## 3 Proposed Converter Topology
### Derivation of Voltage Gain  
The reference voltage is calculated assuming that all the components are perfect. Let V1 be the voltage on the negative side, and V2 is the value on the positive terminal. S3 and S4 are maintained off, whereas S1 and S2 are maintained at a specific rate with a duty cycle of D. S1 and S2 are retained in the OFF state during the buck operations. In contrast, S3 and S4 are driven at a particular intensity with a load current. The suggested bi-directional DC-DC converter's enhanced operating is investigated in three phases. Switches S3 and S4 employ antiparallel diodes D3 and D4 to free up the energy contained in inductances L1 and L2. Mode 1: S1 and S2 are ON  
This operating method is alternative modes 2 and 3. D3 and D4 are reverse-biased during this state, and the capacitance \(C_{0}\) maintains the load. The charges on inductances L1 and L2 grow, as does the power through the inductances. Mode 2: When S1 is ON, and S2 is OFF  
The voltage across the inductance \(L_{a}\) throughout this phase is \(V_{L_{a}}=V_{a}\). Here, \(V_{a}\) is the voltage applied, and \(D\) indicates the platform's duty frequency. D4 is on, and D3 is off. \(C_{a}\) supplies the load, and the corresponding equation is given by,  
\[V_{a}+V_{L_{b}}-V_{b}=0 \tag{4}\]  
The input and output voltage are denoted \(V_{a}\) and \(V_{b}\), and the voltage across the inductor is denoted \(V_{L_{b}}\). Mode 3: When S1 is OFF and S2 is ON  
Transistor D3 is ON, and transistor D4 is OFF. The output voltage equals the inductor voltage and it is represented as \(V_{L_{b}}=V_{b}\). By simply applying KVL, Equation (5) is obtained. \[V_{a}-V_{L_{a}}-V_{C}=0\]The input voltage is denoted \(V_{a}\), the capacitor voltage is denoted \(V_{C}\), and the voltage across the inductor is denoted \(V_{L_{b}}\). The mean voltage across inductances \(L_{a}\) and \(L_{b}\) is zero during a period T, as per voltage-second balance. Inductor \(L_{a}\) charges throughout the ON condition of the power switch, as indicated by the current \(i_{L_{a}}\) and discharging during the OFF phase, as indicated by the lowering current \(i_{L_{a}}\). For the inductance \(V_{a}\), volt-sec balancing is used. The condition is expressed in Equation (6). \[V_{a}\times(D*T)-(V_{a}-V_{C})\times(1-D)T=0 \tag{6}\]  
The input and capacitor voltage are denoted \(V_{a}\) and \(V_{C}\). The duty cycle and the propagation time are expressed as \(D\)and\(T\). Changing the phrases around, and expressed in Equation (7)  
\[V_{C}=\frac{V_{a}}{1+D}+\frac{V_{a}}{1-D} \tag{7}\]  
The input voltage is denoted as \(V_{a}\) and the duty cycle is expressed as \(D\). Inductance \(L_{b}\) charged during the ON phase switch S2, as indicated by the growing current \(i_{L_{b}}\) and discharging during the OFF phase, as indicated by the lower current \(i_{L_{b}}\), for the inductance \(L_{b}\), and denoted in Equation (8). \[V_{a}\times(D*T)+[V_{a}+V_{C}+V_{b}]\times(1-D)*T=0 \tag{8}\]  
The voltage across the input and resistor \(b\) is denoted as \(V_{a}\) and \(V_{b}\), and the propagation period is \(T\). Equation (8) can be rearranged and expressed in Equation (9). \[V_{a}\times(D*T)+\left[V_{a}+\frac{V_{a}}{1+D}+V_{b}\right]\times(1-D)*T=0 \tag{9}\]  
Changing the phrases around, the condition is expressed in Equation (10)  
\[\frac{V_{b}}{V_{a}}=\frac{1}{1-D}+\frac{1+D}{D} \tag{10}\]  
The duty cycle is expressed as \(D\). The voltage across the input and resistor \(b\) is denoted as \(V_{a}\) and \(V_{b}\). The state of charging (SoC) is computed for the batteries. The SoC is varied from 0% to 100%. When the battery is fully charged, the SoC is 100%, and when the battery is fully drained, the SoC is 0%. The charging life should be reduced so that the 100% SoC is achieved faster, and the discharging time should be higher so that the SoC takes more time to reach 0% from 100%. Placing the transistor across the switch reduces the flow of current in the cutoff region, which reduces the battery leakage. By adjusting the duty cycle, the SoC of the battery is controlled and adjusted.