## 3 Converter Topology for the Power Modules
### Single Input Double Output Mode  
For the single input double output (SIDO) mode, the PV-module provides the power for the load and battery charging which means for all modes described below, the current in L1 will flow towards the battery. In this mode, the operating modes of the converter which are illustrated in Figure 4, are as follows:  
1. _Stage I_ is illustrated in Figure 4a: switch S3 is turned ON which means the power extracted from the PV will be used to increase the energy stored in L2 i.e., the voltage across inductor L2 is VPV; due to the initial negative current in L1, diode D1 will have to be turned on and if switch S2 is OFF, S1 is conducting through its body diode, the voltage across inductor L1 is \((-\mathrm{V}_{\mathrm{batt}})\) which will cause its current to reduce in amplitude.
2. _Stage II_ is illustrated in Figure 4b: switch S3 is turned OFF while switch S2 is still OFF and S1 is conducting through its body diode. The difference between the currents in inductors L1 and L2 will flow to the output via D2 which means the voltage potential at the common point between L1 and L2 is clamped to Vo. Therefore, the voltage across inductor L1 is \((\mathrm{V}_{\mathrm{O}}-\mathrm{V}_{\mathrm{batt}})\) and the voltage across inductor L2 is \((\mathrm{V}_{\mathrm{PV}}-\mathrm{V}_{\mathrm{O}})\).
3. _Stage III_ is illustrated in Figure 4c: switch S3 is OFF while switch S2 is ON and S1 is OFF. This has the following implications: the battery is disconnected (\(\mathrm{i}_{\mathrm{b}}=0\)), the voltage across the inductor L1 is V\({}_{\mathrm{O}}\), therefore the current in L1 increases in amplitude and the current in L2 continue to decrease as the voltage across the inductor L2 remains \((\mathrm{V}_{\mathrm{PV}}-\mathrm{V}_{\mathrm{O}})\).  
Figure 5 shows the typical waveforms of the converter during SIDO mode. By applying the voltage balance principle to the inductor L2, the following equations can be obtained:  
\[\mathrm{V}_{\mathrm{PV}}\mathrm{d}_{3}+(\mathrm{V}_{\mathrm{PV}}-\mathrm{V}_{ \mathrm{O}})(1-\mathrm{d}_{3})=0 \tag{7}\]  
Therefore,  
\[\mathrm{V}_{\mathrm{PV}}=(1-\mathrm{d}_{3})\mathrm{V}_{\mathrm{O}} \tag{8}\]  
where \(\mathrm{d}_{3}\) is the duty ratio of switch S3, \(V_{\mathrm{PV}}\) is the PV-module's voltage and \(\mathrm{V}_{\mathrm{O}}\) is the converter output voltage.  
Similarly, by applying the voltage balance principle to inductor L1, the following equations can be obtained:  
\[-\mathrm{V}_{\mathrm{batt}}\mathrm{d}_{3}+(\mathrm{V}_{\mathrm{O}}-\mathrm{V} _{\mathrm{batt}})(1-\mathrm{d}_{3}-\mathrm{d}_{2})+\mathrm{V}_{\mathrm{O}} \mathrm{d}_{2}=0 \tag{9}\]  
Figure 4: Current flow paths for operating stages in single input double output mode: (**a**) Stage I; (**b**) Stage II; and (**c**) Stage III.  
Therefore, the relationship between the battery and output voltages and switches duty cycles defined as:  
\[\mathrm{V_{O}}=\frac{(1-\mathrm{d_{2}})}{(1-\mathrm{d_{3}})}\mathrm{V_{batt}} \tag{10}\]  
where \(\mathrm{d_{2}}\) is the duty ratio of switch S2; as the switches S2 and S1 operate complementary, (10) can be reconstructed as:  
\[\mathrm{V_{O}}=\frac{\mathrm{d_{1}}}{(1-\mathrm{d_{3}})}\mathrm{V_{batt}} \tag{11}\]  
where \(\mathrm{d_{1}}\) is the duty cycle of the switch S1. Based on (8), \(\mathrm{d_{3}}\) controls the PV current and therefore the maximum power point operation of the PV-module independent of the output voltage (as long as it is larger than PV voltage), while \(\mathrm{d_{1}}\) controls the output voltage based on Equation (11) for a defined \(\mathrm{d_{3}}\) and an actual value of battery voltage \(\mathrm{V_{batt}}\).