## 2 High Voltage Boost Converter  
The adopted high-voltage boost converter is illustrated in Figure 1 [42]. The rated input and output voltages of the converter are 80 and 400 V, respectively. For simplicity, the voltage drops across each forward-biased diode; the main switch \(S_{1}\) when remaining on and the inductor \(L_{k1}\) are all neglected. The duty cycle D is defined as  
\[D\triangleq\frac{t_{on}}{T} \tag{1}\]  
where \(t_{on}\) represents the period of time that \(S_{1}\) remains on, and \(T\) is the operation period of the converter. (1) Mode 1 (\(t_{0}\sim t_{1}\))  
The circuit conduction situation in this mode is indicated in Figure 2. \(S_{1}\) remains on, and the input voltage \(V_{in}\) is directly applied to the primary side \(N_{1}\) of the coupled inductor. Accordingly, the current through the secondary side \(N_{2}\) charges the capacitor \(C_{1}\), and the diode \(D_{2}\) remains forward-biased until \(S_{1}\) is turned off at the end of this mode. The voltage drops across components and the currents therethrough are governed by  
\[\Delta i_{Lm1(closed)}=\frac{V_{in}}{Lm1}DT \tag{2}\]  
\[v_{N2}=v_{C1}=V_{in}\frac{N_{2}}{N_{1}} \tag{3}\]  
Figure 1: The presented high-voltage step-up converter. (2) Mode 2 (\(t_{1}\sim t_{2}\))  
The circuit conduction situation in this mode is indicated in Figure 3. \(S_{1}\) is switched off, and \(D_{1}\) is forward-biased. The energy stored in \(N_{1}\) is released to \(C_{1}\), and the polarity of \(N_{2}\) is reversed. As a consequence, the diode \(D_{3}\) is forward-biased, and the capacitor \(C_{2}\) is charged until the energy stored in \(N_{1}\) is completely released. The governing equation is expressed as  
\[v_{N1}=v_{Lm1}=v_{N2}\frac{N_{1}}{N_{2}}=v_{C2}\frac{N_{1}}{N_{2}} \tag{4}\]  
(3) Mode 3 (\(t_{2}\sim t_{3}\))  
The circuit conduction situation in this mode is indicated in Figure 4. \(S_{1}\) remains off. The energy stored in \(C_{1}\), \(C_{2}\) and \(L_{m1}\) is released to the load by way of the diode \(D_{4}\), and an operation cycle is completed accordingly. Figure 3: Circuit conduction situation in Mode 2. Figure 2: Circuit conduction situation in Mode 1. The inductor volt-second balance theory states that, in the steady state, the net change in the current flowing through an inductor is zero over a cycle. Therefore, applying the principle to Equations (2) and (4) gives  
Figure 2.0  
Rearranging Equation (5) gives  
Figure 2.1  
The output voltage is given by  
Figure 2.2  
Substituting Equations (3) and (6) into Equation (7) gives the voltage ratio  
Figure 2.3  
Assuming that all the components are lossless, the input power is equal to the output power, i.e., \(P_{in}=P_{out}\), and  
Figure 2.4  
Substituting Equations (8) and (9) into Equation (7) gives  
Figure 2.5  
where \((\frac{1+N-D}{1-D})\) is the voltage ratio of the converter. Derived from Equation (2), the increment of \(i_{Lm1}\), over the period of time \(DT\) that \(S_{1}\) is on, is given by  
Figure 2.6  
Figure 2.7 Circuit conduction situation in Mode 3. Accordingly, the maximum and the minimum magnetizing current of the inductor, \(I_{Lm1(\max)}\) and \(I_{Lm1(\min)}\), are respectively given by  
Figure 2.8  
Figure 2.9  
The inductor current is continuous on the condition that \(I_{Lm1(\min)}\geq 0\), that is,  
\[\Delta i_{Lm1(closed)}=\frac{V_{in}}{Lm1}DT \tag{2}\]0  
The rated output power and the output voltage are specified as 300 W and 400 V, respectively, in Table 1, and the load \(R_{O}\) is evaluated as 533.33 \(\Omega\) accordingly. It is requested that the inductor \(L_{m1}\) works anytime in the continuous conduction mode (CCM), that is, \(I_{Lm1(\min)}\geq 0\) anytime, regardless of any change in the solar irradiance. As illustrated in Figure 5, the peak of the curve \(y=\frac{D(1-D)^{2}}{(1+2-D)^{2}}\) occurs at \(D=0.394\). Substituting \(D=0.394\), \(R_{O}=533.33\)\(\Omega,f=25\) kHz and the turn ratio \(N=2\) into Equation (14) gives \(L_{m1}=227\) uH.