## 2 Bridge with Modular Balancing for Dielectric Loss Factor Measuring  
The synthesis of the non-bridge measuring system will be based on the principle of operation of the bridge system shown in Figure 1. The bridge was first introduced over 60 years ago [18]. At that time, even the analog implementation of such a system was problematic due to the properties of most voltmeters of the time. These were usually analog devices that were characterized by low input resistances. This, in turn, disrupted the flow of currents and the distribution of voltages in the bridge, making it practically impossible to implement. The synthesized non-bridge measurement system will be a completely new one. The processing of measurement signals in impedance bridge is limited by the properties of RLC components forming the bridge. These components are characterized by limited values resulting from their design, the limited range of changes in these values, the resolution of settings, the presence of parasitic quantities (e.g., leakage of capacitors), limited power, and permissible voltages. In addition, conducted and induced interference occurs in such systems. Such problems do not exist in systems in which measurement signal processing is performed. Therefore, it is worth trying to build non-bridge systems that can be derived on the basis of known measurement systems by digitizing and virtualizing them. The processing in non-bridge systems is carried out by performing mathematical operations on signals. They are equivalent only to the voltage and current of the object being tested. The bridge circuit is only the basis for deriving signal processing equations. The new non-bridge circuit can be implemented, for example, as a digital circuit with software signal processing. The bridge, which is the basis of synthesis, is shown in Figure 1. It is designed to measure the dielectric loss factor \(\tan\delta\). The concept of this bridge was presented in ref. [12]. The principle of operation of the bridge is presented in an extremely concise way, making the analysis of the system somewhat difficult. The transfiguration of the bridge from the original concept is presented in [19]. The object of research in this study is a lossy capacitor, which has been modeled in a simplified way as a branch consisting of \(C_{X}\) capacitance and \(R_{X}\) resistance connected in series. For such a model, the dielectric loss factor can be expressed as follows:  
\[\tan\delta=\omega R_{X}C_{X}. \tag{1}\]  
The bridge has two adjustable elements: the \(R_{P}\) resistor and the \(R_{CC^{\prime}}\) resistor. The resistors in the CB branch have resistances equal to \(R/2\), while the \(R_{CC^{\prime}}\) resistor is part of the resistor switched on between the CE points and in practice can be implemented as a potentiometer, in which the C\({}^{\prime}\) point is attached to the slider. The resistance of this resistor can take values from 0 to \(R/2\). The system has two measurement steps, in which balancing of the system is performed twice. The possibility of building the above system and the results of simulation studies of its basic properties are presented in ref. [20]. This study confirmed the purposefulness of building such a system. Figure 2 shows a phasor diagram of bridge voltages. When constructing the diagram, it was assumed that the supply voltage had a zero phase. The method of constructing graphs using the homographic function (Mobius transformation) presented in [2] was used. Note that points A, B, C, and D are loci on the Gaussian plane of the potentials of the corresponding bridge points from Figure 1.  
Figure 1: The diagram of the dual-module balancing bridge. The bridge requires two measurement steps. In the first measurement step, the \(R_{P}\) resistor is the adjustable element. In this step, the \(V_{EB}\) and \(V_{ED}\) voltage modules are compared. Such a comparison can be made, for example, by comparing the RMS values of the voltages.