example, using the 4:1 ratio, a thermistor with a tolerance of ±0.2ºC should be tested on a system with an overall uncertainty of (±0.2ºC)/4 or ±0.05ºC. If a 10:1 ratio were required, the overall system uncertainty would need to be ±0.02ºC.

To calculate the uncertainty of the overall test system, the uncertainties of the individual components are combined using a statistical approach [12-15]. Each component is represented as an estimated standard deviation, or the standard uncertainty. The two statistical methods most commonly used by NIST are the combined standard uncertainty and the expanded uncertainty [12].

The combined standard uncertainty (NIST suggested symbol u_c) is obtained by combining the individual standard uncertainties using the usual method for combining standard deviations. This method is called the law of propagation of uncertainty, commonly known as the root-sum-of-squares (square root of the sum of the squares).

The expanded uncertainty, suggested symbol U, is obtained by multiplying the combined standard uncertainty by a coverage factor, suggested symbol k, which typically has a value between 2 and 3 (i.e., U = ku_c). For a normal distribution and k = 2 or 3, the expanded uncertainty defines an interval having a level of confidence of 95.45% or 99.73%, respectively. The stated NIST policy is to use the expanded uncertainty method with the coverage factor k = 2 for all measurements other than those to which the combined uncertainty method traditionally has been applied.

The expanded uncertainty of a system thus can be determined once the uncertainties of the bath, the temperature standard, and the resistance measuring instrument are known. The following is an example of the process used for calculating the U of a system with these equipment specifications.

   bath uniformity = ±0.01ºC
   bath stability = ± 0.01ºC
   uncertainty of the temperature standard = ±0.01ºC
   uncertainty of the resistance measuring instrument = ± 0.01% or ± 0.003ºC

   U = 2{(±0.01º C)² + (±0.01ºC)² + (±0.01ºC)² + (±0.003ºC)² }^1/2
   U = ±0.035ºC

Therefore, if an application allows the use of the 4:1 uncertainty ratio, the system illustrated above could measure a thermistor requiring an estimated expanded uncertainty specification no tighter than or equal to 4(±0.035ºC) or ±0.14ºC. If an application required a thermistor measurement capability with a tighter expanded uncertainty, equipment with reduced uncertainties would be necessary. For example, if a system had a bath uniformity of ± 0.005ºC, a bath stability of ±0.005ºC, a temperature standard with a standard uncertainty of ±0.005ºC, and a resistance measuring instrument standard uncertainty of ±0.003ºC, the expanded uncertainty of the system would be ±0.018ºC. The system described in the latter example would be capable of testing thermistors requiring an estimated expanded uncertainty of 4(±0.018ºC) or ±0.07ºC. These two examples illustrate how the individual equipment uncertainties affect the overall system uncertainty.

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