# Non-Contact High Voltage Measurement in the Online Partial Discharge Monitoring System
## 4 Determining the Voltage Ratio and Measurement Uncertainty  
In laboratory conditions, the determination of the voltage ratio of the measuring system of a capacitive probe can be performed in various ways. Each of them should use the standard voltage source for calibration purposes or one of the traditional reference high voltage measurement methods, e.g., measuring sphere gaps (according to IEC 60060-2:2000) [36]. In field conditions, the value of the voltage ratio \(v\) is estimated based on the knowledge of the instantaneous voltage values (recorded by the substation SCADA system) measured by the voltage transformers operating in the transformer bay. In this case, the accuracy of determining the voltage ratio will depend on the measurement uncertainty introduced by the voltage transformer used in the substation and on the parameters of the measurement system. These issues will be discussed in more detail later in this section. In order to determine the value of the voltage ratio, it is necessary to find the relation between the value of the voltage measured on the monitored transformer and the value of the voltage measured by the measuring system of the capacitive probe. It is enough to do this for at least one measuring point because this relation is linear, as shown in Section 6. Knowing the maximum values of the voltages \(U_{1}\) and \(U_{2}\), as well as the parameters of the measuring system, i.e., the elements included in the impedance \(Z_{2}\), the value of the coupling capacitance \(C_{1}\) can be determined using Equation (5), which results from the transformation of Equations (1) and (2). It is assumed that the impedance \(Z_{1}\) is purely capacitive. \[C_{1}=\frac{U_{2}}{\omega\cdot Z_{2}(U_{1}-U_{2})} \tag{5}\]  
In the high voltage measurement method with the use of a capacitive probe, the assessment of the measurement uncertainty requires taking into account several factors. In particular, it concerns the determination of the value of the coupling capacitance \(C_{1}\) between the tested object and the probe. By transforming Equations (1) and (4) with respect to the voltage \(U_{1}\), the formula for the absolute value (modulus) of the voltage \(U_{1}\) can be written as  
\[U_{1}=\left|\left(\frac{C_{1}+C_{2}}{C_{1}}-j\frac{1}{\omega R_{s}C_{1}} \right)\cdot U_{2}\right| \tag{6}\]  
Figure 10: Example of two-stage filtering of a raw voltage signal using an averaging filter and a Butterworth low-pass filter. It can be seen that the voltage \(U_{1}\) depends on five quantities: \(C_{1}\), \(C_{2}\), \(R_{s}\), \(\omega\), and \(U_{2}\). The individual parameters in Equation (6) are independent, and therefore to determine the standard uncertainty of the voltage measurement, the formula for the combined standard uncertainty can be used [6]:  
\[u(U_{1})=\left(\frac{\partial U_{1}}{\partial C_{1}}\right)^{2}u^{2}(C_{1})+ \left(\frac{\partial U_{1}}{\partial C_{2}}\right)^{2}u^{2}(C_{2})+\left( \frac{\partial U_{1}}{\partial\omega}\right)^{2}u^{2}(\omega)+\left(\frac{ \partial U_{1}}{\partial R_{s}}\right)^{2}u^{2}(R_{s})+\left(\frac{\partial U _{1}}{\partial U_{2}}\right)^{2}u^{2}(U_{2}) \tag{7}\]  
Individual derivatives are the result of differentiating the Equation (7) with respect to a given variable, and the expressions \(u^{2}(C_{1})\), \(u^{2}(C_{2})\), \(u^{2}(\omega)\), \(u^{2}(R_{s})\), and \(u^{2}(U_{2})\) are the appropriately determined standard measurement uncertainties of the individual components of Equation (6). The expanded uncertainty of the voltage measurement given as the final result can be written as:  
\[U(U_{1})=k\cdot u(U_{1}) \tag{8}\]  
where \(k\) is a coverage factor equal to 2 for the 95% confidence level.