## 3 Power Loss Analysis of the Cell Balancing System
The main voltage and current waveforms of the bidirectional flyback converter operating in buck mode can be obtained in Figure 6, where \(v_{gs1}\) is the MOSFET \(Q_{1}\) gate signal, \(v_{gs2}\) is the MOSFET \(Q_{2}\) gate signal, \(v_{ds1}\) is the voltage across the MOSFET \(Q_{1}\) from drain to source, \(v_{ds2}\) is the voltage across the MOSFET \(Q_{2}\) from drain to source, \(i_{L}\) is the current passing through the \(L_{m}\), and \(i_{D}\) is the current flowing through the MOSFET Q2. Figure 5: Equivalent circuit diagram of bidirectional flyback converter in (**a**) Interval I (**b**) Interval II. First, the switching loss is loss caused by the current-voltage overlap region during the switching. Because this loss is very difficult to calculate accurately, this research uses the estimation method in the analysis of the MOSFET switching loss proposed by [PERSON] to switch loss estimation [27], as shown in Figure 7 and Equation (1)  
\[P_{sw}=\frac{1}{2}I_{DS}V_{DS}\big{(}t_{off}+t_{on}\big{)}f_{sw}+\frac{1}{2}C_{oss }V_{DS}^{2}f_{sw} \tag{1}\]  
where \(I_{DS}\) is the current on the MOSFET when turning on, \(V_{DS}\) is the voltage across the drain-source of the MOSFET, \(t_{on}\) and \(t_{off}\) are the time required for the MOSFET to switch on and off, \(f_{sw}\) is the frequency of the MOSFET operation, and \(C_{oss}\) is the parasitic capacitor of the MOSFET. In buck mode, due to the synchronous rectification technology in the bidirectional flyback converter, only the MOSFET \(Q_{1}\) will generate switching losses. Figure 6: Illustrated main voltage and current waveforms of the bidirectional flyback converter operating in buck mode. Figure 7: Power loss of MOSFET under switching. Secondly, the conduction loss is caused by the energy transferring through the transformer, the wire resistance, and the switch's on-resistance. The followings are two operating modes in a cycle. Utilizing the switching of MOSFET \(Q_{1}\) and \(Q_{2}\), these two modes will estimate the conduction loss caused by the energy transfer, as shown in Figure 6.  
Interval I [\(t_{0}\)-\(t_{1}\)]:  
In Interval I, the MOSFET \(Q_{1}\) is on, and the field inductor \(L_{m}\) of the transformer stores energy from battery module \(V_{pack}\) via the current \(i_{L}\). Since the current \(i_{L}\) rises linearly, in the loss calculation in this mode, the current \(i_{L}\) is averaged. As shown in Figure 5a, the current will pass through the wire resistance \(R_{s}\) and the on-resistance \(R_{ds1(on)}\) of the MOSFET \(Q_{1}\). The loss formula is given in Equations (2) and (3)  
\[W_{R_{s}}=i_{L}^{2}R_{s}(t-t_{0}) \tag{2}\]  
\[W_{R_{ds1(on)}}=i_{L}^{2}R_{ds1(on)}*(t-t_{0}) \tag{3}\]  
when \(t=t_{1}\), the MOSFET \(Q_{1}\) turns off, and the MOSFET \(Q_{2}\) turns on. This mode ends. Interval II [\(t_{1}\)-\(t_{2}\)]:  
In Interval II, the MOSFET \(Q_{2}\) is on, and the field inductor \(L_{m}\) of the transformer charges the cell \(V_{Bn}\) via the current \(i_{L}\). Since the current \(i_{L}\) decreases linearly, in the loss calculation in this mode, the current \(i_{L}\) is averaged. As shown in Figure 5b, the current will pass through the on-resistance \(R_{ds2(on)}\) of the MOSFET \(Q_{2}\). The loss formula is given in Equation (4). \[W_{R_{ds2(on)}}=\left(\frac{i_{L}}{N}\right)^{2}*R_{ds2(on)}*(t-t_{1}) \tag{4}\]  
when \(t=t_{2}\), the MOSFET \(Q_{2}\) turns off, and the MOSFET \(Q_{1}\) turns on. This mode ends. Through the analysis above, the conduction loss in a cycle can be calculated as follows:  
Figure 3:0  
Figure 3:1  
Figure 3:2  
Figure 3:3  
Figure 3:4  
Besides, the loss of the transformer is divided into the copper loss (\(P_{copper}\)) and the iron loss (\(P_{core}\)). The copper loss is mainly caused by the current passing through the AC impedance \(R_{L(AC)}\) of the transformer. However, \(R_{L(AC)}\) is hard to estimate with a simple mathematical model because it is affected by the skin effect and the proximity effect.