# Hybrid Harmonic Suppression Method at DC Link of Series-Connected 18-Pulse Rectifier
## 2 The Proposed Rectifier with Harmonic Injection Circuit
### Operating Modes Analyse  
In Figure 1, according to the KCL, the currents through the primary winding of the injection transformers are described as  
\[\left\{ \begin{array}{l}i_{\rm x1}=i_{\rm Rec2}-i_{\rm Rec1}\\ i_{\rm x2}=i_{\rm Rec3}-i_{\rm Rec2}\end{array} \right. \tag{5}\]  
where \(i_{\rm x1}\) and \(i_{\rm x2}\) are the currents through the primary windings of injection transformers 1 and 2, respectively. Under constant-voltage load, based on the topology of the three-phase diode bridge rectifier, Figure 3 describes the output currents (\(i_{\rm Rec1}\), \(i_{\rm Rec2}\), \(i_{\rm Rec3}\)) of the rectifier bridges; and from Equation (4), the currents \(i_{\rm x1}\) and \(i_{\rm x2}\) can also be obtained, as shown in Figure 3.  
According to Figure 1, when the input voltage of the converter is harmonic-free, and Equation (4) is calculated as follows. \[\left\{ \begin{array}{l}u_{\rm a1}=\frac{U_{\rm MS}}{\sqrt{3}}\angle 10^{\circ}= \frac{U}{\sqrt{3}}\sin(\omega t+10^{\circ})\\ u_{\rm B1}=\frac{U_{\rm MS}}{\sqrt{3}}\angle 10^{\circ}=\frac{U}{\sqrt{3}}\sin( \omega t+130^{\circ})\\ u_{\rm C1}=\frac{U_{\rm MS}}{\sqrt{3}}\angle 10^{\circ}=\frac{U}{\sqrt{3}}\sin( \omega t-110^{\circ})\end{array} \right\} \begin{array}{l}u_{\rm a2}=\frac{U_ {\rm MS}}{\sqrt{3}}\angle 30^{\circ}=\frac{U}{\sqrt{3}}\sin(\omega t+30^{\circ})\\ u_{\rm B3}=\frac{U_{\rm MS}}{\sqrt{3}}\angle 50^{\circ}=\frac{U}{\sqrt{3}}\sin( \omega t+150^{\circ})\\ u_{\rm C2}=\frac{U_{\rm MS}}{\sqrt{3}}\angle 30^{\circ}=\frac{U}{\sqrt{3}}\sin( \omega t-90^{\circ})\end{array} \right.\left\{ \begin{array}{l}u_{\rm a3}= \frac{U_{\rm MS}}{\sqrt{3}}\angle 50^{\circ}=\frac{U}{\sqrt{3}}\sin(\omega t+50^{ \circ})\\ u_{\rm B3}=\frac{U_{\rm MS}}{\sqrt{3}}\angle 50^{\circ}=\frac{U}{\sqrt{3}}\sin( \omega t+170^{\circ})\\ u_{\rm C3}=\frac{U_{\rm MS}}{\sqrt{3}}\angle 50^{\circ}=\frac{U}{\sqrt{3}}\sin( \omega t-70^{\circ})\end{array} \right. \tag{6}\]  
From Equation (6), the output voltages of the rectifier bridges are described as Figure 4. In Figure 4, the voltages \(u_{\rm KF}\), \(u_{\rm FS}\), and \(u_{\rm SG}\) have 6 pulses per supply cycle with a phase difference of 20 degrees, and each pulse lasts 60 degrees. From Figure 1, the injection voltages \(u_{\rm FP}\) and \(u_{\rm ST}\) are calculated as  
\[\left\{ \begin{array}{l}u_{\rm FP}=\frac{u_{\rm FS}+u_{\rm SG}-u_{\rm KF}}{2 }-\frac{u_{\rm o}}{6}\\ u_{\rm ST}=\frac{u_{\rm SG}-u_{\rm KF}-u_{\rm FS}}{2}+\frac{u_{\rm o}}{6}\end{array} \right. \tag{7}\]  
where \(u_{\rm FP}\) and \(u_{\rm ST}\) are the primary winding voltages of the two injection transformers, respectively. Figure 4: Output voltages of the rectifier bridges. Figure 3: Operating waveforms of the converter. From Figure 4 and Equation (6), the injection voltages \(u_{\text{FP}}\) and \(u_{\text{ST}}\) can be described as in Figure 5.  
From [12; 13; 14], almost all the harmonic reduction methods generate multistep waveforms which are equivalent to the ideal injection waveform (as shown in Figure 5) to suppress harmonics. That is because the cost and difficulty of generating multistep waves are much lower than these of the triangular waves. Besides, the harmonic reduction effects of the multistep wave and the triangular wave are similar. From Figure 5, there are three combinations of currents \(i_{\text{x1}}\) and \(i_{\text{x2}}\), which are 1 \(i_{\text{x1}}>0\), \(i_{\text{x2}}>0\), 2 \(i_{\text{x1}}>0\), \(i_{\text{x2}}<0\), 3 \(i_{\text{x1}}<0\), \(i_{\text{x2}}>0\). On the basis of the on-off state of VD\({}_{1}\), VD\({}_{2}\), and the currents \(i_{\text{x1}}\) and \(i_{\text{x2}}\), the harmonic suppression circuit has seven modes, described as Figure 6.  
Figure 5: Primary winding voltages of the injection transformers when input voltage of the rectifier is harmonic-free. Figure 6: Operating modes of the harmonic injection circuit. (**a**) Mode. 1. (**b**) Mode. 2. (**c**) Mode. 3. (**d**) Mode. 4. (**e**) Mode. 5. (**f**) Mode. 6. (**g**) Mode. 7.