## 2 Bridge with Modular Balancing for Dielectric Loss Factor Measuring
Note that in this balancing step, the current \(I_{X}\) does not change during the measurement, which means that the \(V_{EB}\) voltage remains unchanged throughout the measurement. The loci of points C and E do not change. Changing the \(R_{P}\) resistor setting results in the D point moving along the ADB semicircle. The measurement step completes the state in which the voltage preference lengths \(V_{EB}\) and \(V_{ED}\) are equal to each other. This means that the \(V_{ED}\) voltage module is equal to the \(V_{EB}\) voltage module:  
\[|V_{ED}|=|V_{EB}|. \tag{2}\]  
Figure 3 shows the phasor chart after the first balancing step is completed. On the diagram, it can be seen that due to the similarity of the triangles, the angle between the \(V_{DB}\) and \(V_{BC}\) voltages is equal to the angle \(\delta\) of the object under control. You can also notice that the \(V_{AC}\) voltage is in phase with \(V_{AD}\). Analyzing the above diagram, it can be seen that it is possible to determine the sine of the angle \(\delta\) as a voltage relationship between \(V_{CD}\) and V\({}_{CB}\). The author of ref. [12] suggested another way to measure this relationship, as the resistance relationship between \(R_{CC^{\prime}}\) and \(R\). A second measurement step can be performed for this purpose. In this step, the adjustable element is the \(R_{CC^{\prime}}\) resistor. The value of the \(R_{P}\) resistance from the first step is maintained. The \(V_{CD}\) and \(V_{CC^{\prime}}\) voltage modules are compared. The bridge is brought to a state in which the voltage module \(V_{CD}\) is equal to the voltage module \(V_{CC^{\prime}}\):  
\[|V_{CD}|=|V_{CC^{\prime}}|\]  
Figure 3: Phasor diagram after the first balancing step is completed. Figure 2: Phasor diagram of bridge voltages. From the phasor diagram in this condition, it follows that the sine of the angle \(\delta\) can be calculated as follows:  
\[\sin\delta=\frac{R_{CC^{\prime}}}{R}. \tag{4}\]  
From the sine, the tangent of the angle \(\delta\) can be calculated as follows:  
Figure 1.0  
For low-loss capacitors, Equation (5) can be written in approximate form as follows:  
Figure 1.1  
The relative error \(\varepsilon\) of the approximation (6) can be determined as follows:  
Figure 1.2  
Figure 5 shows the dependence of error (7) on the measured dielectric loss factor \(\tan\delta\). From the assessment of the approximation error, it appears that for typical \(\tan\delta\) dielectric loss factors, the approximation error (6) is vanishingly small. Figure 1.3 Phasor diagram after the second balancing step is completed. Figure 1.4 Relative error \(\varepsilon\) vs. dielectric loss factor \(\tan\delta\). The analysis of the system shows that the second balancing step is actually used to determine the relationship (4). It allows you to determine the sine (or tangent) of the loss angle as a relation of two resistances. Perhaps the author of the concept intended this to facilitate the measurement by, for example, scaling the potentiometer in tan \(\delta\) values. The measured dielectric loss factor can also be determined as a relationship between modules or values of effective voltages \(V_{CD}\) and \(V_{CB}\), which can be easily implemented today. We should remember that voltage measurements in the period when the idea of the bridge was born were difficult due to the relatively low input resistance of devices and their low accuracy. The author states that voltmeter errors are eliminated because it is used only to determine the equality of voltages. However, it does not take into account the impact of module equality detection error on the measurement result.