## 2 Proposed Modular Cell Balancing Circuit
### Analysis of the Previous Cell Balancing Circuits
#### 2.1.2 Cell Balancing Circuit Using SCC  
A cell balancing circuit based on an SCC and operation modes of the circuit are shown in Figure 5 [8; 15; 16; 21]. In order to simplify the analysis for circuit operation mode, it is assumed that the circuit has two cells and \(V_{Cell1}>V_{Cell2}\). When two of four switches (\(S_{a1}\) and \(S_{a2}\) or \(S_{b1}\) and \(S_{b2}\)) in the balancing circuit are turned on or off, the energy is transferred from a cell to another cell via a balancing capacitor (\(C_{b}\)). The switches are controlled by pulse-width-modulated (PWM) signals with fixed duty cycle (50% \(T_{s}\)), as shown in Figure 6. The analysis of the circuit is based on a proposition of voltage of \(Cell_{1}\) and \(Cell_{2}\), as follows:  
Mode 1 [\(t_{0}\), \(t_{1}\)] (See Figure 5b): At \(t_{0}\), \(S_{a1}\) and \(S_{a2}\) are turned on, while \(S_{b1}\) and \(S_{b2}\) are turned off simultaneously. In this mode, the switches \(S_{a1}\) and \(S_{a2}\) conduct. Thus, \(Cell_{1}\) is connected in parallel with \(C_{b}\), as shown in Figure 5b. \(C_{b}\) starts to be charged by \(Cell_{1}\). The voltage across \(C_{b}\) (\(v_{Cb}\)) starts increasing, and current flows of \(C_{b}\) (\(i_{Cb}\)) starts decreasing, as shown in Figure 6. The instantaneous voltage and current of the balancing capacitor in this mode can be expressed [30] by Equations (10) and (11)  
\[v_{Cb}(t)=(v_{Cell1}-v_{Chmin}).\left(1-e^{\frac{-t}{R_{Ch}+b}}\right)+v_{Chmin} \tag{10}\]  
\[i_{Cb}(t)=\frac{v_{Cell1}-v_{Chmin}}{R_{Ch}}e^{\frac{-t}{R_{Ch}+b}} \tag{11}\]  
where \(R_{Ch}\) is the sum of the equivalent series resistance of balancing capacitor \(R_{\text{ESR}}\), the turn-on resistance of two switches \(S_{\text{a1}}\), \(S_{\text{a2}}\).  
Mode 2 [\(t_{1}\), \(t_{2}\)] (See Figure 5c): At \(t_{1}\), \(S_{\text{a1}}\) and \(S_{\text{a2}}\) are turned off, and \(S_{\text{b1}}\) and \(S_{\text{b2}}\) are turned on simultaneously. In this mode, the switches \(S_{\text{b1}}\) and \(S_{\text{b2}}\) conduct. Thus, \(Cell_{2}\) is connected in parallel with \(C_{b}\), as shown in Figure 5c. The energy of \(C_{b}\) starts to be discharged to \(Cell_{2}\). The current of \(C_{b}\) begins to boost in the opposite direction as represented in Figure 6. The instantaneous voltage across and the current of \(C_{b}\) in this mode are respectively expressed in Equations (12) and (13).  
Figure 5: The switched capacitor circuit for two cells [21]: (**a**) Circuit description; (**b**) Mode 1; (**c**) Mode 2.  
Figure 6: Operational waveforms of the switched-capacitor circuit.  
\[v_{Cb}(t)=v_{Chmin}-(v_{Cbmax}-v_{Cdl2}).\cdot\left(1-e^{\frac{-t}{R_{dis} \cdot C_{b}}}\right) \tag{12}\]  
\[i_{Cb}(t)=-\frac{v_{Cbmax}-v_{Cdl2}}{R_{dis}}e^{\frac{-t}{R_{dis}\cdot C_{b}}} \tag{13}\]  
where \(R_{dis}\) is the total of the equivalent series resistance of balancing capacitor \(R_{\text{ESR}}\), and the turn-on resistance of two switches \(S_{\text{b1}}\) and \(S_{\text{b2}}\)  
This circuit has some advantages [8]; for example, it does not require voltage sensors or closed-loop control. Therefore, this circuit is simple to implement with low loss. However, in the case of a large number of cells and/or the high unbalance of voltage between the cells, the equalization time is long [21]. Therefore, the novel MCBC needs to be considered carefully to reduce the mentioned disadvantages as shown in Section 2.2.