# Output Filter Design for a Novel Dual-Input PV-Wind Power Converter by Energy Balance Principle
## 4 Inductance Derivation  
Conceptual waveforms of gate signals and output inductor voltage and current are illustrated in Figure 6, in which the switches \(SW_{1,wind}\) and \(SW_{2,wind}\) are closed for \(D_{wind}T_{s}\) and \(SW_{1,pv}\) and \(SW_{2,pv}\) are closed for \(D_{pv}T_{s}\), respectively. Since the voltage ripple at each capacitor is much less than the average capacitor voltage, the voltage across the output inductor can be regarded as constant under all switch statuses. Accordingly, the current flowing through the output inductor will be piecewise linear over one switching cycle. The derivation of the output inductance of the DIPWC is sophisticated. For a clear description, the related procedure is summarized as follows. Following the previous seven steps, a detailed derivation of output inductance is presented below. From Figure 2a, it can be observed that the voltage \(V_{CI,wind}\) is \(n\) times the magnitude of \(V_{wind}\) if the leakage inductor \(L_{k,wind}\) is neglected. That is,  
\[V_{\text{CI},wind}=n_{wind}V_{wind} \tag{1}\]  
Meanwhile, the current increment on magnetizing inductor \(L_{m,wind}\) can be estimated by  
\[\Delta i_{Lm,wind,+}=\frac{V_{wind}}{L_{m,wind}}\,D_{wind}T_{s} \tag{2}\]  
Figure 6: The waveforms for the understanding of inductance derivation. When switches \(SW_{1,wind}\) and \(SW_{2,wind}\) are both turned off, the voltage polarity of \(L_{m,wind}\) reverses. The energy stored in \(L_{m,wind}\) will be forwarded to the secondary of the coupled inductor \(T_{1}\) to charge the capacitor \(C_{2,wind}\), and then the current \(i_{Lm,wind}\) decreases. The descent is calculated as:  
\[\Delta i_{Lm,wind,-}=\frac{V_{C2,wind}-V_{C1,wind}}{n_{wind}L_{m,wind}}(1-D_{ wind})T_{s} \tag{3}\]  
Based on the energy balance principle, that is, net current change in \(L_{m,wind}\) being zero, the voltage across \(C_{2,wind}\) can be expressed as:  
\[V_{C2,wind}=\frac{n_{wind}V_{wind}}{1-D_{wind}} \tag{4}\]  
Similarly, from Figure 2d, in which the switches \(SW_{1,pv}\) and \(SW_{2,pv}\) are closed, the voltage \(V_{C1,pv}\) and the current increment on the magnetizing inductor \(L_{m,pv}\) can be represented by:  
\[V_{C1,pv}=n_{pv}V_{pv} \tag{5}\]  
and  
\[\Delta i_{Lm,pv,+}=\frac{V_{pv}}{L_{m,pv}}D_{pv}T_{s} \tag{6}\]  
respectively. Once \(SW_{1,pv}\) and \(SW_{2,pv}\) are turned off, the capacitor \(C_{2,pv}\) begins absorbing energy from \(C_{1,pv}\), resulting in current decrease in \(L_{m,pv}\). The current drop is estimated as follows:  
\[\Delta i_{Lm,pv,-}=\frac{V_{C2,pv}-V_{C1,pv}}{n_{pv}L_{m,pv}}(1-D_{pv})T_{s} \tag{7}\]  
In steady state, net current change is zero, which yields:  
\[V_{C2,pv}=\frac{n_{pv}V_{pv}}{1-D_{pv}} \tag{8}\]  
Subsequently, the finding for \(V_{C0}\) is discussed. When the switches \(SW_{1,wind}\) and \(SW_{2,wind}\) are in the on state, the output inductor \(L_{o}\) will absorb energy from \(C_{2,wind}\) via the loop of \(C_{o}\)-\(C_{2,wind}\)-\(n_{2,wind}\)-\(D_{5,wind}\)-\(L_{o}\). Thus, the voltage across \(L_{o}\) over the time interval \(D_{wind}T_{s}\) can be given by:  
\[V_{Lo,wind}=n_{wind}V_{wind}\frac{2-D_{wind}}{1-D_{wind}}-V_{C0} \tag{9}\]  
During the interval \(D_{pv}T_{s}\), the switches \(SW_{1,pv}\) and \(SW_{2,pv}\) are in the on state, which results in the output inductor \(L_{o}\) absorbing energy from \(C_{2,pv}\) via the loop of \(C_{o}\)-\(C_{2,pv}\)-\(n_{2,pv}\)-\(D_{5,pv}\)-\(L_{o}\). In this time interval, the voltage across \(L_{o}\) becomes:  
\[\Delta i_{Lm,wind,+}=\frac{V_{wind}}{L_{m,wind}}\,D_{wind}T_{s} \tag{2}\]0  
As shown in Figure 2c, the statuses of the four switches are open in the remaining time of a switching period, \((1-D_{wind}-D_{pv})T_{s}\).