Accurate measurements: repeatable factors

Date of publication: 18-12-2025 🕒 6 min read

In a previous article, we discussed the economic constraints and the issue of element value tolerance. We presented basic information on the impact of temperature changes. In this article, further aspects of the problem of accuracy and stability limits are signaled.

Piotr Górecki - popularizer of electronics. Currently publishes his own magazine "Understanding Electronics". Previously, for many years he was the Editor-in-Chief of a popular Polish magazine (Elektronika dla Wszystkich). He is also the author of hundreds of articles and educational projects. Until 1993, he worked in the telecommunications industry.

This is the sixth article in a series on metrology and measurement accuracy, which began with an article entitled Accuracy and range of measurements in electronics. In the previous article Accurate measurements: basic limitations we dealt with the issue of tolerances and the effect of temperature. Now we are expanding on this important topic.

The title "repeatable factors" refers to properties and parameters, that are associated with changes that are repeatable, predictable, whose impact can be taken into account and compensated for. However, this is not as simple, as it might seem. Here is further information on the subject.

(In)linearity of TCR coefficient and hysteresis

In most catalogs, even the best, reputable manufacturers, one, specific value of the thermal coefficient. In the case of resistors, it is denoted as TCR (Temperature Coefficient of Resistance), usually expressed in ppm/°C. This is one specific value, and this suggests, that the dependence of resistance on temperature is linear, and therefore that the resistance linearly increases or decreases with increasing temperature. However, the reality is different.

An interesting curiosity is the question, how the heat factor is defined and determined. This is a separate issue, which is poorly understood and leads to misunderstandings. Now we show only an outline of the issue. We talk about the limitations of the accuracy of measurements and the possibilities of correction and compensation. Well, if the temperature affects the properties of the element linearly, then digital correction can be done relatively simply. You need to know the value of the linear thermal coefficient and the difference between the current temperature and the reference temperature. Digital correction in total will mean multiplying the "raw" result by a value proportional to the temperature difference.

In reality, the relationship is nonlinear. For example, the resistance of metals varies nonlinearly with temperature. The thermal coefficient is determined by the Callendar-Van Dusen formula, where temperature occurs in the second and third powers of the.

For an accurate description, you need the coefficients A, B, C (or α, β, δ), which are determined from measurements at at least three temperatures. The nonlinearity and unsteadiness of the heat factor hinder the most accurate measurements.

The signaled nonlinearity of the TCR coefficient makes compensation difficult, but it can be partially corrected by analog or digital means. However, hysteresis remains a problem - an element, such as a resistor or reference voltage source, remembers the effect of temperature and after cooling down does not exactly return to the previous value. Hysteresis is a serious limitation for the most accurate measurements, and that's not the end of the challenge.

The problem of pressure and humidity

The performance of electronic components also depends on atmospheric pressure, although only to a small extent and we generally ignore this. However, the problem of humidity cannot be ignored. In general, atmospheric humidity is one of the biggest enemies of all electronic circuits. In this article we discuss only the issue of repetitive changes, reversible, and variations in humidity cause changes in parameters, which are generally reversible. What is much worse, water also causes oxidation, corrosion, so moisture seriously increases (accelerates) aging, which we will discuss in the next article. Contrary to popular perceptions, the vast majority of electronic components are not hermetic at all. Neither varnish, neither paint, nor plastics are effective barriers to moisture. Only a few electronic components have truly hermetic, well-sealed enclosures, eliminating the influence of moisture on the parameters, which, of course, raises the price a lot. These are metal and-glass. For example, the most stable resistors have metal enclosures, and their interior is filled with oil. Also, the most precise integrated voltage standards (LTZ1000, ADR1000, LM399 ) do not have typical plastic enclosures, but are placed in "archaic" and having other significant disadvantages, expensive metal enclosures. Mainly because, that such enclosures provide full airtightness.

Another, still in use today, solution (it can be debated whether simpler, or more complicated) is placing not individual components, but entire precision circuits in sealed enclosures, containers filled with oil. For now we do not go into details, but only signaling a very serious problem of humidity. And here is another very serious problem and challenge with the most accurate measurements.

The problem of thermoelectric cells

Not very well known is the abbreviation EMF, or more precisely thermal EMF (Thermal Electromotive Force), denoting voltages, which can be called "thermal voltages". Well, almost everyone has heard of thermocouples and it is widely known, that the voltages they produce are very small, of the order of microvolts. For example,, the average thermal coefficient of the most popular measuring thermocouple type K (NiCr-NiAl), commonly used in multimeters, is about 41 microvolts per degree Celsius. However, it is important to remember, that a thermocouple is also the contact of any other conductors (and Semiconductors). Each thermocouple produces a small voltage when, when there is a temperature difference between its ends.

