*2.1. Steady-State Analysis: Voltage Source*

In this section, we widen the steady-state analysis of the proposed converter, where the input port of the converter is connected to a voltage supply. In next subsection, the contribution of the PV module and converter is discussed.

The operating modes of the converter are defined by the conditions of the active switches, where the following assumptions are stated:


$$q = \begin{cases} 1 & \rightarrow & 0 < t \le t\_{\text{on}} \\ 0 & \rightarrow & t\_{\text{on}} < t \le T\_{\text{s}} \end{cases} \qquad\qquad \overline{q} = \begin{cases} 0 & \rightarrow & 0 < t \le t\_{\text{on}} \\ 1 & \rightarrow & t\_{\text{on}} < t \le T\_{\text{s}} \end{cases} \tag{1}$$

where *ton* is the conducting time of the active switches and *T<sup>s</sup>* is the switching period.

The operation modes of the converter are described by the on- and off-state of the active switches. During the switch on-state (Figure 4a), inductor *L*<sup>1</sup> is connected to the source voltage *E*, whereas inductor *L*<sup>2</sup> exhibits voltage *vC*<sup>1</sup> − *E*. In this state, capacitor *C*<sup>1</sup> transfers energy to inductor *L*2, whereas capacitor *C*<sup>2</sup> supports the power demanded by load *R*. In this operating mode, the differential equations that describe the behavior of the system are:

$$\begin{aligned} L\_1 \frac{di\_{L1}}{dt} &= E\_\prime \\ L\_2 \frac{di\_{L2}}{dt} &= v\_{C1} - E\_\prime \\ C\_1 \frac{dv\_{C1}}{dt} &= -i\_{L2} \\ C\_2 \frac{dv\_{C2}}{dt} &= -\frac{v\_{C2}}{R} \end{aligned} \tag{2}$$

When the switches are in the turn-off state (Figure 4b), inductor *L*<sup>1</sup> transfers energy to capacitor *C*<sup>1</sup> through diode *S*2, whereas inductor *L*<sup>2</sup> supplies energy to conjunct *R*–*C*<sup>2</sup> via diode *S*4. The corresponding differential equations are given by:

**Figure 4.** Operating modes of the proposed converter: (**a**) function *q* = 1; (**b**) function *q* = 0.

The set of Equations (2) and (3) defines the voltages at the terminals of the inductors and the currents through the capacitors in one switching period. For these voltages and currents, the principles of inductor volt-second balance and capacitor-charge balance describe a steady-state operation of the converter, which are given by Equations (4) and (5).

$$
\langle v\_L(t) \rangle = 0 = \frac{1}{T\_s} \int\_0^{T\_s} v\_L(t)dt,\tag{4}
$$

$$
\langle i\_{\mathbb{C}}(t) \rangle = 0 = \frac{1}{T\_s} \int\_0^{T\_s} i\_{\mathbb{C}}(t)dt. \tag{5}
$$

Using these principles allows us to express the averaged inductor currents and averaged capacitor voltages in the steady-state condition, with the result in Equations (6)–(9).

$$I\_{L1} = \frac{ED^3}{(1 - D)^4 R'} \tag{6}$$

$$I\_{L2} = \frac{ED^2}{(1 - D)^3 R'} \tag{7}$$

$$V\_{\mathbb{C}1} = \frac{E}{(1 - D)'} \tag{8}$$

$$V\_{\mathbb{C}2} = \frac{ED^2}{(1-D)^2} \tag{9}$$

where *D* is the nominal duty ratio of the converter that defines the on-state time of the active switches by *ton* = *DT<sup>s</sup>* . Then, the voltage gain of the converter has a quadratic dependence on the duty ratio, which implies a wide output step-up or step-down voltage conversion characteristic. Therefore, the voltage gain is expressed in Equation (10).

$$M = \frac{V\_{\rm C2}}{E} = \frac{D^2}{(1 - D)^2}.\tag{10}$$

It is important to notice that the voltage conversion ratio and the steady-state operating condition are valid only during CCM operation of the converter. In addition, the conversion ratio is positive and the converter presents a common ground between the input and output ports. In this operating mode, two passive networks are formed (Figure 4), in contrast to the discontinuous conduction mode (DCM), where three passive networks are formed, losing the quadratic voltage gain characteristic of the converter.

