## 2 Proposed Voltage Fluctuation Compensator  
### Bipolar AC/AC Chopper  
As a power electronic converter for generating compensation voltages in the proposed compensator of voltage fluctuations, a bipolar AC/AC chopper with the topology shown in Figure 1 is proposed [31; 32]. To obtain a bipolar output voltage, the chopper is powered by a transformer with a split, symmetrical output winding. The voltage ratio of the transformer depends on the level of compensation voltages. In this paper, the voltage ratio for both output windings is 1:1. For such connection of the transformer on the secondary windings, voltages equal to \(v_{i}\) and \(-v_{i}\), as marked in Figure 1, are obtained. The midpoint of the secondary winding of the transformer is switched through two bidirectional switches, \(S_{1}\) and \(S_{2}\), generating a bipolar impulse voltage. In order to eliminate the high-frequency components of the currents drawn by the chopper, low-pass filters \(L_{1}\), \(C_{1}\) and \(L_{2}\), \(C_{2}\) were used. The bipolar output voltage is filtered out by the low-pass output filter \(L_{L}\), \(C_{1}\) before being fed to the series transformer. The frequency of the PWM signal is 10 kHz and is a compromise between power losses and the quality of the shaped output waveforms. The resonant frequency of the LC filters should be in the range of about 1/10 to 1/3 of the PWM frequency. In simulation studies, the values of the elements are set at \(L_{1}=L_{2}=0.5\) mH, \(C_{1}=C_{2}=10\) uF, \(L_{L}=0.5\) mH, \(C_{L}=50\) uF.  
The AC/AC choppers are controlled by a complementary pair of PWM signals using dead time. In the dead time interval, when switches \(S_{1}\) and \(S_{2}\) are off, a short-term overvoltage \(L(di/dt)\) is induced. The dead time has been permanently set to 0.8 uS, which allows for the lowest overvoltages on the power switches. RC suppressors (\(R_{smub}\), \(C_{smub}\)) are used to protect against overvoltages (Figure 1). Parameters of RC dampers depend on the value of \"dead time\" and overvoltage energy.In the case of the analyzed circuit, \(R_{smub}=15\)\(\Omega\), \(C_{smub}=47\) nF. These elements will be needed mainly in the experimental implementation, which is in the preparatory phase. For the purposes of simulation studies, the snubber elements were omitted.  
Both bi-directional switches are controlled by the PWM (Pulse Width Modulation) signal with variable duty cycle \(D\) which is defined as \(t_{1}/T_{S}\). A complementary pair of PWM signals is fed simultaneously to each of the two transistors of individual switches \(S_{1}\) and \(S_{2}\). During the switching sequence period \(T_{S}\), the \(S_{1}\) switch is closed in the relative time \(t_{1}/T_{S}=D\) and the \(S_{2}\) switch is closed in the relative time \(t_{2}/T_{S}=1-D\). When the \(S_{1}\) switch is on, the output voltage is \(v_{i}\), while when the \(S_{2}\) switch is on, the output voltage is \(-v_{i}\). The average value of the output voltage can be described by the following equation:  
\[v_{o}=v_{i}D+(-v_{i})(1-D)=(2D-1)v_{i}. \tag{1}\]  
The voltage gain is then  
\[\left|\frac{v_{o}}{v_{i}}\right|=2D-1. \tag{2}\]  
Substituting the \(D\) coefficient from 0 to 0.5 into Equation (1), the voltage \(v_{o}\) change from the maximum value equal to \(V_{i}\) to 0 is obtained, where \(V_{i}\)--the amplitude of the input voltage \(v_{i}\). In addition, the voltage \(v_{o}\) is in antiphase with the voltage \(v_{i}\). This means that the phase shift between the input and output voltages is \(180^{\circ}\) (or \(\pi\) rad). On the other hand, for changes in the \(D\) coefficient value from 0 to 1.0, the output voltage \(v_{o}\) change from 0 to \(V_{i}\) is obtained. For such values of \(D\), the voltage \(v_{o}\) is in phase with the voltage \(v_{i}\). The idealized control characteristics obtained on the basis of (2) are shown in Figure 2.