When there is no temperature difference - there is no such problem. But electronic circuits heat up more or less during operation and it is difficult to avoid even small temperature differences, or - as experts say - temperature gradient. Figure 1 (from Fluke materials) shows the thermoelectric coefficients of the connections of copper tracks and wires with Others materials. Surprisingly, even a copper-to-copper connection yields a thermocouple (among other things because of the various minor additives in copper). And the worst is at the copper-copper oxide connector.

Figure_1

Copper can oxidize and if there is no good metal-to-metal contact, a "strong thermocouple" is formed, which is also a cuprite rectifier, giving a "gigantic" voltage of as much as one millivolt for each degree of temperature difference. Such thermocouples can be formed at cold solder joints, which, however, applies only to obvious errors and defects, and not normal operation. But in the most precise circuits, the practical problem is posed by normal, proper connectors - connections and contacts.

What is important, the ubiquitous copper connectors with tin solder-lead solder yield voltages of up to 3...5 microvolts for each degree Celsius of temperature difference between the ends of such a thermocouple. Much better, by an order of magnitude, is for solder containing tin and (poisonous) cadmium - only about a 0,2 microvolts per degree. This is only seemingly not much. However, let's remember, that we are talking about the most accurate measurements. The best multimeters have an indication of 8,5-digital and on the lowest range 100,000000mV the resolution of the indication is 1nV (1 nanovolt). If in the measuring circuit there would be only one connector (thermocouple) copper - lead solder with a ratio of 3μV/°C or 3000nV/°C, then with a temperature difference between the ends of such a thermocouple of only half a degree, it will produce its own voltage (EMF thermoelectric force) of the magnitude of a 1,5 microvolts (1500nV) and by that much the voltage measurement result will be falsified.

Let's discuss the problem with an example 8,5-digital Multimeters when measuring DC voltage on the range of 200mV, which, according to the specifications, has a long-term accuracy or rather uncertainty of the result of 20ppm = 0,002% of the measured value (plus about 10ppm of the range value). If poorly selected cables are used for measurements - probes or other connections, being thermocouples, then thermoelectric voltages will be generated when temperature differences occur in these measurement connections, possibly much greater than 1 microvolt, which will even ruin the accuracy of the instrument itself. The resolution of the readings will be 1nV, but the last four or even five digits of the result may turn out to be worthless, insignificant. The problem also applies to other measurements, not only voltage, but also resistance, mainly the small and very small ones.

As you can see, significant errors can arise already through a seemingly insignificant temperature difference and a single thermoelectric copper - lead solder connector. For example, the best LTZ1000 and ADR1000 Integrated circuits in hermetic Enclosures use leads from the covar with a very high coefficient - tens of microvolts per degree! This can be a major source of errors in voltage patterns. Ways to reduce this problem is a broad separate topic.

Already here you can see, that not only designing the most precise circuits, but also making accurate measurements is a difficult art, because in addition to other important issues, you also need to minimize the impact of inevitable thermocouples, and, above all, to minimize the temperature differences of individual elements of the measuring system. In any case, for minimizing thermal voltages, in addition to the appropriate design of the circuit board, you also need to avoid drafts, or even any air movement.

The appearance of boards for precision measurement systems, especially voltage standards may surprise with cutouts or gilding, which are supposed to testify to the highest accuracy and precision. There is also no shortage of voices, that this is just a matter of aesthetics and fashion, and comparable effects can be obtained in simpler, much less spectacular realizations of printed circuit board. Others interesting comparisons and examples from high-end factory equipment can be given in this regard.

Photo 2 (by Dave Jones CC BY-SA 2,0) shows a factory module from a Keysight 34470 A multimeter.

Most importantly, however, the discussed problem of EMF voltages applies not only to the connections and thermocouples inside the measuring instrument, but also any connectors on the outside. Therefore, special connectors and Cable Assemblies are used for precise measurements, that provide low thermal coefficients. Some are made of pure copper or copper with additions of selected elements, often gold plated. They reduce the problem, but do not eliminate the harmful effects of random thermocouples. It is also worth signaling, that some errors resulting from thermal voltages can be eliminated, by swapping the polarity of the probe cables and averaging the results obtained with the two polarities.

In any case, it is necessary to take into account and minimize the impact of EMF "thermal voltages", but that's not all. The worst is yet to come! For now, we have only discussed the reversible factors limiting accuracy. In the next article of this series, titled Accurate Measurements: Irreversible Limits, i will discuss irreversible factors.

© Piotr Górecki

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