## 4 Errors and Uncertainties  
In the solution discussed, several sources of error and measurement uncertainty can be indicated. One source of uncertainty is the uncertainty of determining the measured dielectric loss factor from Equation (13). This coefficient is determined in the equilibrium state as a relationship of the values of the voltage modules \(V_{CD}\) and \(V_{CB}\), but this relation can be determined as the ratio of the effective values of the voltages in question. Thus, the relative measurement uncertainty of the dielectric loss factor \(\tan\delta\), denoted as \(u_{rel}\) (\(\tan\delta\)), can be estimated as  
\[u_{rel}(\tan\delta)=\sqrt{\left(u_{rel}(V_{CD})\right)^{2}+u_{rel}(V_{CB})^{2}}, \tag{19}\]  
where \(u_{rel}(V_{CD})\) and \(u_{rel}(V_{CB})\) are the uncertainty of determining the RMS value of the \(V_{CD}\) and \(V_{CB}\) voltages, respectively. Other sources of error in the system in question are related to the incorrect position of the D point during measurement. This can be caused by an error in the detection of equilibrium and a loss of the \(C_{P}\) capacitor. In the first case, the equilibrium state of the bridge means the equality of modules and thus the values of the effective voltages \(V_{EB}\) and \(V_{ED}\). The inequality of the mentioned voltages, caused by the incorrect detection of their equality, results in the adoption of an equilibrium state despite the inequality of the \(V_{EB}\) and \(V_{ED}\) voltage modules. Then, on the phasor diagram, the point D\({}^{\prime}\) is taken as the potential of point D, which results in a change in the value of the voltage \(V_{CD}\) to the value of \(V_{CD^{\prime}}\). Figure 7 shows an illustration of the loci of points D and D\({}^{\prime}\) on the Gaussian plane. For readability, a fragment of the phasor chart is presented. As a result of the incorrect position of the locus D, the dielectric loss coefficient \(\tan\delta\) determined from Equation (15) will have the value \(\tan\delta^{\prime}\) equal to the following:  
\[\tan\delta^{\prime}=\frac{\left|V_{CD^{\prime}}\right|}{\left|V_{CB}\right|} \tag{20}\]  
Figure 7: D point offset due to detection error. The absolute error in determining the dielectric loss coefficient \(\Delta\) tan \(\delta_{D}\), caused by the shift in the point D, will have the following value:  
\[\Delta\tan\delta_{D}=\frac{|V_{CD^{\prime}}|}{|V_{CB}|}-\frac{|V_{CD}|}{|V_{CB}|} =\frac{|V_{CD^{\prime}}|-|V_{CD}|}{|V_{CB}|}. \tag{21}\]  
The analytical calculation of this error is complicated. This is due to the need to calculate the modules of complex voltages. However, the error in determining the dielectric loss factor can be estimated by analyzing the phasor diagram shown in Figure 7. On the diagram, it can be seen that the difference in voltage modules \(V_{CD}\) and \(V_{CD^{\prime}}\), for a small difference in voltage modules subject to detection \(V_{ED}\) and \(V_{ED^{\prime}}\), is approximately equal to this difference:  
\[|V_{DD^{\prime}}|=|V_{CD^{\prime}}|-|V_{CD}|\ \approx\ |V_{ED}|-|V_{ED^{\prime}}| \tag{22}\]  
The error Equation (21) can then be written as follows:  
\[\Delta\tan\delta_{D}\ \approx\ \frac{|V_{CD^{\prime}}|}{|V_{CB}|}-\frac{|V_{CD}|} {|V_{CB}|}=\frac{|V_{ED}|-|V_{ED^{\prime}}|}{|V_{CB}|}=\frac{|\Delta V_{ED}|} {|V_{CB}|} \tag{23}\]  
The difference between \(V_{ED}\) and \(V_{ED^{\prime}}\) voltage modules, denoted as \(\Delta V_{ED}\), is an absolute error of modular detection. In the same way, the error related to the loss of the \(C_{P}\) capacitor can be determined. In this case, the displacement of the D point will involve a change in the phase angle of the voltage across the lossy capacitor. This offset can be estimated as the \(V_{DD^{\prime}}\) voltage from Equation (22).