# A New SEPIC-Based DC-DC Converter with Coupled Inductors Suitable for High Step-Up Applications
## 2 Steady State Analysis of the Proposed Hybrid SEPIC-Based Converter with Coupled Inductors  
In [24], Prof. [PERSON] et al. proposed a new hybrid SEPIC converter with a switching Up 3 structure. The architecture of this topology was analyzed in [24; 62] and is presented in Figure 1.  
Figure 1: SEPIC hybrid converter with a switching Up 3 structure presented in [24]. The idea of developing the novel converter proposed in this paper started by coupling the inductors \(L_{1}\) and \(L_{2}\). The authors also used this technique in other published research [63; 64; 65; 66]. The coupling is assumed to be ideal and the numbers of turns corresponding to \(L_{1}\) and \(L_{2}\) are denoted as \(N_{1}\) and \(N_{2}\) respectively. The transformer ratio \(n\) defined as:  
\[n=\frac{N_{2}}{N_{1}} \tag{1}\]  
can be higher, lower, or equal to unity. In this paper, the case when the transformer ratio is assumed to be lower than unity will be analyzed. It can be shown that, in this situation, diode \(D_{3}\) is always off and therefore it can be removed from the circuit. Taking this into account, the proposed topology is presented in Figure 2. It is obvious that it is simpler and cheaper than the previous one because one magnetic core and one diode are eliminated. The novel converter obtained consists of one active switch, three passive switches, an individual inductor, and a pair of coupled inductors, combined with one internal and one output capacitor. In order to analyze the converter, all components are assumed to be ideal. Because the inductors are perfectly coupled, they will be modelled by an ideal transformer (\(IT\)) that has the transformer ratio \(n\), together with a magnetizing inductor, \(L_{M}\), equal to \(L_{1}\). In Figure 3, the equivalent schematic is presented, and all the analyses will be developed on it considering the CCM operation. Noncapital letters will denote instantaneous variables, and capital letters will specify dc values or maximum values. Considering the notations used in Figure 3, the ideal transformer equations are:  
\[v_{2}=n\times v_{1} \tag{2}\]  
Figure 3: Equivalent schematic of the proposed hybrid SEPIC-based dc-dc converter with coupled inductors. Figure 2: The proposed hybrid SEPIC-based converter with coupled inductors. \[i_{1}+n\times i_{2}=0 \tag{3}\]  
The transistor is controlled by a pulse-width modulated (PWM) signal with a fixed frequency \(f_{\rm s}\) and a variable duty cycle \(D\). The converter goes through two topological states, depicted in Figure 4a,b. In Figure 4a, in the first topological state, which lasts from 0 to \(D\cdot T_{\rm s}\), both transistor \(S\) and diode \(D_{1}\) are on, while in Figure 4b, associated to the second topological state, which lasts from \(D\cdot T_{\rm s}\) to \(T_{\rm s}\), the diodes \(D_{2}\) and \(D_{4}\) are on. The dc analysis is performed considering the small ripple assumption for the capacitor voltages and inductor currents. To calculate the dc voltages for the inner and output capacitors, the volt-second balance principle is invoked, and it can be written as in the following equations:  
\[D\times\frac{V_{g}}{n}+(1-D)\times\left(\frac{V_{g}-V_{\rm C1}-V_{\rm C2}}{1+n }\right)=0 \tag{4}\]  
\[D\times V_{\rm C1}+(1-D)\times(-V_{\rm C2})=0 \tag{5}\]  
Figure 1:0 The equivalent circuits of the proposed hybrid SEPIC-based dc-dc converter with coupled inductors in CCM: (**a**) first topological state; (**b**) second topological state. By solving Equations (4) and (5), the dc voltages across the internal capacitor, \(V_{\text{C1}}\), and across the output capacitor, \(V_{\text{C2}}\), are found as:  
Figure 1:1  
Figure 1:2  
Because \(V_{o}=V_{\text{C2}}\), the ideal static conversion ratio can be obtained from (7):  
Figure 1:3  
The next step is to invoke the charge balance principle to calculate the dc magnetizing inductor current, \(I_{LM}\), and the dc output inductor current, \(I_{L3}\):  
Figure 1:4  
Figure 1:5  
From (9) and (10), it follows that:  
Figure 1:6  
Figure 1:7