# A Transformer-Less Buck-Boost Grid-Tied Inverter with Low Leakage-Current and High Voltage-Gain
## 4 Loss Analysis
The switching on losses of the MOSFETs \(S_{Bo1}\) and \(S_{Bo2}\) are able to be expressed as in Equation (31):  
\[P_{con,L3}=P_{con,L4}=\frac{2}{T_{ac}}\left[\sum\limits_{n=1}^{N_{ tem}}[i_{L3,Bu}(\frac{n}{N_{s}}\cdot T_{ac})]^{2}\cdot R_{ESR,L1}\cdot T_{\rm S}+\sum\limits_{n=N_{ tem}}^{N_{s}/4}[i_{L3,Bo}(\frac{n}{N_{s}}\cdot T_{ac})]^{2}\cdot R_{ESR,L1}\cdot T_{\rm S}\right], \tag{21}\]1  
in which \(R_{Cau}\) is the resistance of gate-loop during the on state, \(Q_{GS}\) is the charge between gate and source, \(Q_{GD}\) is the charge between gate and drain, \(V_{mp,on}\) is the Miller voltage of the on state, and \(V_{th}\) is the threshold voltage of gate. Additionally, the switching off losses of the MOSFETs \(S_{Bo1}\) and \(S_{Bo2}\) are able to be calculated as in Equation (32):\[P_{con,L3}=P_{con,L4}=\frac{2}{T_{ac}}\left[\sum\limits_{n=1}^{N_{ tem}}[i_{L3,Bu}(\frac{n}{N_{s}}\cdot T_{ac})]^{2}\cdot R_{ESR,L1}\cdot T_{\rm S}+\sum\limits_{n=N_{ tem}}^{N_{s}/4}[i_{L3,Bo}(\frac{n}{N_{s}}\cdot T_{ac})]^{2}\cdot R_{ESR,L1}\cdot T_{\rm S}\right], \tag{21}\]2  
in which \(R_{Gmf}\) is the resistance of the gate-loop during off state, and \(V_{mp,off}\) is the Miller voltage of off state. Furthermore, using similar calculations, the losses of the switch \(S_{Bu1}\) (\(S_{Bu2}\)) during switching on and switching off are able to be calculated respectively as in Equations (33) and (34):  
\[P_{con,L3}=P_{con,L4}=\frac{2}{T_{ac}}\left[\sum\limits_{n=1}^{N_{ tem}}[i_{L3,Bu}(\frac{n}{N_{s}}\cdot T_{ac})]^{2}\cdot R_{ESR,L1}\cdot T_{\rm S}+\sum\limits_{n=N_{ tem}}^{N_{s}/4}[i_{L3,Bo}(\frac{n}{N_{s}}\cdot T_{ac})]^{2}\cdot R_{ESR,L1}\cdot T_{\rm S}\right], \tag{21}\]3  
\[P_{con,L3}=P_{con,L4}=\frac{2}{T_{ac}}\left[\sum\limits_{n=1}^{N_{ tem}}[i_{L3,Bu}(\frac{n}{N_{s}}\cdot T_{ac})]^{2}\cdot R_{ESR,L1}\cdot T_{\rm S}+\sum\limits_{n=N_{ tem}}^{N_{s}/4}[i_{L3,Bo}(\frac{n}{N_{s}}\cdot T_{ac})]^{2}\cdot R_{ESR,L1}\cdot T_{\rm S}\right], \tag{21}\]4  
The total switch switching loss is able to be calculated from Equations (31)-(34) and presented as in Equation (35):  
\[P_{con,L3}=P_{con,L4}=\frac{2}{T_{ac}}\left[\sum\limits_{n=1}^{N_{ tem}}[i_{L3,Bu}(\frac{n}{N_{s}}\cdot T_{ac})]^{2}\cdot R_{ESR,L1}\cdot T_{\rm S}+\sum\limits_{n=N_{ tem}}^{N_{s}/4}[i_{L3,Bo}(\frac{n}{N_{s}}\cdot T_{ac})]^{2}\cdot R_{ESR,L1}\cdot T_{\rm S}\right], \tag{21}\]5  
in which \(P_{Coss,Bo1}\) and \(P_{Coss,Bu1}\) are the losses of output capacitance of MOSFETs \(S_{Bo1}\) and \(S_{Bu1}\), respectively. In accordance with the voltage between drain and source, the characteristic curve illustrated in the datasheet of MOSFET is able to be used for the estimation of both \(P_{Coss,Bo1}\) and \(P_{Coss,Bu1}\). Based on above loss analyses, an example with 220 V\({}_{\text{rms}}\) output voltage and 500 W output power (\(V_{M}=312\) V, \(R=96.8\)\(\Omega\)) is taken to draw the curves of power losses. The curves of inductor conduction loss, switch conduction loss, switch switching loss, and diode conduction loss are illustrated in Figure 9, in which input voltage is between 100 and 200 V. For evaluation, Figure 9 also shows the curves of a grid-tied system with two-stage energy processing (cascading a boost converter and an H-bridge SPWM inverter). \[P_{con,L3}=P_{con,L4}=\frac{2}{T_{ac}}\left[\sum\limits_{n=1}^{N_{ tem}}[i_{L3,Bu}(\frac{n}{N_{s}}\cdot T_{ac})]^{2}\cdot R_{ESR,L1}\cdot T_{\rm S}+\sum\limits_{n=N_{ tem}}^{N_{s}/4}[i_{L3,Bo}(\frac{n}{N_{s}}\cdot T_{ac})]^{2}\cdot R_{ESR,L1}\cdot T_{\rm S}\right], \tag{21}\]6 \(Cont.\)  
It is able to be understood from Figure 9a; since the investigated inverter does not need to boost the voltage to 400 V, it has lower total conduction loss of inductors. Figure 9b shows the curves of the total conduction loss of the switches based on Equations (24)-(27). When the input voltage is low, the high input current needs to go through the switch \(S_{Bu1}\) or \(S_{Bu2}\) in the step-up mode, resulting in slightly higher conduction loss of the switches.