## 4 Design Guidelines and Loss Analysis
### Power Loss Analysis  
The efficiency of the transformer is generally assessed using weighted efficiency, using measures such as CEC efficiency or EU efficiency. Therefore, the power loss in the power semiconductors of the proposed inverter is addressed in this part. Losses can be classified into conduction and switching losses. The average conduction power \(P_{con}\) can be expressed as the voltage drop across the device multiplied by the passing current over the switching period. A comprehensive power loss analysis was carried out in [34; 35]. It is recommended to use MOSFETs in the proposed inverter, and so  
\[V_{on\_mos}=i(t)~{}R_{ds}=\langle i_{g}(t)\rangle~{}R_{ds} \tag{23}\]  
where \(V_{on\_mos}\) is the voltage drop in the MOSFET during conduction, \(i(t)\) is the current passing through the MOSFET, and \(R_{ds}\) is the MOSFET resistance. Hence, the power conduction losses are  
\[P_{con\_S1}=\frac{1}{T_{g}}\int_{0}^{\frac{T_{g}}{2}}\lvert V_{on\_mos1}~{}i_ {S1}(t)\rvert dt \tag{24}\]  
where \(i_{S1}(t)\) is the current in \(S_{1}\) during conduction and \(P_{con\_S1}\) is the average power conduction loss in \(S_{1}\); since \(i_{S1}(t)\) is almost unchanged in a switching cycle, the switching cycle averaging method can be utilized, and thus by substituting \(\langle i_{S1}(t)\rangle\), this yields  
\[P_{con\_S1}=\frac{1}{T_{g}}\int_{0}^{\frac{T_{g}}{2}}\lvert V_{on\_mos1}~{} \langle i_{S1}(t)\rangle\rvert dt \tag{25}\]  
Substituting (12) into (13), then (13) and (23) into Equation (25), we obtain  
\[P_{con\_S1}=\frac{1}{T_{g}}\int_{0}^{\frac{T_{g}}{2}}\lvert D_{m}I_{m}{}^{2}R _{ds}~{}\text{sin}^{3}(\omega t)\rvert d\omega t \tag{26}\]Solving (26) and substituting \(T_{g}=2\pi\) yields  
\[P_{con\_S1}=\frac{2D_{m}I_{m}{}^{2}R_{ds}}{3\pi} \tag{27}\]  
A similar analysis was carried out for the power switches \(S_{2}\), \(S_{3}\), and \(S_{4}\), which yields (28), (29), and (30), respectively, as follows  
\[P_{con\_S2}=\frac{1}{T_{g}}\int_{\frac{T_{g}}{2}}^{T_{g}}\lvert V_{on\_mos2}\;i _{S2}(t)\;d(t)\rvert dt=\frac{2D_{m}{}I_{m}{}^{2}R_{ds}}{3\pi} \tag{28}\]  
\[P_{con\_S3}=\frac{1}{T_{g}}\int_{\frac{T_{g}}{2}}^{T_{g}}\lvert V_{on\_mos3}\;i _{S3}(t)\;d(t)\rvert_{zero}\rvert dt={I_{m}}^{2}R_{ds}\left\{\frac{1}{4}-\frac {2D_{m}}{3\pi}\right\} \tag{29}\]  
\[P_{con\_S4}=\frac{1}{T_{g}}\int_{0}^{\frac{T_{g}}{2}}\lvert V_{on\_mos4}\;i_{ S4}(t)\;d(t)\rvert_{zero}\rvert dt={I_{m}}^{2}R_{ds}\left\{\frac{1}{4}-\frac {2D_{m}}{3\pi}\right\} \tag{30}\]  
Furthermore, in the freewheeling period, the current passed through the diodes \(D_{2}\) in the positive cycle and \(D_{3}\) in the negative cycle, and the conduction loss of the diode can be obtained as  
\[V_{on\_diode}=V_{f}+i(t)\;R_{d} \tag{31}\]  
\[P_{con\_d}=\frac{1}{T_{g}}\int_{0}^{\frac{T_{g}}{2}}\lvert V_{on\_diode}\;i(t )\;d(t)\rvert_{zero}\rvert dt \tag{32}\]  
where \(V_{on\_diode}\) is the diode voltage drop during conduction, \(V_{f}\) is the diode forward voltage, \(R_{d}\) is the diode resistance, \(i(t)\) the current through the device, and \(P_{con\_d}\) the average diode conduction power loss. Substituting (31) into (32) yields Equation (33), and solving for \(T_{g}=2\pi\) and \(i(t)=\langle i_{g}(t)\rangle\) results in (34) as follows  
\[P_{con\_S1}=\frac{1}{T_{g}}\int_{0}^{\frac{T_{g}}{2}}\lvert V_{on\_mos1}~{}i_ {S1}(t)\rvert dt \tag{24}\]0  
\[P_{con\_S1}=\frac{1}{T_{g}}\int_{0}^{\frac{T_{g}}{2}}\lvert V_{on\_mos1}~{}i_ {S1}(t)\rvert dt \tag{24}\]1  
The other losses experienced by the devices are the switching losses caused by the MOSFET's transition states and the reverse recovery of the antiparallel body diodes. However, here, the MOSFET's switching losses are considered while the reverse recovery diode loss is ignored by assuming SiC MOSFETs are used, which have a very fast recovery time.