# Output Filter Design for a Novel Dual-Input PV-Wind Power Converter by Energy Balance Principle
## 4 Inductance Derivation
The inductor \(L_{o}\) releases energy and its voltage is valued as:  
\[\Delta i_{Lm,wind,+}=\frac{V_{wind}}{L_{m,wind}}\,D_{wind}T_{s} \tag{2}\]1  
By applying VSBC to \(L_{o}\) and deriving with Equations (9)-(11), the following relationship holds:  
\[\Delta i_{Lm,wind,+}=\frac{V_{wind}}{L_{m,wind}}\,D_{wind}T_{s} \tag{2}\]2Rearranging equation (12), one can obtain the following representation of \(V_{Co}\):  
\[\Delta i_{Lm,wind,+}=\frac{V_{wind}}{L_{m,wind}}\,D_{wind}T_{s} \tag{2}\]3  
The values of \(V_{a}\), \(V_{b}\) and \(V_{c}\) shown in Figure 6 can be found by substituting (13) into (9), (10), and (11),  
\[\Delta i_{Lm,wind,+}=\frac{V_{wind}}{L_{m,wind}}\,D_{wind}T_{s} \tag{2}\]4  
\[\Delta i_{Lm,wind,+}=\frac{V_{wind}}{L_{m,wind}}\,D_{wind}T_{s} \tag{2}\]5  
and  
\[\Delta i_{Lm,wind,+}=\frac{V_{wind}}{L_{m,wind}}\,D_{wind}T_{s} \tag{2}\]6  
Once \(V_{a}\), \(V_{b}\) and \(V_{c}\) have been determined, the inductor currents at the time points, \(t=D_{1}T_{s}\), \(D_{2}T_{s}\), and \(D_{3}T_{s}\), can be readily calculated. The current magnitudes at these time points are \(I_{a}\), \(I_{b}\), and \(I_{c}\), in turn, which are estimated as follows:  
\[\Delta i_{Lm,wind,+}=\frac{V_{wind}}{L_{m,wind}}\,D_{wind}T_{s} \tag{2}\]7  
\[\Delta i_{Lm,wind,+}=\frac{V_{wind}}{L_{m,wind}}\,D_{wind}T_{s} \tag{2}\]8  
and  
\[\Delta i_{Lm,wind,+}=\frac{V_{wind}}{L_{m,wind}}\,D_{wind}T_{s} \tag{2}\]9  
In Step 6, the average current of \(i_{Lo}\), \(I_{Lo,avg}\), is the integral over the period \(T_{s}\). That is,  
\[ \begin{array}{ll}I_{Lo,avg}&=\frac{1}{t_{s}}(\int_{0}^{D_{1}T_{s}}\frac{I_{b }}{D_{1}t_{s}}dx+\int_{0}^{(D_{2}-D_{1})T_{s}}\left[\frac{I_{b}-I_{a}}{(D_{2} -D_{1})T_{s}}x+I_{a}\right]dx+\int_{0}^{(D_{3}-D_{2})T_{s}}\left[\frac{I_{b}- I_{b}}{(D_{3}-D_{2})T_{s}}x+I_{b}\right]dx\\ &+\int_{0}^{(1-D_{3})T_{s}}\left[\frac{-I_{c}}{(1-D_{3})T_{s}}x+I_{c}\right]dx \end{array}  \tag{20}\]  
Calculating and simplifying the equation (20) results:  
\[I_{Lo,avg}=\frac{I_{a}D_{2}+I_{b}(D_{2}+D_{pv}-D_{wind})+I_{c}(1-D_{2})}{2} \tag{21}\]  
According to Ampere Second Balance (ASBC), the average current of the output capacitor \(C_{o}\) should be zero. As a result, the current \(I_{Lo,avg}\) is equal to the load current \(I_{o}\). Then, substituting (14), (15), (16), (17), (18), and (19) into (21), one can obtain the minimum inductance of \(L_{o}\) for CCM as follows:  
\[L_{o}=\frac{(-2+D_{pv})D_{pv}(-1+2D_{2}+D_{pv})n_{pv}T_{s}V_{pv}-(-1+D_{pv})( D_{pv}(-1+2D_{2}+D_{pv})+D_{wind})n_{wind}T_{s}V_{wind}}{2(1-D_{pv})I_{o}} \tag{22}\]  
With respect to capacitor design, voltage ripple dominates capacitance determination. Voltage ripple across a capacitor, \(\Delta v_{c}\), can be found by:  
\[\Delta v_{c}=\frac{\Delta Q}{C}=\frac{i_{c}\Delta t}{C} \tag{23}\]  
where \(\Delta v_{c}\) stands for charge variation during time interval \(\Delta t\), \(i_{c}\) is the current flowing through the capacitor, and \(C\) is the corresponding capacitance. The currents of capacitors \(C_{1,wind}\), \(C_{2,wind}\), \(C_{1,pv}\), \(C_{2,pv}\) and \(C_{o}\) are \(i_{D3,wind}\), \(i_{D4,wind}\), \(D_{3,pv}\), \(D_{4,pv}\) and \(i_{Lo}\)-\(I_{o}\), respectively. According to the operation principle discussed in Section 3, the voltage ripples across capacitors \(C_{1,wind}\), \(C_{2,wind}\), \(C_{1,pv}\), \(C_{2,pv}\) and \(C_{o}\) can be estimated as follows:  
\[\Delta v_{c1,wind}=\frac{i_{D3,wind}D_{wind}}{C_{1,wind}f_{s}} \tag{24}\]\[\Delta v_{c2,wind}=\frac{i_{\text{D4,}wind}(1-D_{wind})}{C_{2,wind}f_{s}} \tag{25}\]  
\[\Delta v_{c1,pv}=\frac{i_{\text{D3,}pv}D_{pv}}{C_{1,pv}f_{s}} \tag{26}\]  
\[\Delta v_{c2,pv}=\frac{i_{\text{D4,}pv}(1-D_{pv})}{C_{2,pv}f_{s}} \tag{27}\]  
and  
\[\Delta v_{co}=\frac{(i_{Lo}-I_{o})D_{wind}}{C_{o}f_{s}} \tag{28}\]  
From (24)-(28), the capacitances of \(C_{1,wind}\), \(C_{2,wind}\), \(C_{1,pv}\), \(C_{2,pv}\) and \(C_{o}\) can be readily computed.