*2.2. Steady-State Analysis: Current Source*

The consideration of a current source in the converter is related to PV applications, since a PV module/array is modeled as a current source. As a consequence, the voltage source of the converter in Figure 3 is changed by a PV module and a coupling capacitor *Ci* . In this system, the switching function in Equation (1) describes the operation of the active and passive switches. Figure 6 shows the resulting networks by the operation of the converter according to this switching function.

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**Figure 6.** Operating modes of the proposed converter using a PV module. (**a**) Function *q* = 1. (**b**) Function *q* = 0.

In PV applications, during the switch on-state (Figure 6a), the inductor *L*<sup>1</sup> is connected to coupling capacitor *C<sup>i</sup>* and exhibits voltage *vCi*, whereas inductor *L*<sup>2</sup> exhibits voltage *vC*<sup>1</sup> − *vCi*. In this scenario, both inductors are in charge mode. The PV module transfers energy to capacitor *C<sup>i</sup>* ; in turn, capacitor *C*<sup>1</sup> transfers energy to inductor *L*<sup>2</sup> and *C*<sup>2</sup> supports the load power demand. Equations (17) describe the behavior of the system in this state.

$$\begin{aligned} L\_1 \frac{di\_{L1}}{dt} &= v\_{\text{Ci}\prime} \\ L\_2 \frac{di\_{L2}}{dt} &= v\_{\text{C1}} - v\_{\text{Ci}\prime} \\ \mathcal{C}\_i \frac{dv\_{\text{Ci}}}{dt} &= i\_{pv} - i\_{L1} + i\_{L2\prime} \\ \mathcal{C}\_1 \frac{dv\_{\text{C1}}}{dt} &= -i\_{L2\prime} \\ \mathcal{C}\_2 \frac{dv\_{\text{C2}}}{dt} &= -\frac{v\_{\text{C2}}}{R} .\end{aligned} \tag{17}$$

Conversely, in the turn-off state of the active switches (Figure 6b), inductor *L*<sup>1</sup> and capacitor *C<sup>i</sup>* transfer energy to capacitor *C*1, while inductor *L*<sup>2</sup> supplies energy to conjunct *R*–*C*2. The differential equations in this condition are given by Equations (18).

$$\begin{aligned} L\_1 \frac{di\_{L1}}{dt} &= v\_{\text{Ci}} - v\_{\text{C1}} \\ L\_2 \frac{di\_{L2}}{dt} &= -v\_{\text{C2}} \\ \mathbf{C}\_i \frac{dv\_{\text{Ci}}}{dt} &= i\_{pv} - i\_{L1} \\ \mathbf{C}\_1 \frac{dv\_{\text{C1}}}{dt} &= i\_{L1} \\ \mathbf{C}\_2 \frac{dv\_{\text{C2}}}{dt} &= i\_{L2} - \frac{v\_{\text{C2}}}{R} \end{aligned} \tag{18}$$

The above expressions define the voltages at the terminals of the inductors and the currents through the capacitors in one switching period. The steady-state condition of the converter can by obtained by using the principles of volt-second and charge balance, which result in Equations (19)–(23).

$$I\_{L1} = \frac{I\_{pv}}{D} \, \tag{19}$$

$$I\_{L2} = \frac{(1 - D)I\_{pv}}{D^2},\tag{20}$$

$$V\_{\rm Ci} = \frac{(1 - D)^4 I\_{pv} R}{D^4} \,\mathrm{}$$

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$$V\_{\mathbb{C}1} = \frac{(1 - D)^3 I\_{pv} R}{D^4} \,\mathrm{},\tag{22}$$

$$V\_{\rm C2} = \frac{(1 - D)^2 I\_{\rm pv} R}{D^2}. \tag{23}$$

The voltage conversion gain of the converter can be obtained using Expressions (21) and (23), where the relation between the output voltage and the coupling capacitor voltage maintain a quadratic dependence, given by Equation (24).

$$M = \frac{V\_{\rm C2}}{V\_{\rm Ci}} = \frac{D^2}{(1 - D)^2}.\tag{24}$$

The power generated by the PV module/array depends on weather conditions, that is, incident solar irradiance and temperature. In addition, the voltage at the terminals permits the adjustment of the power supplied by the PV module. Since the maximum power of a PV module is related to a specific voltage, Equation (21) shows that the voltage in the PV module can be selected by using the duty ratio signal of the converter, that is

$$D = \frac{\left[I\_{p\upsilon}R\right]^{\frac{1}{4}}}{\left[I\_{p\upsilon}R\right]^{\frac{1}{4}} + \left[V\_{\text{Ci}}\right]^{\frac{1}{4}}}.\tag{25}$$

Finally, current *I<sup>o</sup>* injected to load (*R*) is given by Equation (26). This implies that the output voltage of converter (*VC*<sup>2</sup> ) is not fixed and it depends on the duty ratio, the generated current by the PV module, and the value of load *R*.

$$I\_o = \frac{(1 - D)^2 I\_{pv}}{D^2} = \frac{I\_{pv}}{M}. \tag{26}$$

The inequality (13), which is the boundary condition between the CCM and DCM operation of the converter, holds independently of the PV application. Additionaly, the inductance values are given by Expressions (16).

PV systems can be applied in different ways; however, there are some applications in which the output voltage of the power converter is clamped. Figure 7a shows the proposed converter topology, where the output power condition is given in terms of the output current and output voltage. Figure 7b shows the implementation of the PV system in a DC microgrid, where the output voltage of the converter corresponds to the bus voltage of the microgrid. Finally, Figure 7c shows a grid-connected application, where the output voltage of the converter is clamped or regulated by the inverter (DC-AC converter). In this kind of scenario, the operating point of the converter and the semiconductor stresses are different than those presented by a pure resistive load.

**Figure 7.** *Cont*.

**Figure 7.** Structure of PV applications. (**a**) Proposed converter. (**b**) DC network application. (**c**) Gridconnected PV system.

A similar analysis applied to the converter topology in Figure 7a shows that the steady-state condition of the converter results in Equations (27)–(31).

$$I\_{L1} = \frac{I\_{pv}}{D},$$
 
$$\left(\begin{array}{c} \\ \end{array}\right) \tag{27}$$

$$I\_{L2} = \frac{(1 - D)I\_{pv}}{D^2},\tag{28}$$

$$V\_{\rm Ci} = \frac{(1 - D)^2 v\_o}{D^2} \,\tag{29}$$

$$V\_{\rm C1} = \frac{V\_{\rm Ci}}{(1 - D)'} \tag{30}$$

$$V\_{\mathbb{C2}} = v\_o.\tag{31}$$

Equation (29) shows that the voltage conversion ratio (*M* = *VC*2/*VCi*) of the converter maintains a quadratic dependence on the duty ratio. Additionally, to achieve a specific voltage at the terminals of the PV module requires the duty ratio to be

$$D = \frac{\sqrt{V\_{\rm C2}}}{\sqrt{V\_{\rm C2}} + \sqrt{V\_{\rm Ci}}}.\tag{32}$$

In addition, the current injected to the next system *I<sup>o</sup>* corresponds to the expression given in Equation (26).
