## 3 Harmonics of Inverter Outputs  
Single-phase output equivalent circuit of grid-tied inverters can be performed as Figure 8 when the three-phase output filter is symmetrical. The parameters of output circuits of PV inverters can be found in Table 1.  
Figure 8: Distribution of 202.90 Hz inter-harmonic in the output filter. Figure 7: Phase sequences of voltage and current signals at PV PCC: (**a**) phase sequences of voltage signals at PV PCC; and (**b**) phase sequences of current signals at PV PCC. \(L_{\text{O}}\) and \(C\) are filtering inductor and capacitor, respectively; \(L_{\text{g}}\) and \(R_{\text{g}}\) are line impedances of the micro-grid, and the ratio of isolating transformer in the PV inverters is 400/215. When the line resistor \(R_{\text{g}}\) can be ignored, the output impedance of PV inverter can be depicted as  
\[Z=\text{j}\omega\frac{L_{\text{O}}+L_{\text{E}}-\omega^{2}L_{\text{O}}L_{ \text{E}}C}{1-\omega^{2}L_{\text{E}}C} \tag{3}\]  
where \(L_{\text{E}}\) is the equivalent inductance of the isolating transformer in the PV inverters. The \(LCL\) resonance frequencies of output equivalent impedance \(Z\) are related to inductors \(L_{\text{O}}\) and \(L_{\text{E}}\) and capacitor \(C\). Based on [PERSON]'s current law and the reference directions of branch currents of PV inverter output filter, the current equation below can be satisfied:  
\[-\dot{I_{\text{Lo}}}\ +\dot{I_{\text{Le}}}+\dot{I_{\text{C}}}=0 \tag{4}\]  
where \(I_{\text{Lo}}\) is the RMS of inverter output inter-harmonic current, \(I_{\text{C}}\) is the RMS of capacitor branch current of inverter output filters, and \(I_{\text{LE}}\) is the RMS of the injected inter-harmonic current into micro-grid. When LCL resonances take place, the RMS of the currents above satisfy the following equation:  
\[I_{\text{C}}=I_{\text{Lo}}\ +I_{\text{Le}} \tag{5}\]  
The current vectors of LCL resonance are depicted in Figure 9.  
In Figure 8, considering the 202.90 Hz inter-harmonic, the red data are branch currents and the data in parentheses are the real-time phases of inter-harmonic currents. The inter-harmonic RMS current \(I_{\text{C}}\) of 202.90 Hz in capacitor branch was 2.20 A, which was less 1/15 of the RMS of inter-harmonic current \(I_{\text{LO}}\) or \(I_{\text{LE}}\) with the same frequency. \(U_{\text{Inverters}}\) was the inverter inter-harmonic voltage at insulated gate bipolar transistor (IGBT)-bridge output of 202.90 Hz, which was the significant one of 202.90 Hz in the PV inverter output circuit. Other frequency inter-harmonics at the PV PCC presented the same characteristics in the inverter output circuits. The analysis results are shown in Table 2. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter Symbols** & **Values** \\ \hline \(L_{\text{O}}\) & 0.25 mH \\ \(C\) & 47 uF \\ \(L_{\text{g}}\) & 0.15 mH (Calculated) \\ \(R_{\text{g}}\) & 0.15 Ohm (Calculated) \\ \(V_{\text{grid}}\) & 380 V (RMS) \\ \(V_{\text{DC}}\) & 400 V \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of photovoltaic (PV) Inverter Filter Circuits. RMS: root mean square  
Figure 9: Current vectors of inductor branches and capacitor branch as LCL resonance occurring. All branch harmonic currents and node voltages presented in Table 2 had the following characteristics: the capacitor inter-harmonic currents (\(I_{\rm C}\)) are very small and much smaller than inductor branch harmonic currents (\(I_{\rm LO}\) and \(I_{\rm LE}\)); and harmonic voltages at inverter bridge (\(U_{\rm Inverters}\)) are twice of the harmonic voltages at filter capacitor (\(U_{\rm C}\)). As illustrated in Equation (5), the LCL resonance of PV inverter filter usually generates a significant inter-harmonic current in capacitor branch, and the voltage RMS of filtering capacitor is the biggest one of all the node inter-harmonic voltages in the inverter output circuits.