*3.2. Steady-State Conditions*

For calculating the inductances *L*<sup>1</sup> and *L*2, the period when the active switches are turned ON is analyzed, see Figure 5. For large enough inductances, the currents rise linearly from their minimum value to their maximum value during *tON*; then, the voltages in the inductor terminals of *L*<sup>1</sup> and the inductor terminals of *L*<sup>2</sup> are, respectively:

$$\begin{aligned} V\_{L\_1} &= L\_1 \frac{\Delta I\_{L\_1}}{t\_{\text{ON}}} = E\\ V\_{L\_2} &= L\_2 \frac{\Delta I\_{L\_2}}{t\_{\text{ON}}} = -V\_{\mathbb{C}\_p} + V\_0 \end{aligned} \tag{4}$$

Substituting *tON* = *DT<sup>s</sup>* , *VC<sup>p</sup>* = *DV*0, and *V*<sup>0</sup> = *E*/(1 − *D*) 2 into Equation (4), the expressions for the inductor current ripples are:

$$
\begin{aligned}
\Delta \dot{i}\_{L\_1} &= \frac{ED}{L\_1 f\_s} \\
\Delta \dot{i}\_{L\_2} &= \frac{ED}{(1 - D)L\_2 f\_s}
\end{aligned}
\tag{5}
$$

Considering the equality between the input and output power (*EIL*<sup>1</sup> = *V* 2 0 /*R*), the average current through *L*<sup>1</sup> can be computed. On the other hand, the average current of *L*<sup>2</sup> can be computed using the relationship *IL*<sup>2</sup> = (1 − *D*)*IL*<sup>1</sup> ; then, the corresponding expressions for *IL*<sup>1</sup> and *IL*<sup>2</sup> are:

$$\begin{aligned} I\_{L\_1} &= \frac{E}{R(1-D)^4} \\\\ I\_{L\_2} &= \frac{E}{R(1-D)^3} \end{aligned} \tag{6}$$

The maximum and minimum current values reached by the first and second inductors can be obtained as follows:

$$\begin{aligned} I\_{L\_{1MAX}} &= I\_{L\_1} + \frac{\Delta I\_{L\_1}}{2}, \ I\_{L\_{1MIN}} = I\_{L\_1} - \frac{\Delta I\_{L\_1}}{2} \\\\ I\_{L\_{2MAX}} &= I\_{L\_2} + \frac{\Delta I\_{L\_2}}{2}, \ I\_{L\_{2MIN}} = I\_{L\_2} - \frac{\Delta I\_{L\_2}}{2} \end{aligned} \tag{7}$$

In switching converters, it is essential to find out the critical inductance values for operation in CCM [22]. The critical inductance values of the proposed configuration can be computed using the relationships:

$$\begin{aligned} 0 &= I\_{L\_1} - \frac{\Delta I\_{L\_1}}{2} \\ 0 &= I\_{L\_2} - \frac{\Delta I\_{L\_2}}{2} \end{aligned} \tag{8}$$

then, the inductances for the CCM operation are:

$$\begin{aligned} L\_1 &> \frac{DR(1-D)^4}{2f\_s} \\ L\_2 &> \frac{DR(1-D)^2}{2f\_s} \end{aligned} \tag{9}$$

The charge variation in a capacitor is defined as ∆*Q<sup>C</sup>* = *C*∆*vC*, where ∆*Q<sup>C</sup>* depicted the area under the current curve when the capacitor stores energy, *C* denotes the capacitance, and ∆*v<sup>C</sup>* is the voltage ripple (see Figure 7). For *tON*, the capacitor *C<sup>p</sup>* is charged by the current *iL*<sup>2</sup> (*iL*<sup>2</sup> = *iC<sup>p</sup>* ). On the other hand, the capacitor *C*<sup>0</sup> is charged by the current of *L*<sup>1</sup> at time *tOFF*. In this interval, the current through *C*<sup>0</sup> is *iC*<sup>0</sup> = *iL*<sup>1</sup> − *I*0, that is:

$$\begin{aligned} \Delta \mathbf{Q}\_{\mathbb{C}\_p} &= \int\_{t\_0}^{t\_1} \dot{\mathbf{i}}\_{L\_2}(t) dt = \mathbf{C}\_p \Delta v\_{\mathbb{C}\_p} \\\\ \Delta \mathbf{Q}\_{\mathbb{C}\_0} &= \int\_{t\_1}^{t\_2} [\dot{\mathbf{i}}\_{L\_1}(t) - I\_0] dt = \mathbf{C}\_0 \Delta v\_{\mathbb{C}\_0} \end{aligned} \tag{10}$$

where *tON* = *t*<sup>1</sup> − *t*<sup>0</sup> and *tOFF* = *t*<sup>2</sup> − *t*1. Calculating ∆*Q<sup>C</sup>* for *C<sup>p</sup>* and *C*0, the following expressions for the capacitor ripples are obtained:

$$\begin{aligned} \Delta v\_{\mathbb{C}\_p} &= \frac{V\_{\mathbb{C}\_p}}{R(1 - D)^3 \mathbb{C}\_p f\_s} \\\\ \Delta v\_0 &= \frac{V\_0 D (2 - D)}{R(1 - D) \mathbb{C}\_0 f\_s} \end{aligned} \tag{11}$$

According to Equations (5) and (11), the inductor current and capacitor voltage ripples can be chosen for a specific amplitude. Selecting a ripple value is essential, especially in the first inductor current (∆*iL*<sup>1</sup> ). The above is due to the requirements of renewable sources as in fuel and photovoltaic cells, which do not allow high ripples on the demanded currents. Large magnitude of current ripple in high frequency (>10 kHz) cause degradation of the catalyst of fuel cell plates. Moreover, fuel-cell currents should not have a high pulsating either negative profile.
