## 2 The LVC Converter in Their Traditional Operation
### Mathematical Model in the Traditional Operation  
The converter's mathematical model in the traditional operation was introduced in [18]. It was obtained through the averaging technique with the circuits in Figures 3 and 4 [4,18]. The averaging technique is a method used to simplify complex circuits by reducing them to their average behavior over a certain time period. This is achieved by considering the circuit's average values for current and voltage instead of their individual variations and then representing it with a simplified equivalent circuit. The dynamical model presented in [18] is (2)-(5).  
\[L_{a}\frac{di_{La}}{dt}=dv_{in}-(1-d)v_{Ca} \tag{2}\]  
\[L_{b}\frac{di_{LB}}{dt}=dv_{in}-(1-d)v_{Cb} \tag{3}\]  
\[C_{a}\frac{dv_{Ca}}{dt}=(1-d)i_{La}-i_{out} \tag{4}\]  
\[C_{b}\frac{dv_{Cb}}{dt}=(1-d)i_{LB}-i_{out} \tag{5}\]  
where the output current \(i_{out}\) can be calculated in terms of the output voltage (1) and the load resistance \(R\) as (6).  
\[i_{out}=\frac{v_{out}}{R} \tag{6}\]  
Figure 4: Equivalent circuits or switching states of the LVC converter in the traditional operation: (**a**) Transistors on; and (**b**) transistors off.  
Figure 3: LVC converter topology schematic drawn in a vertical manner.  
The model which was previously introduced in [18] is a fourth-order nonlinear model. From the dynamic model, the equilibrium operation or steady state can be obtained. In order to do so, we must consider the small ripple approximation [4].  
By taking into account that the state variables in Equations (2)-(5) are at equilibrium, where their derivatives (with respect to the time) are equal to zero, we can determine the voltage across the capacitors and the current through the inductors in a steady state, resulting in (7)-(10)  
\[V_{Ca}=\frac{D}{1-D}V_{in} \tag{7}\]  
\[V_{Cb}=\frac{D}{1-D}V_{in} \tag{8}\]  
\[I_{Lb}=\frac{1}{(1-D)}\frac{V_{out}}{R} \tag{9}\]  
\[I_{La}=\frac{1}{(1-D)}\frac{V_{out}}{R} \tag{10}\]  
The output voltage \(V_{out}\), can be expressed from (1), (7), and (8), as (11).  
\[V_{out}=V_{in}\frac{1+D}{1-D} \tag{11}\]  
The converter has a voltage gain larger than the traditional boost converter (for the same duty cycle), which is an advantage, as described in [18]. However, their main advantage can be corroborated by (7) and (8), as the voltage across the capacitors is lower than the output voltage (11), making it an additional advantage over other state-of-the-art converters.  
The energy stored in the capacitors can be expressed using Equation (12).  
\[E_{C}=\frac{CV_{C}^{2}}{2} \tag{12}\]  
Since the capacitor's stored energy is a function of its voltage squared, a relatively small reduction in the voltage results in a significant reduction in the stored energy, which is an advantage since the volume of the capacitor is proportional to its stored energy [14; 15; 16].  
The selection of inductors and capacitors is performed with the traditional method based on the voltage ripple for capacitors and current ripple for inductors [4], as introduced in [18]; this can be summarized as (13)-(16).  
\[L_{a}=\frac{V_{in}}{2\Delta i_{La}}DT_{S} \tag{13}\]  
\[L_{b}=\frac{V_{in}}{2\Delta i_{Lb}}DT_{S} \tag{14}\]  
\[C_{a}=\frac{I_{out}}{2\Delta v_{Ca}}(1-D)T_{S} \tag{15}\]  
\[C_{b}=\frac{I_{out}}{2\Delta v_{Cb}}(1-D)T_{S} \tag{16}\]  
Reference [18] also describes the selection of semiconductors and the real gain of the converter considering losses. The traditional method also considers both inductors as equal \(L_{a}=L_{b}\) and both capacitors as also equal \(C_{a}=C_{b}\).