# Hybrid Harmonic Suppression Method at DC Link of Series-Connected 18-Pulse Rectifier
## 2 The Proposed Rectifier with Harmonic Injection Circuit
### Operating Modes Analyse
The red arrows represents the directions of currents \(i_{\text{x1}}\) and \(i_{\text{x2}}\). In Figure 6a, VD\({}_{1}\) is turned on, VD\({}_{2}\) is turned off, \(i_{\rm{A2}}>0\), D\({}_{1}\), D\({}_{2}\) and D\({}_{3}\) are forward-biased, D\({}_{4}\) is revise-biased, and the injection voltages \(u_{\rm{FP}}\) and \(u_{\rm{ST}}\) are calculated as  
\[\left\{ \begin{array}{l}u_{\rm{FP}}=0\\ u_{\rm{ST}}=\frac{2N_{7}}{3N_{8}}u_{\rm{o}}\end{array} \right. \tag{8}\]  
In Figure 6b, VD\({}_{1}\) and VD\({}_{2}\) are turned off, \(i_{\rm{x1}}>0\), \(i_{\rm{x2}}>0\), D\({}_{1}\) and D\({}_{3}\) are forward-biased, D\({}_{2}\) and D\({}_{4}\) are revise-biased, and the injection voltages \(u_{\rm{FP}}\) and \(u_{\rm{ST}}\) are calculated as  
\[\left\{ \begin{array}{l}u_{\rm{FP}}=\frac{2N_{7}}{3N_{8}}u_{\rm{o}}\\ u_{\rm{ST}}=\frac{2N_{7}}{3N_{8}}u_{\rm{o}}\end{array} \right. \tag{9}\]  
In Figure 6c, VD\({}_{1}\) is turned off, VD\({}_{2}\) is turned on, \(i_{\rm{x1}}>0\), D\({}_{1}\), D\({}_{3}\) and D\({}_{4}\) are forward-biased, D\({}_{2}\) is revise-biased, and the injection voltages \(u_{\rm{FP}}\) and \(u_{\rm{ST}}\) are calculated as  
\[\left\{ \begin{array}{l}u_{\rm{FP}}=\frac{2N_{7}}{3N_{8}}u_{\rm{o}}\\ u_{\rm{ST}}=0\end{array} \right. \tag{10}\]  
In Figure 6d, VD\({}_{1}\) and VD\({}_{2}\) are turned off, D\({}_{1}\) and D\({}_{4}\) are forward-biased, D\({}_{2}\) and D\({}_{3}\) are revise-biased, and the injection voltages \(u_{\rm{FP}}\) and \(u_{\rm{ST}}\) are calculated as  
\[\left\{ \begin{array}{l}u_{\rm{FP}}=\frac{2N_{7}}{3N_{8}}u_{\rm{o}}\\ u_{\rm{ST}}=-\frac{2N_{7}}{3N_{8}}u_{\rm{o}}\end{array} \right. \tag{11}\]  
In Figure 6e, VD\({}_{1}\) is turned on, VD\({}_{2}\) is turned off, D\({}_{1}\), D\({}_{2}\) and D\({}_{4}\) are forward-biased, D\({}_{3}\) is revise-biased, and the injection voltages \(u_{\rm{FP}}\) and \(u_{\rm{ST}}\) are calculated as  
\[\left\{ \begin{array}{l}u_{\rm{FP}}=0\\ u_{\rm{ST}}=-\frac{2N_{7}}{3N_{8}}u_{\rm{o}}\end{array} \right. \tag{12}\]  
In Figure 6f, VD\({}_{1}\) is turned off, VD\({}_{2}\) is turned on, D\({}_{2}\), D\({}_{3}\) and D\({}_{4}\) are forward-biased, D\({}_{1}\) is revise-biased, and the injection voltages \(u_{\rm{FP}}\) and \(u_{\rm{ST}}\) are calculated as  
\[\left\{ \begin{array}{l}u_{\rm{FP}}=-\frac{2N_{7}}{3N_{8}}u_{\rm{o}}\\ u_{\rm{ST}}=0\end{array} \right. \tag{13}\]  
In Figure 6g, VD\({}_{1}\) and VD\({}_{2}\) are turned off, D\({}_{2}\) and D\({}_{3}\) are forward-biased, D\({}_{1}\) and D\({}_{4}\) are revise-biased, and the injection voltages \(u_{\rm{FP}}\) and \(u_{\rm{ST}}\) are calculated as  
\[\left\{ \begin{array}{l}u_{\rm{FP}}=-\frac{2N_{7}}{3N_{8}}u_{\rm{o}}\\ u_{\rm{ST}}=\frac{2N_{7}}{3N_{8}}u_{\rm{o}}\end{array} \right. \tag{14}\]  
From Equations (7)-(13), and Figures 5 and 6, the injection voltage of the converter can be obtained, as shown in Figure 7. In Figure 7, there are 48 combinations per power supply cycle; and in order to suppress the (18\(k\pm\) 1)th and (36\(k\pm\) 1)th harmonics, the duration of combinations 6, 14, 22, 30, 38, and 46 is 1/27 power supply cycle, and the duration of the other combinations is 1/54 power supply cycle.