# A Transformer-Less Buck-Boost Grid-Tied Inverter with Low Leakage-Current and High Voltage-Gain
## 3 Operation Principles
### Step-Down Mode  
When the absolute value of the mains voltage \(\mid v_{s}\)(t) \(\mid\) is lower than the input voltage \(V_{in}\), the step-down mode is performed. During the positive half-cycle (\(0\leq\omega t\leq\pi\)), the switch \(S_{Bu2}\) and the switches \(S_{Bo1}\), \(S_{Bo2}\) remain in off state, and the switch \(S_{1}\) remains in on state to provide a path for energy transfer. At this time, the power switch \(S_{Bu1}\) is controlled by high-frequency SPWM switching, and its duty ratio \(d_{Bu1}\)(t) is able to be presented as in Equation (1):  
\[d_{Bu1}(t)=d_{Bu}(t)=\frac{V_{P}\sin\omega t}{V_{in}},\ (V_{P}\sin\omega t\leq V _{in}). \tag{1}\]  
when the switch \(S_{Bu1}\) is switched on, the diode \(D_{1}\) turns off. Currently, the input voltage \(V_{in}\) charges the inductors \(L_{1}\) and \(L_{3}\) in series through the switches \(S_{Bu1}\) and \(S_{1}\), and it transmits energy directly to the output. Figure 5a shows the corresponding circuit. When the switch \(S_{Bu1}\) is switched off and the diode \(D_{1}\) is turned on, the inductors \(L_{1}\) and \(L_{3}\) in series release energy to the output. Figure 5b shows the corresponding circuit. The voltage gain of continuous conduction mode (CCM) is able to be presented as in Equation (2):  
\[\frac{\mid v_{s}(t)\mid}{V_{in}}=d_{Bu}(t). \tag{2}\]  
It is worth mentioning that since the switch \(S_{1}\) can connect the mains and the negative end of the input source, the leakage current of the PV array is able to be suppressed.  
Figure 4: The timing diagram of working modes and switch driving signals.  
Figure 5: During the positive half-cycle, the corresponding circuits of step-down mode: (**a**) \(S_{Bul}\) ON, \(S_{Bul2}\) OFF, \(S_{Bol}\) OFF, \(S_{Bol}\) OFF, \(S_{Bul2}\) OFF, \(S_{1}\) ON, and \(S_{2}\) OFF; (**b**) \(S_{Bol}\) OFF, \(S_{Bul2}\) OFF, \(S_{Bol}\) OFF, \(S_{Bul2}\) OFF, \(S_{1}\) ON, and \(S_{2}\) OFF.  
When the mains voltage \(v_{s}\)(t) becomes negative (\(\pi\leq\omega t\leq 2\pi\)), the switches \(S_{Bu1}\), \(S_{Bo1}\), and \(S_{Bo2}\) remain off, and the switch \(S_{2}\) is turned on to provide a path for energy transmission. At this moment, the switch \(S_{Bu2}\) is controlled by high-frequency SPWM. Its duty ratio is the same as that of the switch \(S_{Bu1}\) in the positive half-cycle, which can be presented as in Equation (3):  
\[d_{Bu2}(t)=d_{Bu}(t)=\frac{|V_{P}\sin\omega t|}{V_{in}},\ (|V_{P}\sin\omega t| \leq V_{in}). \tag{3}\]  
when the switch \(S_{Bu2}\) is switched on, the diode \(D_{2}\) is switched off. At this moment, the input source \(V_{in}\) charges the inductors \(L_{2}\), \(L_{4}\) in series through the power switches \(S_{Bu2}\) and \(S_{2}\), and it transfers energy directly to the output. Figure 6a shows the corresponding circuit. When the switch \(S_{Bu2}\) is switched off, the diode \(D_{2}\) is switched on, and the inductors \(L_{2}\), \(L_{4}\) in series release energy to the output. Figure 6b shows the corresponding circuit, and the voltage gain of CCM is the same as Equation (2). In this state, the path of connecting the mains and the negative end of the input voltage is provided by the power switch \(S_{2}\), which can also achieve the effect of suppressing the leakage current of the PV array.