## 3 Synthesis of a Non-Bridge System  
In this bridge circuit, two signals from the impedance under study are processed: voltage \(V_{X}\) and current \(I_{X}\). These signals require the application of the \(V_{Z}\) supply voltage. In the case of digital processing, the \(I_{X}\) signal should be converted to voltage, e.g., at a resistor, which acts as a shunt. Note that the supply voltage \(V_{Z}\), together with the branch containing the impedance under test \(Z_{X}\) and resistor \(R\), is half of the bridge in Figure 1. The other half can be realized in a different way, using transducers that allow the construction of a non-bridge system. In the case of building a non-bridge equivalent to a bridge, the second measurement step can be omitted. In this case, the measurement will require only one measurement step. The bridge signals subject to detection are \(V_{EB}\) and \(V_{ED}\). In a non-bridge system, these will be the voltages marked as \(w_{1}\) and \(w_{2}\), described by the following equations:  
\[w_{1}=V_{EB}=\frac{V_{CB}}{2}=\frac{R}{2}I_{X}. \tag{8}\]  
The \(w_{1}\) signal can be realized by dividing the voltage drawn from resistor \(R\) by 2. The \(w_{2}\) signal can be written with a dependency:  
\[w_{2}=V_{EB}-\frac{(V_{X}\ +\ V_{EB})}{Z_{P}}R_{P}, \tag{9}\]  
where the impedance \(Z_{P}\) is equal to the following:  
\[Z_{P}=R_{P}-j\frac{1}{\omega C_{P}}=\ R_{P}-jX_{P}, \tag{10}\]  
The relationship (9) can be rearranged to  
\[w_{2}=V_{EB}-\frac{V_{X}}{Z_{P}}R_{P}-\frac{V_{EB}}{Z_{P}}R_{P}, \tag{11}\]  
By substituting the \(V_{EB}\), determined by Equation (8), into Equation (11), it can be transformed into the form  
\[w_{2}=\frac{R}{2}I_{X}-\frac{V_{X}}{Z_{P}}R_{P}-\frac{\frac{R}{2}I_{X}}{Z_{P}} R_{P}, \tag{12}\]  
After ordering Equation (12), the dependence on the output signal \(w_{2}\) is obtained:  
\[w_{2}=\frac{R}{2}\bigg{(}1-\frac{R_{P}}{Z_{P}}\bigg{)}I_{X}-\bigg{(}\frac{R_{P }}{Z_{P}}\bigg{)}V_{X}. \tag{13}\]The \(w_{2}\) signal can be obtained by multiplying the voltage and current signals by the appropriate coefficients. Notice that both coefficients contain the relation \(R_{P}\) to \(Z_{P}\), which can be written as follows:  
\[\frac{R_{P}}{Z_{P}}=\frac{R_{P}}{(R_{P}-jX_{P})} \tag{14}\]  
In relation (12), there is the setting parameter \(R_{P}\) and the constant \(X_{P}\), which is the reactance of the capacitor \(C_{P}\). This relation is a complex number. In an analog measurement system, the multiplication by an imaginary unit j means a phase shift of \(90^{\circ}\). This can be accomplished using a phase shifter. In digital realization, the multiplication by an imaginary unit can be accomplished by shifting the signal by the number of samples corresponding to \(1/4\) of the period. After omitting the second measurement step, the measured dielectric loss factor can be determined from the voltage relation:  
\[\tan\delta=\frac{|V_{CD}|}{|V_{CB}|}, \tag{15}\]  
The corresponding voltages depending on (15) are equal to  
\[V_{CB}=RI_{X} \tag{16}\]  
and  
\[w_{1}=V_{EB}=\frac{V_{CB}}{2}=\frac{R}{2}I_{X}. \tag{8}\]0  
The \(V_{CD}\) voltage can be written definitively as follows:  
\[w_{1}=V_{EB}=\frac{V_{CB}}{2}=\frac{R}{2}I_{X}. \tag{8}\]1  
Figure 6 shows the structure of a non-bridge system for measuring the dielectric loss coefficient \(\tan\delta\), derived from the equations describing the bridge in \[w_{1}=V_{EB}=\frac{V_{CB}}{2}=\frac{R}{2}I_{X}. \tag{8}\]2 The non-bridge system consists of two parts. The analog part contains the power source, the \(Z_{X}\) impedance under test, and the \(R\) resistor of a known value. The output signals of the analog part are the voltage \(V_{X}\) across the impedance \(Z_{X}\) and the voltage proportional to its current \(I_{X}\). The second part of the system is the arithmetic part, in which the detected voltages \(w_{1}\) and \(w_{2}\) are determined according to Equations (8) and (13), respectively. The modules or RMS values of the \(w_{1}\) and \(w_{2}\) signals are compared in the DET detector.