## 2 Background  
### Existing Hybrid Power Converters  
A wide variety of DC-DC converter topologies have been developed for power electronics over the past few years [32; 33]. Interleaved converters, like the interleaved boost converter, have proven successful among the various converters. An important aspect of interleaved converters is that all parallel converters need to have an identical duty cycle or voltage gain. Parallel converters share the same input voltage. Therefore, a severe power imbalance occurs when one converter increases the voltage more than the other. A key feature of interleaved converters is that they have a zero input-current ripple at their operation points. For instance, a double-phase interleaved boost converter exhibits zero input-current ripple. Furthermore, hybrid interleaved converters such as the one described in [34] are being investigated. It is feasible to select a zero input-current ripple for this converter. This enables it to operate in the best operating range. Unlike the conventional interleaving converters with an identical duty cycle for all converters, a hybrid converter [34] employ a PWM strategy to assign a duty cycle to each converter. The duty cycles are dependent. Although hybrid converters may have independent duty cycles, an equal gain in voltage may be obtained by any set of duty cycles. This results in various input ripple currents. The hybrid converters [24; 27] are illustrated in Figure 1a. It is a combination of a Cuk converter and a boost converter. Consequently, their input voltages are identical, and the load is connected differentially from the Cuk output to the boost converter output. Owing to the Cuk converter's negative output voltage, the voltage gain is more significant when the two boost converters are interleaved. This is because they have independent outputs and may have different duty cycles. The PWM strategy is illustrated in Figure 1b. Meanwhile, Figure 1c shows the current waveform in a duty cycle where the ripple in the input-current is zero. The mathematical model of the hybrid power converter can be summarized as follows:  
\[V_{\text{C1}}=V_{in}\frac{1}{1-D_{1}};\quad I_{L1}=I_{O}\frac{1}{1-D_{1}} \tag{1}\]  
\[V_{\text{C2}}=V_{in}\frac{1}{1-D_{2}};\quad I_{L2}=I_{O}\frac{D_{2}}{1-D_{2}} \tag{2}\]  
\[V_{\text{C3}}=V_{in}\frac{1}{1-D_{2}};\quad I_{L3}=I_{O} \tag{3}\]  
The duty cycle for each transistor s\({}_{\text{x}}\) is determined by \(D_{\text{x}}\) (x = 1, 2). The voltage \(V_{\text{Cy}}\) (y = 1, 2, 3) represents the voltage across each capacitor \(C_{\text{y}}\). The current \(I_{Lx}\) (z = 1, 2, 3) is equal to every inductor's current \(L_{\text{z}}\). \(I_{o}\) is the output current. A series connection of \(C_{1}\) and \(C_{3}\) yields the output voltage. The output results are expressed as follows, based on the load resistance \(R\):  
\[V_{o}=V_{in}\frac{1}{1-D_{1}}+V_{in}\frac{D_{2}}{1-D_{2}};\quad I_{o}=\frac{ V_{O}}{R} \tag{4}\]  
As part of the PWM strategy of a hybrid converter [31], the duty cycle of the converter is s\({}_{\text{2}}\), d = d\({}_{\text{2}}\). Therefore, the duty cycle of s\({}_{\text{1}}\) (d\({}_{\text{1}}\)) is expressed as a fraction of d multiplied by k, as follows:  
\[\text{d}=\text{d}_{\text{2}};\text{d}_{\text{1}}=\text{kd} \tag{5}\]  
The following equation can be obtained by applying similar properties to inductors:  
\[\text{L}_{\text{1}}=\text{kL}_{\text{2}} \tag{6}\]k is determined by the duty cycle wherein zero input-current ripples are preferred. The duty cycle is called \(D_{Z}\), in which a mathematical equation is written as the following:  
\[k=\frac{1-D_{Z}}{D_{Z}} \tag{7}\]  
A detailed description of the PWM strategy is available in [34]. It is generally straightforward. However, it has drawbacks because it loses a degree of freedom when a duty cycle is proportional to another over the entire range of operations. Considering this limitation in [34], a new PWM strategy was proposed in [31]. The duty cycles can be represented as in Equation (8). Here, k is a variable and not a constant.