## 2 Proposed Modular Cell Balancing Circuit  
### Analysis of the Previous Cell Balancing Circuits  
#### 2.1.1 Active Cell Balancing Circuit Using the MWT  
Figure 2 shows an active cell balancing circuit using the MWT with the forward converter structure and operating modes [17; 18; 21; 25; 28]. This circuit consists of \(N\) series-connected cells. Each cell in the SCBS is associated with a primary winding of the MWT (\(T\)) and a power switch \(S_{k}\) (\(k=1,\ldots,4\)). A diode (D) on the secondary side is used to reset the magnetizing inductance of the transformer. The MWTFC-based balancer equalizes the voltage of cells in the SCBS by selectively turning the switches \(S_{k}\) on or off according to the constant duty cycle (\(D\)) simultaneously. Three operational modes in one switching period (\(T_{S}\)) of the circuit and the operation waveforms are shown in Figures 2b,c and 3, respectively. In order to simply analyze the operating modes for the circuit, it is assumed that this circuit includes four series-connected cells; The relationship among the voltage of the battery cells is assumed as follows: \(V_{\text{cell1}}>V_{\text{cell2}}>V_{\text{aver}}>V_{\text{cell3}}>V_{\text{ cell4}}\), the sum voltage is \(V_{\text{Total}}=V_{\text{cell1}}+V_{\text{cell2}}+V_{\text{cell3}}+V_{\text{ cell4}}\), and the average voltage is \(V_{\text{aver}}\) where \(V_{\text{cell1}}\), \(V_{\text{cell2}}\), \(V_{\text{cell3}}\), and \(V_{\text{cell4}}\) represent the voltage across \(Cell_{1}\), \(Cell_{2}\), \(Cell_{3}\), and \(Cell_{4}\), respectively. The operating modes of the circuit are presented as follows:  
Mode 1 [\(t_{0}\), \(t_{1}\)] (see Figure 2b): At \(t_{0}\), four switches (\(S_{1}\), \(S_{2}\), \(S_{3}\), and \(S_{4}\)) are turned on simultaneously as shown in Figure 3. In this mode, the energy is transferred from high voltage level cells (i.e., \(Cell_{1}\), \(Cell_{2}\)) to the low one (i.e., \(Cell_{3}\), \(Cell_{4}\)) through the transformer \(T\). In mode 1, there are four loops forming four equations, and each loop equation can be expressed as Equation (1). Figure 4 shows an equivalent circuit. The relationship among the current of a magnetizing inductance \(L_{\text{m}}\), \(i_{\text{Lm}}\), and cell currents \(i_{1}\), \(i_{2}\), \(i_{3}\) and \(i_{4}\), and voltage across \(L_{\text{m}}\), \(V_{\text{Lm}}\), can be expressed as Equations (2) and (3):  
\[V_{TPi}=V_{aver}=V_{Celli}-L_{iki}\frac{d_{i}}{dt} \tag{1}\]  
\[i_{Lm}=i_{1}+i_{2}+i_{3}+i_{4}, \tag{2}\]  
Figure 2: The MWTFC based on the balancing circuit [21]: (**a**) Circuit diagram; (**b**) Mode 1; (**c**) Mode 2. \[V_{Lm}=L_{m}\frac{di_{Lm}}{dt} \tag{3}\]  
where \(i_{1}\), \(i_{2}\), \(i_{3}\) and \(i_{4}\) are the currents of four switches of \(S_{1}\), \(S_{2}\), \(S_{3}\) and \(S_{4}\), respectively. In this mode, four switches are turned on. The value of the leakage inductance and the magnetizing inductance can be used to apply the average voltage. From Equation (1) to Equation (3), the \(V_{TP}\) can be expressed as follows:  
Figure 4: Laplace transform circuit of mode 1. Figure 3: Operation waveforms of the cell balancing circuit. \[V_{TP}=L_{m}\frac{di_{m}}{dt}=L_{m}\frac{d}{dt}(i_{1}+i_{2}+i_{3}+i_{4}), \tag{4}\]  
\[\frac{d}{dt}i_{1}=\frac{V_{Cell1}-V_{Lm}}{L_{lk1}} \tag{5}\]  
\[\frac{d}{dt}i_{2}=\frac{V_{Cell2}-V_{Lm}}{L_{lk2}} \tag{6}\]  
\[V_{TPi}=V_{aver}=V_{Celli}-L_{iki}\frac{d_{i}}{dt} \tag{1}\]0  
\[V_{TPi}=V_{aver}=V_{Celli}-L_{iki}\frac{d_{i}}{dt} \tag{1}\]1  
In this mode, the inrush currents likely to flow in Figure 4 need to be considered. In the case, if there is a high voltage difference among the cells, the inrush currents likely to flow are large. Therefore, NTC thermistor [29] in series with the primary is suggested to limit the current. The NTC thermistor offers high resistance at the beginning of switching and limits the inrush current. After a short time, the NTC thermistor resistance decreases to a low value due to self-heating and it does not affect normal operation.