# A Transformer-Less Buck-Boost Grid-Tied Inverter with Low Leakage-Current and High Voltage-Gain
## 3 Operation Principles
### Step-Up Mode  
When the absolute value of the mains voltage \(\mid v_{s}\)(t) \(\mid\) is higher than the input voltage \(V_{in}\), the inverter works in step-up mode. The working principle of this mode is similar to that of a boost converter. The switch \(S_{Bo1}\) (\(S_{Bo2}\)) is controlled by high-frequency SPWM. When the mains voltage \(v_{s}\)(t) is positive (\(0\leq\omega t\leq\pi\)), the switch \(S_{Bu1}\) stays on, and the switch \(S_{Bu2}\) stays off. When the switch \(S_{Bo1}\) is switched on, the diode \(D_{3}\) is switched off, and switch \(S_{1}\) remains on to provide a path for energy delivering. At this moment, the input voltage \(V_{in}\) delivers energy to the inductor \(L_{1}\), and the capacitor \(C_{f}\) provides the energy required by the output. Figure 7a shows the corresponding circuit. The voltage across the inductor \(L_{1}\) is able to be expressed as in Equation (4):  
\[v_{L1}=V_{in}. \tag{4}\]  
When the switch \(S_{Bo1}\) is switched off, the diode \(D_{3}\) is switched on. At this moment, the input source \(V_{in}\) transfers energy to the output terminal and charges the capacitor \(C_{f}\) through the coupled inductors \(L_{1}\), \(L_{3}\) and the switch \(S_{1}\). Figure 7b shows the corresponding circuit. Since the conduction of the switch \(S_{1}\) can connect the mains and the negative end of the input source, the leakage current of the PV array can be suppressed. The voltage across the inductor \(L_{1}\) is able to be presented as in Equation (5):  
\[v_{L1}=\frac{V_{in}-v_{s}(t)}{1+N}, \tag{5}\]  
in which \(N\) is the turn ratio of the coupled inductor and is defined as (\(n_{1}\):\(n_{3}\) = 1:\(N\)). Using the volt-second balance theorem, Equation (6) is able to be found. \[\int\limits_{0}^{d_{B0}T_{S}}V_{in}\;dt+\int\limits_{d_{B0}T_{S}}^{T_{S}}\left( \frac{V_{in}-v_{s}(t)}{1+N}\right)\;dt=0, \tag{6}\]  
where \(d_{B0}\) is the duty ratio of the switch \(S_{B01}\). By rearranging Equation (6), the voltage gain in CCM is shown in Equation (7):  
\[\frac{v_{s}(t)}{V_{in}}=\frac{|V_{P}\sin\omega t|}{V_{in}}=\frac{1+Nd_{B0}}{1- d_{B0}}\;. \tag{7}\]  
The duty ratio \(d_{B0}\) is able to be presented as in Equation (8):  
\[d_{B01}(t)=d_{B0}(t)=\frac{|V_{P}\sin\omega t|-V_{in}}{|V_{P}\sin\omega t|+N \cdot V_{in}}. \tag{8}\]  
When the current of the inductor \(L_{1}\) (\(L_{2}\)) drops to zero during \(S_{B01}\) switched off, the inverter works in discontinuous conduction mode (DCM), and the capacitor \(C_{f}\) provides the energy required by the load. Equation (9) defines the inductance constant \(\tau_{L}\):  
\[\tau_{L}=\frac{L_{p}f_{S}}{R_{L}}, \tag{9}\]  
where the inductor \(L_{1}\) (\(L_{2}\)) is assumed to be the same inductance \(L_{p},f_{S}\) is the frequency of MOSFET switching, and \(R_{L}\) is the resistance of equivalent output load. The voltage gain of DCM is presented as in Equation (10):  
\[\frac{v_{s}(t)}{V_{in}}=\frac{1}{2}+\sqrt{\frac{1}{4}+\frac{d_{B0}^{2}}{2\tau _{L}}}\;. \tag{10}\]  
Figure 7: During the positive half-cycle, the corresponding circuits of step-up mode: (**a**) \(S_{B01}\) ON, \(S_{B02}\) OFF, \(S_{B01}\) ON, \(S_{B02}\) OFF, \(S_{1}\) ON, and \(S_{2}\) OFF; (**b**) \(S_{B01}\) ON, \(S_{B02}\) OFF, \(S_{B01}\) OFF, \(S_{B02}\) OFF, \(S_{1}\) ON, and \(S_{2}\) OFF. If the gain shown in Equation (7) is equal to Equation (10), the inverter will operate in the boundary conduction mode (BCM). Derived from this condition, the inductance constant \(\tau_{LB}\) of BCM is able to be found and expressed as in Equation (11):  
\[\tau_{LB}=\frac{d_{Bo}\left(1-d_{Bo}\right)^{2}}{2(1+N)(1+Nd_{Bo})}. \tag{11}\]  
If the inductance constant \(\tau_{L}\) is smaller than its boundary value \(\tau_{LB}\), the inverter will work in DCM. When the main voltage \(v_{s}\)(t) becomes negative (\(\pi\leq\omega t\leq 2\pi\)) and working in the step-up mode, the roles of the switches \(S_{Bu1}\) and \(S_{Bu2}\) are exchanged. The switch \(S_{Bu1}\) stays off, and the switch \(S_{Bu2}\) stays on.