## 2 System Description and Its Mathematical Modeling
### Small-Signal Model of the Inverter under the Differential Peak Current Mode Control  
The control of a power converter is generally undertaken in the frequency domain. Examination of a system loop gain in this domain easily allows the determination of stability margins. The design methodology requires that a transfer function be available. For the boost inverter, this is generally complicated since it is a six dimensional system. Since the grid frequency is much smaller than  
Figure 5: Steady-state waveforms of the average inductor currents and capacitor voltages of the differential boost inverter from the quasi-steady-state analytical expressions (8)–(11) for two different values of power \(P=670\) W and \(P=1.4\) kW by assuming a unity power factor. the switching frequency, a simplified analysis will be performed by considering the inverter like a DC-DC converter in which the grid voltage is considered constant at the switching time scale. The averaged model under CMC can finally be expressed as follows  
\[\dot{\mathbf{x}} = \mathbf{f}(\mathbf{\overline{x}},d) \tag{17}\] \[d = \frac{R_{s}(i_{\text{ref}}(t)-(\vec{i}_{1}-\vec{i}_{2}))}{(m_{a}+ R_{sL}\frac{\overline{v}_{o2}}{2L})T}, \tag{18}\]  
where \(\mathbf{\overline{x}}=(\vec{i}_{1},\quad\vec{i}_{2},\quad\overline{v}_{o1}, \quad\overline{v}_{o2},\quad\hat{i}_{g})^{\intercal}\) and function \(\mathbf{f}\) is as follows  
\[\mathbf{f}(\mathbf{\overline{x}})=\left( \begin{array}{c}\frac{d\vec{i}_{1}} {dt}\\ \frac{d\vec{i}_{2}}{dt}\\ \frac{d\overline{v}_{o1}}{dt}\\ \frac{d\overline{v}_{o2}}{dt}\\ \frac{d\underline{i}_{g}}{dt}\end{array} \right) \tag{19}\]  
The coordinates of the quasi-equilibrium point can be derived by substituting \(D\) by its expression in (8)-(13). For a practically constant input voltage (PV voltage \(V_{\text{dc}}\approx V_{\text{mpp}}\approx 154\) V) and sinusoidally time varying output voltage with RMS value \(V_{\text{g,rms}}=\)230 V, according to (15), the duty cycle varies in the range (0.28,0.72). Therefore the system will be studied in this range of duty cycle. To make the system operate in this range with outer loop open, the reference current is fixed according to the following equation which is obtained by solving in steady-state (18) for \(i_{\text{ref}}\)  
\[i_{\text{ref}}=\frac{m_{a}DT}{R_{sL}}-\frac{D(1-D)Tv_{g}}{2L(1-D)}+\frac{3I_{ \text{mpp}}}{2D-1} \tag{20}\]  
Because the duty cycle could be larger than 0.5, the system may exhibit subhamronic oscillation if the slope of the compensating signal is not appropriately selected. The slope of the compensating ramp is selected in such a way to guarantee a system free from subharmonic oscillation for the entire range of \(D\in(0.28,0.72)\). The design of the controller is conventionally based on a small-signal averaged model which can be obtained from (17) and (18) after performing a conventional perturbation and linearization close to the operation point which is supposed to be the MPP. Let \(\hat{v}_{1}=\overline{v}_{o1}-V_{o1}\), \(\hat{v}_{2}=\overline{v}_{o2}-V_{o2}\), \(\hat{i}_{1}=\vec{i}_{1}-I_{1}\), \(\hat{i}_{2}=\vec{i}_{2}-I_{2}\) and \(\hat{d}=d-D\) the small deviations of the average state variables \(\overline{v}_{o1},\overline{v}_{o2},\vec{i}_{1},\vec{i}_{2}\), and the average duty cycle \(d\) with respect to their steady-state values \(V_{o1},V_{o2}\), \(I_{1}\), \(I_{2}\) and \(D\) respectively.