# Common-Mode Voltage Analysis and Reduction for the Quasi-Z-Source Inverter with a Split Inductor
## 4 Modified qZSI for CMV Reduction  
The qZSI can achieve the boost capability by utilizing the ST vectors. The introduction of the ST vectors will also change the CMV. According to Table 1, the CMV is zero when the ST vector state is adopted, introducing high-frequency harmonics. Thus, an input-split-inductor qZSI is adopted to reduce the CMV when the ST vector state is utilized in the conventional CMV reduction strategies. According to the vectors, the average CMV value during the non-ST state and the CMV during the ST state is applied, which can reduce the high-frequency components in the CMV. The CMV reduction in the qZSI can be achieved in the following. According to Figure 2, the voltage of the inductor \(L_{1}\) during the non-ST state can be obtained as  
\[v_{L_{1}}=-\mathrm{D}V_{\mathrm{DC}} \tag{10}\]  
while the voltage of the inductor \(L_{1}\) during the ST state can be described as  
\[v_{L_{1}}=(1-\mathrm{D})V_{\mathrm{DC}} \tag{11}\]  
The CMV and the inductor voltage in the qZSI are presented in Figure 5, in which \(v_{\mathrm{cmv}}\) is the CMV, \(v_{\mathrm{cmv\_avg}}\) is the average CMV during the non-ST state when DPWM, AZPWM, RSPWM and NSPWM are used. The average CMV during the non-ST state can be obtained as  
\[v_{\mathrm{cmv\_avg}}=\frac{0\cdot t_{zero\_0}+\frac{V_{\mathrm{DC}}}{3}\cdot t _{add}+\frac{2V_{\mathrm{DC}}}{3}\cdot t_{even}+V_{\mathrm{DC}}\cdot t_{zero \_7}}{t_{zero}+t_{odd}+t_{even}} \tag{12}\]  
in which \(t_{odd}\) is the time interval of all the odd vector states, i.e., \(\mathbf{V_{1}}\), \(\mathbf{V_{3}}\) and \(\mathbf{V_{5}}\), \(t_{even}\) is the time interval of all the even vector states, i.e., \(\mathbf{V_{2}}\), \(\mathbf{V_{4}}\) and \(\mathbf{V_{6}}\), \(t_{zero\_0}\) is the time interval of the zero vector state \(\mathbf{V_{0}}\), \(t_{zero\_7}\) is the time interval of the zero vector state \(\mathbf{V_{7}}\) and \(t_{zero}\) is the sum of \(t_{zero\_0}\) and \(t_{zero\_7}\). As observed in Figure 5, the voltage of the inductor \(L_{1}\) is positive with its maximum value being \((1-D)V_{\mathrm{DC}}\) during the ST state when \(v_{\mathrm{cm}}\) is zero. Meanwhile, the voltage of the inductor \(L_{1}\) is negative with its minimum value being \(-DV_{\mathrm{DC}}\) during the ST state when \(v_{\mathrm{cm\_avg}}\) is positive. According to the phenomenon seen from the CMV and the inductor voltage, the input inductor voltage can be used to compensate for the CMV by splitting the inductor and maintaining the same total inductance. The inductor \(L_{1}\) can be divided into \(L_{11}\) and \(L_{12}\), as shown in Figure 6. The proportional relationship among \(L_{1}\), \(L_{11}\) and \(L_{12}\) can be obtained as  
\[\left\{ \begin{array}{l}L_{11}=xL_{1}\\ L_{12}=(1-x)L_{1}\end{array} \right. \tag{13}\]  
where \(x\) is a coefficient that is used to describe the ratio of the split inductor, and then:  
\[\left\{ \begin{array}{l}v_{L_{11}}=xv_{L_{1}}\\ v_{L_{12}}=(1-x)v_{L_{1}}\end{array} \right. \tag{14}\]  
in which \(v_{L_{11}}\) and \(v_{L_{12}}\) are the voltages of \(L_{11}\) and \(L_{12}\) and \(v_{L_{1}}\) is the voltage of the original inductor \(L_{1}\). Referring to (1), the CMV can be derived as  
\[v_{\mathrm{cmv}}=v_{L_{11}}+\frac{v_{uN^{\prime}}+v_{vN^{\prime}}+v_{wN^{ \prime}}}{3} \tag{15}\]  
Subsequently, the average CMV during the non-ST state can be described as  
\[v_{\mathrm{cmv}}=v_{\mathrm{cmv\_avg}}-xDV_{\mathrm{DC}} \tag{16}\]while the CMV during the ST state can be given as  
\[v_{\text{cmv}}=x(1-D)V_{\text{DC}} \tag{17}\]  
To reduce the CMV in the split-inductor qZSI, the average CMV during the ST state and the CMV during the non-ST state should be equal.