## 3 Modeling the Inverter in the Frequency-Domain
### Synchronous Rotating Frame Based Current Control Method  
The block diagram of a typical dq-transformation based current control single-phase inverter [23] is illustrated in Figure 8. Its hardware topology is the same as that of the PR-based current control inverter. The dq components of the inverter output current \(\mathrm{i_{dq}}\) are obtained by dq transformation. Then the dq components of the current are controlled separately. The reference value of the dq components of the current \(\mathrm{i^{*}_{dq}}\) is obtained by a DC voltage regulator or/and calculated by the active/reactive power requirements. The output of the current regulator is converted into a reference of the inverter output voltage \(\mathrm{u^{*}_{dq}}\) after decoupling and inverse dq transformation. As shown in Figure 8, the output phase angle of the PLL \(\mathrm{\theta_{PLL}}\) affects the results of the dq transformation and inverse dq transformation. With a small-signal perturbation, due to the PLL output difference \(\mathrm{\delta_{PLL}}\), there is an influence on the value associated with the dq transformation. In order to present this influence, a small signal model is shown in Figure 9.  
Figure 8: The block diagram of a typical synchronous rotating frame based current control inverter. In Figure 9, the hardware system (in the grey box) is the same as in Figure 6. G\({}_{\rm{IPL}}\)(s) is the transfer function matrix from the grid voltage disturbance to the inverter current disturbance in the controller, as expressed in Equation (19). G\({}_{\rm{uPLL}}\)(s) is the transfer function matrix from the grid voltage disturbance to the inverter output reference voltage disturbance in the controller, as expressed in Equation (20). G\({}_{\rm{deco}}\)(s) is the decoupling control matrix and G\({}_{\rm{C}}\)(s) is the current regulator matrix, as expressed in Equations (21) and (22), separately [27]. \[\widetilde{\rm{i}}_{\rm{dq}}^{\rm{PLL}}={\rm{G}}_{\rm{iPLL}}({\rm{s}}) \widetilde{\rm{v}}_{\rm{dq}}+\widetilde{\rm{i}}_{\rm{dq}} \tag{19}\]  
\[{\rm{\widetilde{u}}}_{\rm{dq}}^{*\rm{PLL}}={\rm{G}}_{\rm{uPLL}}\widetilde{ \rm{v}}_{\rm{dq}}+\widetilde{\rm{u}}_{\rm{dq}}^{*} \tag{20}\]  
\[{\rm{G}}_{\rm{deco}}({\rm{s}})=\left[ \begin{array}{cc}0&-\omega{\rm{L}}\\ \omega{\rm{L}}&0\end{array} \right] \tag{21}\]  
\[{\rm{G}}_{\rm{C}}({\rm{s}})=\left[ \begin{array}{cc}{\rm{K}}_{\rm{P-C}}+ \frac{{\rm{K}}_{\rm{I-C}}}{{\rm{s}}}&0\\ 0&{\rm{K}}_{\rm{P-C}}+\frac{{\rm{K}}_{\rm{I-C}}}{{\rm{s}}}\end{array} \right] \tag{22}\]  
By applying the same derivation method in Section 3.1, \(Y_{0}\) and G\({}_{\rm{id}}\) can be derived, as expressed in Equation (23). \[\left\{ \begin{array}{cc}{\rm{Y}}_{0}({\rm{s}})=-\frac{1}{{\rm{L}}(\omega^{2} +{\rm{s}}^{2})}\left[\begin{array}{cc}{\rm{s}}&\omega\\ -\omega&{\rm{s}}\end{array}\right]\\ {\rm{G}}_{\rm{id}}({\rm{s}})=\frac{1}{{\rm{L}}(\omega^{2}+{\rm{s}}^{2})}\left[ \begin{array}{cc}{\rm{s}}&\omega\\ -\omega&{\rm{s}}\end{array}\right]\end{array} \right. \tag{23}\]  
Small-signal perturbation is added to Equations (19), (20), and (10) to derive Equations (24) and (25). \[\left[ \begin{array}{cc}{\rm{I}}_{\rm{d}}^{\rm{PLL}}+\widetilde{\rm{i}}_{ \rm{d}}^{\rm{PLL}}\\ {\rm{I}}_{\rm{q}}^{\rm{PLL}}+\widetilde{\rm{i}}_{\rm{q}}^{\rm{PLL}}\\ \end{array} \right]\approx\left[ \begin{array}{cc}1&\widetilde{\rm{\delta}}_{ \rm{PLL}}\\ -\widetilde{\rm{\delta}}_{\rm{PLL}}&1\end{array} \right]\cdot\left[ \begin{array} []{cc}{\rm{I}}_{\rm{d}}+\widetilde{\rm{i}}_{\rm{d}}\\ {\rm{I}}_{\rm{q}}+\widetilde{\rm{i}}_{\rm{q}}\end{array} \right] \tag{24}\]  
Figure 9: The block diagram of a typical synchronous rotating frame based current control inverter. By eliminating the steady state values in Equations (24) and (25), we obtain Equations (26) and (27). \[\left[ \begin{array}{c}\widetilde{\mathrm{i}}_{\mathrm{d}}^{\mathrm{PLL}}\\ \widetilde{\mathrm{i}}_{\mathrm{q}}^{\mathrm{PLL}}\end{array} \right]\approx \left[