# Instrumental Uncertainty of Conductance Transducers for
## 5 Instrumental Measurement Uncertainty  
Based on the results obtained from the performed experimental activities described in Section 4, several uncertainty components were identified, each one contributing for the instrumental measurement uncertainty of the conductance transducer. A first group of uncertainty components is related to the input quantities--output voltage and linear parameters of the calibration curve--which can be propagated to the output quantity--water level--by application of the GUM Law of Propagation of Uncertainty [9], resulting in the following expression:  
\[u(h)_{\text{cur}}=\sqrt{c_{m}^{2}\cdot u^{2}(m)+c_{V}^{2}\cdot u^{2}(V)+c_{b}^ {2}\cdot u^{2}(b)+2\cdot c_{m}\cdot c_{b}\cdot u(m)\cdot u(b)\cdot r(m,b)} \tag{2}\]  
taking into account the uncertainty contribution, \(u(h)_{\text{cur}}\), of the adopted linear calibration curve of the conductance transducer  
\[h=m\cdot V+b, \tag{3}\]  
where \(h\) is the water level (in mm), \(m\) is the slope linear parameter (expressed in mm\(\cdot\)V\({}^{-1}\)), \(V\) is the output voltage of the conductance transducer (expressed in V) and \(b\) is the intercept linear parameter (in mm). The sensitivity coefficients shown in Expression (2) are given by the following expressions:  
\[c_{m}=\frac{\partial h}{\partial m}=V, \tag{4}\]  
\[c_{V}=\frac{\partial h}{\partial V}=m, \tag{5}\]  
\[c_{b}=\frac{\partial h}{\partial b}=1, \tag{6}\]  
allowing to write Expression (3) as  
\[u(h)_{\text{cur}}=\sqrt{V^{2}\cdot u^{2}(m)+m^{2}\cdot u^{2}(V)+u^{2}(b)+2 \cdot V\cdot u(m)\cdot u(b)\cdot r(m,b)}. \tag{7}\]  
Both the standard uncertainties related to linear parameters, \(u(m)\) and \(u(b)\), as well as the correlation factor between parameters, \(r(m,b)\), can be obtained from the application of the Least Squares Method (examples shown in Tables 1 and 2 of Section 4.2). The standard uncertainty related to the output voltage, \(u(V)\), results from the combination of the identified uncertainty components: (i) calibration and drift of the used multimeter, \(u(V)_{\text{cal}}\) and \(u(V)_{\text{drt}}\), respectively; (ii) resolution of the selected DC voltage measurement scale, \(u(V)_{\text{res}}\); (iii) stability (noise) of the output voltage signal, \(u(V)_{\text{stb}}\). Therefore, the application of the Law of Propagation of Uncertainty allows determining the standard uncertainty of the output voltage by  
\[u(V)=\sqrt{u(V)_{\text{cal}}^{2}+u(V)_{\text{drt}}^{2}+u(V)_{\text{res}}^{2}+ u(V)_{\text{stb}}^{2}}. \tag{8}\]  
In addition to the water level uncertainty component related with the calibration curve, \(u(h)_{\text{cur}}\) (given by Expression 7), the metrological characterization of the conductance transducers allowed to identify the following uncertainty components: linearity, \(u(h)_{\text{lin}}\); dimensional reversibility, \(u(h)_{\text{rev}}\); dimensional stability, \(u(h)_{\text{stb}}\); temperature influence, \(u(h)_{\theta,t}\); and thermal reversibility, \(u(h)_{\theta,r}\). Therefore, the combined measurement uncertainty of the water level obtained by the conductance transducer is given by  
\[u(h)=\sqrt{u(h)_{\text{cur}}^{2}+u(h)_{\text{lin}}^{2}+u(h)_{\text{rev}}^{2 }+u(h)_{\text{stb}}^{2}+u(h)_{\theta,t}^{2}+u(h)_{\theta,r}^{2}}. \tag{9}\]Based on the presented formulation, and taking into account the experimental results shown in Section 4, Table 6 presents the measurement uncertainty budget, which supported the determination of the instrumental uncertainty of the short-range (\(\pm\)140 mm) conductance transducer, for a nominal output voltage equal to 5 V (DC), and considering a correlation factor of \(-\)0.88 between linear parameters of the calibration curve. Based on the results presented in Table 6, a combined measurement uncertainty of 1.7 mm was obtained for the conductance transducer. The calculated number of effective degrees of freedom is equal to 123, which gives a coverage factor of 2.02, considering a 95% level of confidence.