Further Effects

Many factors affect the measurement uncertainty of a system. Here is a selection:

Repeatability and reproducibility

A measurement is easily repeatable if several successive measurements under the same conditions deviate only slightly from each other. Reproducibility means that the measurement also gives similar results under changed conditions (e.g. a different person, a different instrument, a different location). Ideally, the results of several measurements should be the same. In practice, however, this is never the case. If a measurement is poorly repeatable, we cannot trust it. If the results fluctuate even under unchanged conditions, they will probably fluctuate even worse when the conditions vary as well. Repeatability and reproducibility are part of the measurement uncertainty. The worse they are, the greater the measurement uncertainty.

Cross-sensitivity

In physics, many quantities are related to or depend on each other. The ohmic resistance of a wire, for example, depends on the temperature of the wire. A cold copper wire has a lower resistance than a warm wire. This depends on the material – here copper. When we measure the resistance value, we must take into account the temperature at which we measured.

So the wire does not have one fixed resistance value at all, but many different values. How can we then call one value true? The easiest way to solve this problem is to measure every wire at a reference temperature of, for example, 25 °C. This makes different wires comparable again. The challenge then is to set the temperature of the wire with sufficient accuracy. To do this, we can place it in a climatic chamber (a kind of oven), for example. You can already see that the effort is considerable.

We can also determine the temperature dependence of the wire material. Then we measure the resistance value at any temperature and correct the measured value with a formula for the temperature dependence. Let us look at a wire that has the resistance value R₀ = 1 kΩ at 25 °C. What resistance value does it have at T = 100 °C? The following formulas apply to copper:

\[ \begin{gathered} \text{General calculation of the temperature dependence of a resistance:} \\[6pt] R(T) = R_0 \cdot (1 + \alpha \cdot \Delta T) \text{ with } \Delta T = T - 25\,°\mathrm{C} \\[6pt] \text{Material coefficient of copper: } \alpha_{\mathrm{Kupfer}} = 3.93 \cdot 10^{-3}\,\frac{1}{°\mathrm{C}} \\[6pt] \text{Resistance at the reference temperature } T = 25\,°\mathrm{C} \text{ in the example: } R_0 = 1\,\mathrm{k\Omega} \\[6pt] \text{Resistance at } T = 100\,°\mathrm{C}\text{:} \\[6pt] R(100\,°\mathrm{C}) = R_0 \cdot (1 + \alpha \cdot \Delta T) = 1\,\mathrm{k\Omega} \cdot \left(1 + 3.93 \cdot 10^{-3}\,\frac{1}{°\mathrm{C}} \cdot 75\,°\mathrm{C}\right) = 1\,\mathrm{k\Omega} \cdot (1 + 0.295) = 1.295\,\mathrm{k\Omega} \end{gathered} \]

The resistance value of a copper wire depends strongly on temperature. We call this effect cross-sensitivity.

The problem normally presents itself the other way round. In practice, we measure a resistor at T = 100 °C, for example. But we always give resistance values at the reference temperature of T = 25 °C. Otherwise there would be no single “correct” value. That is why we have to convert the measured value to the reference temperature. To do this, we rearrange the formula above. What we are actually looking for is the value R₀. That is the resistance value at 25 °C. We have been given the resistance value at T = 100 °C – we measured it.

\[ \begin{gathered} \text{General formula: } R(100\,°\mathrm{C}) = R_0 \cdot (1 + \alpha \cdot \Delta T) \text{ with } \Delta T = 100\,°\mathrm{C} - 25\,°\mathrm{C} = 75\,°\mathrm{C} \\[6pt] \text{with } R_0\text{: value at } T = 25\,°\mathrm{C} \text{ and } R(100\,°\mathrm{C})\text{: measured value at } 100\,°\mathrm{C} \\[6pt] R_0 = \frac{R(100\,°\mathrm{C})}{1 + \alpha \cdot \Delta T} = \frac{1.295\,\mathrm{k\Omega}}{1.295} = 1\,\mathrm{k\Omega} \end{gathered} \]

By rearranging the formula, we convert the resistance value measured at the “wrong” temperature to the “correct” value at the reference temperature. You can also see from the numerical example that the mathematical compensation has eliminated the influence of temperature on the measurement. After all, the wire had a resistance value of 1 kΩ from the outset. We were just silly enough to measure the value at T = 100 °C instead of 25 °C. As a result, we determined a wrong value.

In general, we call the influence of other physical quantities on the measurement result the cross-sensitivity of a measurement. Almost all measurements depend on temperature. Some also depend on pressure. You must determine (look up) the specific cross-sensitivity of each measurement and its compensation before you carry out a measurement.

Cross-sensitivities increase the measurement uncertainty. They are sometimes stated separately from the measurement uncertainty due to all other influencing quantities. Then there are two specifications of measurement uncertainty. If only a single value is given for a measurement uncertainty, the cross-sensitivities are already included in it.