# A Transformer-Less Buck-Boost Grid-Tied Inverter with Low Leakage-Current and High Voltage-Gain
## 4 Loss Analysis
\[P_{con,L3}=P_{con,L4}=\frac{2}{T_{ac}}\left[\sum\limits_{n=1}^{N_{ tem}}[i_{L3,Bu}(\frac{n}{N_{s}}\cdot T_{ac})]^{2}\cdot R_{ESR,L1}\cdot T_{\rm S}+\sum\limits_{n=N_{ tem}}^{N_{s}/4}[i_{L3,Bo}(\frac{n}{N_{s}}\cdot T_{ac})]^{2}\cdot R_{ESR,L1}\cdot T_{\rm S}\right], \tag{21}\] \[P_{con,Ls}=\frac{4}{T_{ac}}\left[\sum\limits_{n=1}^{N_{ tem}}[i_{Ls,Bu}(\frac{n}{N_{s}}\cdot T_{ac})]^{2}\cdot R_{ESR,Ls}\cdot T_{\rm S}+\sum\limits_{n=N_{ tem}}^{N_{s}/4}[i_{Ls,Bo}(\frac{n}{N_{\rm S}}\cdot T_{ac})]^{2}\cdot R_{ESR,Ls}\cdot T_{\rm S}\right], \tag{22}\]  
where \(N_{tm}\) is the amount of high-frequency switching cycle within the time interval of \(t=0\sim t_{m}\), and \(N_{s}\) is the amount of high-frequency switching cycle within one AC cycle \(T_{ac}\). The equivalent series resistance (ESR) of each inductor is expressed as \(R_{ESR,L1}\), \(R_{ESR,L3}\), and \(R_{ESR,Ls}\), respectively. Finally, the total inductor conduction loss of the inverter is able to be calculated from Equations (20)-(22) and presented as in Equation (23):  
\[P_{con,L\_total}=P_{con,L1}\times 2+P_{con,L3}\times 2+P_{con,Ls}. \tag{23}\]  
The conduction loss of each MOSFET is able to be calculated as in Equations (24)-(26):  
\[P_{con,Bu1}=P_{con,Bu2}=\frac{2}{T_{ac}}\left[\sum\limits_{n=1}^{N_{ tem}}[i_{s}(\frac{n}{N_{s}}\cdot T_{ac})]^{2}\cdot R_{DS,on}\cdot\left[d_{Bu}( \frac{n}{N_{s}}\cdot T_{ac})\right]^{2}\cdot T_{S}+\sum\limits_{n=N_{tm}}^{N_{ s}/4}[i_{L1,Bo}(\frac{n}{N_{s}}\cdot T_{ac})]^{2}\cdot R_{DS,on}\cdot T_{\rm S} \right], \tag{24}\]  
\[P_{con,Bo1}=P_{con,Bo2}=\frac{2}{T_{ac}}\left[\sum\limits_{n=N_{tm}}^{N_{s}/4}[ i_{L1,Bo}(\frac{n}{N_{s}}\cdot T_{ac})]^{2}\cdot R_{DS,on}\cdot d_{Bo}(\frac{n}{N_{s}} \cdot T_{ac})\cdot T_{\rm S}+\sum\limits_{n=1}^{N_{s}/4}[i_{s}(\frac{n}{N_{s}} \cdot T_{ac})]^{2}\cdot R_{DS,on}\cdot T_{\rm S}\right], \tag{25}\]  
\[P_{con,S1}=P_{con,S2}=\frac{2}{T_{ac}}\left[\sum\limits_{n=1}^{N_{s}/4}[i_{s}( \frac{n}{N_{s}}\cdot T_{ac})]^{2}\cdot R_{DS,on}\cdot T_{\rm S}\right], \tag{26}\]  
where \(R_{DS,on}\) is the conduction resistance of each MOSFET. The total switch conduction loss of the inverter is able to be calculated from Equations (24)-(26) and is presented as in Equation (27):  
\[P_{con,S\_total}=P_{con,Bu1}\times 2+P_{con,Bo1}\times 2+P_{con,S1}\times 2. \tag{27}\]  
The conduction losses of the power diodes \(D_{1}\), \(D_{2}\), \(D_{3}\), and \(D_{4}\) are able to be calculated as in Equations (28) and (29):  
\[P_{con,D1}=P_{con,D2}=\frac{2}{T_{ac}}\left[\sum\limits_{n=1}^{N_{tm}}i_{L1, Bu}(\frac{n}{N_{s}}\cdot T_{ac})\cdot V_{\rm F}\cdot\left[1-d_{Bu}(\frac{n}{N_{s}} \cdot T_{ac})\right]\cdot T_{\rm S}\right], \tag{28}\]  
\[P_{con,D3}=P_{con,D4}=\frac{2}{T_{ac}}\left[\sum\limits_{n=1}^{N_{tm}}i_{L3, Bu}(\frac{n}{N_{s}}\cdot T_{ac})\cdot V_{\rm F}\cdot T_{\rm S}+\sum\limits_{n=N_{tm}}^{N_{s}/4}i_{L3, Bo}(\frac{n}{N_{s}}\cdot T_{ac})\cdot V_{\rm F}\cdot\left[1-d_{Bo}(\frac{n}{N_{s}} \cdot T_{ac})\right]\cdot T_{\rm S}\right], \tag{29}\]  
where \(V_{F}\) is the diode voltage during forward biased. The total diode conduction loss of the inverter is able to be calculated from Equations (28) and (29) and presented as in Equation (30)  
\[P_{con,L3}=P_{con,L4}=\frac{2}{T_{ac}}\left[\sum\limits_{n=1}^{N_{ tem}}[i_{L3,Bu}(\frac{n}{N_{s}}\cdot T_{ac})]^{2}\cdot R_{ESR,L1}\cdot T_{\rm S}+\sum\limits_{n=N_{ tem}}^{N_{s}/4}[i_{L3,Bo}(\frac{n}{N_{s}}\cdot T_{ac})]^{2}\cdot R_{ESR,L1}\cdot T_{\rm S}\right], \tag{21}\]0  
In addition to conduction losses, switching losses also affect the conversion efficiency. The switching losses are mainly composed of switching on loss, switching off loss, and output capacitance loss [24].