The boundary conditions between the CCM and DCM operation of the converter are given by the dimensionless parameters as,

$$k\_1 = \frac{2L\_1}{RT}, \quad \quad \quad \quad \quad k\_2 = \frac{2L\_2}{RT}, \tag{11}$$

for inductor *L*<sup>1</sup> and *L*2, respectively, for the limit parameter in the following form,

$$k\_{crit(1)} = \frac{(1 - D)^4}{D^2}, \qquad \qquad \qquad k\_{crit(2)} = (1 - D)^2. \tag{12}$$

Then, the converter operates in CCM if it satisfies the criteria in Equation (13).

$$k\_1 > k\_{\text{crit}(1)'} \qquad \qquad \qquad \qquad \qquad k\_2 > k\_{\text{crit}(2)} \tag{13}$$

Figure 5 graphs the limit parameter for each switching cell in the converter, as well as the operative regions depending on the parameters *k*<sup>1</sup> and *k*2. In CCM operation, it is easy to show that *k*<sup>2</sup> > 1 satisfies the condition for the switching cell *S*3, *S*4, and *L*2. However, the front-end switching cell (*S*1, *S*2, and *L*1) has an operational limit given by *D*0 < *D* < 1, where *D*0 is the duty ratio that satisfies *k*<sup>1</sup> = *kcrit*(1) . Therefore, an increase in *k*<sup>1</sup> corresponds to an increase in the operational limit of the converter in CCM (reduced value of *D*0 ).

**Figure 5.** Boundary operation of the proposed converter under parameter variations: (**a**) parameter *kcrit*(1) ; (**b**) parameter *kcrit*(2) .

A complete operation of the converter in DCM requires both switching cells to have the same operational limit given by 0 < *D* < *D*0 . The duty ratio *D*0 sets the values of *k*<sup>1</sup> and *k*<sup>2</sup> through the equalities given by Equations (14).

$$k\_1 = k\_{\rm crit(1)}(D'), \quad \qquad \qquad k\_2 = k\_{\rm crit(2)}(D'). \tag{14}$$

Then, the complete voltage conversion ratio of the proposed converter is given by Equation (15). In CCM operation, the voltage gain of the converter depends on the quadratic of the duty ratio, whereas in DCM, it depends on the duty ratio and parameter *k*1.

$$M = \frac{V\_{\mathbb{C}\mathbb{Z}}}{E} = \begin{cases} \frac{D^2}{(1-D)^2} & \rightarrow & k\_j > k\_{crit(j)} \text{ with } j = 1, 2, \\\ & \frac{D}{\sqrt{k\_1}} & \rightarrow & k\_j < k\_{crit(j)} \text{ with } j = 1, 2. \end{cases} \tag{15}$$

In CCM operation, it is assumed that current ripples on the inductors are small, that is, ∆*IL*<sup>1</sup> ≤ *α*<sup>1</sup> *IL*<sup>1</sup> and ∆*IL*<sup>2</sup> ≤ *α*<sup>2</sup> *IL*2, where 0 < *α<sup>j</sup>* ≤ 1. In this sense, the inductances required to keep the ripples values below a given threshold are provided by Equation (16).

$$L\_1 = \frac{(1 - D)^4 \mathcal{R} T}{\mathfrak{a}\_1 D^2}, \qquad \qquad \qquad L\_2 = \frac{(1 - D)^2 \mathcal{R} T}{\mathfrak{a}\_2}.\tag{16}$$
