## 3 The Proposed Operation of the LVC Converter  
In the former operation, transistors switch synchronously; they close and open at the same time, which means that they have the same switching function and the same duty cycle. The duty cycle \(d\) (or \(D\) in steady state) is defined as the time in which a transistor remains closed multiplied by the switching frequency \(f_{S}\) (or divided over the switching period \(T_{S}\)). The switching period (inverse of the switching (frequency) can be divided into two different times: the time a transistor remains closed (\(DT_{S}\)), and the time a transistor remains open \((1-D)T_{S}\). The proposed operation does not intend to change the converter's duty cycle but to have the same duty cycle of transistors with a different switching function. The proposed change consists of using two switching signals (instead of one) based on the two-phase interleaved PWM, similar to that used in the interleaved buck or boost. Both switching signals with equal duty ratios are shifted in phase by 180\({}^{\circ}\); the phase shift is produced with two carrier signals (triangular) which shifted 180\({}^{\circ}\). We can call \(s_{a}\) and \(s_{b}\) to the two switching signals and \(s_{att}\) and \(s_{btri}\) to the triangular carriers. This PWM scheme can be generated with two comparators, as shown in Figure 5. The triangular carrier signals have an amplitude of one. This produces the duty cycle that can be represented with a dc signal of the same magnitude as the intended duty ratio. Figure 5: Two switching signals’ PWM generator with two ideal comparators. The former operation, based on a single switching signal, has two equivalent circuits (as can be seen in Figure 3). In the proposed operation, since transistors have a different switching function, all four combinations of two digital signals may be present, which means that the set \(\{s_{a},s_{b}\}\) may take values of {0, 0}, {0, 1}, {1, 0}, or {1, 1}. Figure 6 shows the equivalent circuits of the converters according to this set of possible values for the switching functions. The number of equivalent circuits increases from two to four. Nonetheless, during the operation, the number of equivalent circuits increases from two to three, and in the particular case in which \(D=0.5\), the converter only has two equivalent circuits. We can expect that the duty ratio may change due to disturbances on either the input voltage or the output current since the duty ratio is used to compensate for those perturbations. We can classify the operation in three situations, namely the instance in which the duty cycle is larger than 0.5 (\(D>0.5\)), the instance in which \(D<0.5\), and the particular instance in which the duty cycle is exactly 0.5. The waveforms related to the two switching signals' PWM are shown in Figure 7. Those signals are generated with the logic of Figure 3.  
Figure 7a shows the PWM generator in which the duty cycle is \(D=0.4\), and Figure 7d shows the corresponding waveforms of the operation in Figure 7a. The switching signal \(s_{u}\) and the triangular carrier signal \(s_{atri}\) are black in color while that corresponding to \(s_{b}\) is in dark gray. We can see that this example corresponds to the situation in which \(D<0.5\), and in this situation, sometimes \(s_{a}=1\) while \(s_{b}=0\), and other times we have the opposite situation, \(s_{a}=0\) while \(s_{b}=1\), and may also have both signals as low (\(s_{a}=1\) while \(s_{b}=0\)), which resembles a dead-time in the firing signals. In this operating condition, the simultaneous switching of signals does not coincide at any time. Figure 7b shows the PWM generator in which the duty cycle is \(D=0.5\), whilst the corresponding waveforms of this operation points are shown in Figure 7e. We can see that this example corresponds to the particular situation in which the duty cycle is exactly \(0.5\), and in this situation, sometimes \(s_{a}=1\) while \(s_{b}=0\), and other times we have the opposite situation, \(s_{a}=0\) while \(s_{b}=1\).