## 3 Regulation Range of Compensation Voltage Amplitude
### Regulation Range in Line 1  
For the compensator for line 1, the chopper MC1 is powered from line 2 and the chopper MC2 is powered from line 3. Since each chopper is bipolar, it is possible to obtain two voltages, \(v_{2}\) and \(-v_{2}\), for chopper MC1 and \(v_{3}\) and \(-v_{3}\) for chopper MC2 (according to the characteristics shown in Figure 2). Writing the indicated voltages on the complex plane, and taking into account that the output voltage of the compensator is the sum of the voltages of individual choppers:  
\[v_{\text{C1}}\ =v_{\text{MC1}}+v_{\text{MC2}} \tag{7}\]  
the compensation voltage adjustment range for line 1 was obtained as shown in Figure 5. In consequence, different sectors for each AC/AC converter can be determined. Due to the fact that they are asymmetrical, they can be divided into eight sectors. The regulation range needs to be defined individually for each of them.  
In the description below, the variables are presented without reference to the analyzed phase of the output voltage, so that the relationships are general for each phase. Therefore, the designations \(V_{m}\) and \(V_{ref}\) were used to determine the amplitude of the output and reference voltage.  
Assuming the three-phase balanced system with the phase voltages amplitude \(V_{m}\) and applying the law of sines to a triangle for sector I (Figure 6a), the follow equation is obtained:  
\[\frac{V_{m}}{\sin(60^{\circ}+\varphi)}=\frac{|V_{ref}|}{\sin(60^{\circ})} \tag{8}\]  
where \(\varphi\) is vector phase shift measured from the beginning of the sector. The angle \(\varphi\) for this sector changes from 0 to \(60^{\circ}\). From (8), the regulation range of the amplitude of the output voltage is obtained as follows:  
\[|V_{ref}|\ =\frac{V_{m}\sin(60^{\circ})}{\sin(60^{\circ}+\varphi)}=\frac{\sqrt {3}\ V_{m}}{2\sin(60^{\circ}+\varphi)}. \tag{9}\]  
Equation (8) achieves its minimum for \(\varphi=30^{\circ}\) equal \(|V_{ref\ min}|=\frac{\sqrt{3}}{2}\ V_{m}\) and maximum for \(\varphi=0^{\circ}\) and \(\varphi=60^{\circ}\) equal \(|V_{ref\ max}|=V_{m}\) (Figure 7a).  
For sector II, based on Figure 6b, the following relationships can be described:  
\[|V_{ref}|=\frac{V_{m}\sin(120^{\circ})}{\sin(60^{\circ}-\varphi)}=\frac{\sqrt {3}\ V_{m}}{2\sin(60^{\circ}-\varphi)} \tag{10}\]  
The angle \(\varphi\) for sector II changes from 0 to \(30^{\circ}\) (Figure 7b). Equation (10) achieves its minimum for \(\varphi=0^{\circ}\) equal \(|V_{ref\ min}|=V_{m}\) and maximum for \(\varphi=30^{\circ}\) equal \(|V_{ref\ max}|=\sqrt{3}\ V_{m}\) (Figure 7a).  
Based on the description of the voltage vectors for the subsequent sectors presented in Figure 6, the dependencies for the amplitude of output voltage were calculated, and are listed in Table 1 and illustrated in Figure 7a. Meanwhile, Figure 7b shows the change in \(\varphi\) in each of the sectors.  
Figure 5: Voltage phasor diagram for AC/AC I (Figure 4), where I—VIII are the number of sectors.  
Figure 6: Calculation of the output voltage vector amplitude in: (**a**) sector I, (**b**) sector II, (**c**) sector III, (**d**) sector IV, (**e**) sector V, (**f**) sector VI, (**g**) sector VII, (**h**) sector VIII.  
Figure 7: (**a**) Regulation range of the amplitude of the output voltage vector, (**b**) position angle of the output voltage vector calculated from the beginning of each sector.