## 2 Proposed Modular Cell Balancing Circuit
### Proposed Modular Cell Balancing Circuit
#### 2.2.2 Proposed Modular Cell Balancing Circuit
Mode 1 [\(t_{0}\), \(t_{1}\)] (see Figure 10a): At \(t_{1}\), eight switches in intra-module balancers (module 1: \(S_{11}\), \(S_{12}\), \(S_{13}\), \(S_{14}\); module 2: \(S_{21}\), \(S_{22}\), \(S_{23}\), \(S_{24}\)) and \(S_{\text{a1}}\), \(S_{\text{a2}}\) are turned on, while \(S_{\text{b1}}\), \(S_{\text{b2}}\) are turned off simultaneously, as shown in Figure 9. In this mode, the energy is transferred from a high voltage level cell to a low level one inside each module through two MWTs (\(T_{1}\)) and (\(T_{2}\)), respectively. \(V_{\text{Cell11}}\), \(V_{\text{Cell12}}\) are higher than \(V_{\text{aver1}}\), \(V_{\text{Cell13}}\), \(V_{\text{Cell14}}\) in \(M_{1}\), and \(V_{\text{Cell21}}\), \(V_{\text{Cell22}}\) are higher than \(V_{\text{aver2}}\), \(V_{\text{Cell23}}\), \(V_{\text{Cell24}}\) in \(M_{2}\). Therefore, \(i_{11}\), \(i_{12}\) and \(i_{21}\), \(i_{22}\) flow from cells to \(T_{1}\), \(T_{2}\), respectively, and \(i_{13}\), \(i_{14}\) and \(i_{23}\), \(i_{24}\) flow from \(T_{1}\), \(T_{2}\) to the battery cells. This means that, in module 1, the electric charges in \(Cell_{11}\), \(Cell_{12}\) are transmitted to \(Cell_{13}\), \(Cell_{14}\), and in module 2, the electric charges in \(Cell_{21}\), \(Cell_{22}\) are transmitted to \(Cell_{23}\), \(Cell_{23}\). Figure 10: _Cont._  
At the same time, \(S_{\text{a1}}\) and \(S_{\text{a2}}\) conduct, so the \(C_{\text{m}}\) will be charged by the \(V_{\text{M1}}\) through the \(S_{\text{a1}}\) and \(S_{\text{a2}}\) so that the \(V_{\text{Cm}}\) increases gradually to a steady state period. The instantaneous voltage and current of the balancing capacitor in this mode can be given by Equations (14) and (15). \[v_{Cm}(t)=(V_{\text{M1}}-V_{Cmmin}).\left(1-e^{\frac{-t}{C_{\text{Cm}}\cdot t _{\text{m}}}}\right)+V_{Cmmin} \tag{14}\]  
\[i_{Cm}(t)=\frac{V_{M1}-V_{Cmmin}}{R_{Ch}}e^{\frac{-t}{C_{\text{Cm}}\cdot t_{ \text{m}}}} \tag{15}\]  
Mode 2 [\(t_{1}\), \(t_{2}\)] (see Figure 10b): At \(t_{1}\), eight switches in the intra-modules balances are turned off simultaneously, as shown in Figure 9. The voltages across \(L_{\text{m1}}\) and \(L_{\text{m2}}\) are negative as expressed by Equations (16) and (17). The diodes \(D_{1}\) and \(D_{2}\) conduct. Then, \(V_{\text{D1}}\), \(V_{\text{D2}}\) become zero because \(D_{1}\), \(D_{2}\) are ideal devices. The \(L_{\text{m1}}\) and \(L_{\text{m2}}\) are reset through the secondary diodes \(D_{1}\), \(D_{2}\) resulting in a decrease of the magnetizing currents of transformers 1 and 2 (\(i_{\text{Lm1}}\) and \(i_{\text{Lm2}}\)). Figure 7:0  
Figure 7:1  
Mode 3 [\(t_{2}\), \(t_{3}\)] (see Figure 10c): Mode 3 starts when the currents through the magnetizing inductance of two transformers (\(i_{\text{Lm1}}\)) and (\(i_{\text{Lm2}}\)) are equal to zero. No current flows in the MWTs from \(t_{2}\) to \(t_{3}\). Mode 4 [\(t_{3}\), \(t_{4}\)] (see Figure 10d): At \(t_{3}\), the \(S_{\text{a1}}\), \(S_{\text{a2}}\) are turned off, and the \(S_{\text{b1}}\), \(S_{\text{b2}}\) starts to be turned on as shown in Figure 7:2 The voltage \(V_{\text{Cm}}\) is higher than the \(V_{M2}\) such that the energy of \(C_{\text{m}}\) is in a  
Figure 7:3 Operating modes of the proposed MCBC: (**a**) Mode 1; (**b**) Mode 2; (**c**) Mode 3; (**d**) Mode 4.  
discharging condition. Module 2 receives stored energy in \(C_{\text{m}}\) gradually. The instantaneous voltage and current of the module balancing capacitor in this mode are respectively given in Equations (18) and (19). At \(t_{4}\), this operating mode is finished and returns to mode 1 in the next switching period. Figure 7:4  
Figure 7:5  
According to the analysis of the above operation modes, it can be seen that the MWTFC-based intra-module balancer and SCC-based outer-module balancer work separately. The energy is transferred from a high to a low level of both the cells in each module and the modules simultaneously. Thus, the short balancing time, simple control scheme and the low mismatch between the leakage inductances can be obtained due to a small number of cells.