# Multi-Mode Lithium-Ion Battery Balancing Circuit Based on Forward Converter with Resonant Reset
## 2 Balancing Circuit
### Circuit Analysis  
Figure 4 shows the equivalent circuit of the balancing main circuit in Figure 1. In Figure 4, ignoring the internal resistors of MOSFETs, the internal resistor of the primary and secondary side coils of the forward converter is expressed by RTi\({}_{\text{Ia}}\), RTi\({}_{\text{2a}}\), RTi\({}_{\text{lb}}\), and RTi\({}_{\text{2b}}\). The output resistors of the forward converters are represented by R\({}_{\text{01}}\) and R\({}_{\text{02}}\); U\({}_{\text{H}}\) and U\({}_{\text{L}}\) denote the voltage of the high-energy cell and the voltage of the low-energy cell, respectively. The excitation inductors of the primary side of the forward converter are reflected by L\({}_{\text{m1}}\) and L\({}_{\text{m2}}\). C\({}_{\text{r1}}\) and C\({}_{\text{r2}}\) denote the resonant capacitors. Figure 3: Three-cell-to-two-cell balancing. Figure 4: Equivalent circuit and current-flow paths: (**a**) equivalent circuit, (**b**) Stage 1, (**c**) Stage 2, (**d**) Stage 3, (**e**) Stage 4, (**f**) Stage 5. Because the balancing main circuit is symmetrical, operational waveforms of one forward converter during one cycle are shown in Figure 5. According to the different balancing model, they are divided into Figure 5a,b. The voltage of the high- energy cells and the low-energy cells is represented by U\({}_{\text{B}}\), and the current of the primary side circuit of the forward converter is reflected by i\({}_{\text{a}}\). The balancing process in one switching cycle is divided into five steps as follows. For the convenience of analysis, resistance parameters in Figure 4 will be replaced by \(R_{T\!a}\), \(R_{T\!b}\), and \(R_{o}\). Stagel [t\({}_{0}\)-t\({}_{1}\)]: At t\({}_{0}\), the MOSFETs are turned on. The transformer's secondary side maps out N times the primary side voltage, and the transmission current is produced on the energy transmission line. N times transmission currents are generated in the primary side coil of the forward converters, and the currents of the excitation inductors \(L_{m1}\) and \(L_{m2}\) increase at the slopes of \(\frac{d\dot{t}_{I\!a}}{dt}\) and \(\frac{d\dot{t}_{I\!a}}{dt}\). At t\({}_{0}\)-t\({}_{1}\), the voltages mapped out on the secondary side of the two forward converters are presented as follows:  
\[\left\{ \begin{array}{l}U_{T1b}=N\dot{U}_{T1a}=N(U_{L}+NR_{T\!a}I_{TRANS})\\ U_{T2b}=N\dot{U}_{T1b}=N(U_{H}-NR_{T\!a}I_{TRANS})\end{array} \right. \tag{1}\]  
where \(N=\frac{N_{2}}{N_{1}}\) is the inverse of the forward converter turns ratio, and the expression of the transmission current is:  
\[I_{TRANS}=\frac{U_{T2b}-U_{T1b}}{2(R_{T\!b}+R_{o})} \tag{2}\]  
Combining (1) and (2), the final equation of the transmission current is:  
\[I_{TRANS}=\frac{N(U_{H}-U_{L})}{2(N^{2}R_{T\!a}+R_{T\!b}+R_{o})} \tag{3}\]  
Figure 5: Operational waveforms of the balancing main circuit voltage and current: (a) n-cell-to-n-cell balancing mode, (b) n-cell-to-(n-1)-cell balancing mode. Thus, the average current of energy transfer during one switching cycle is achieved by:  
\[I_{AVG}=\frac{\int_{0}^{T_{m}}tI_{TRANS}dt}{T_{S}} \tag{4}\]  
Substituting in \(T_{on}=DT_{S}\), \(I_{AVG}\) can be expressed as:  
\[I_{AVG}=DI_{TRANS}=\frac{DN(U_{H}-U_{L})}{2(N^{2}R_{T_{2}}+R_{T_{2}}+R_{o})} \tag{5}\]  
Therefore, the primary side currents \(i_{T1a}\), \(i_{T2a}\) of the forward converter are presented as:  
\[\left\{ \begin{array}{l}i_{T1a}=-NI_{TRANS}=-\frac{N^{2}(U_{H}-U_{L})}{2(N^{2 }R_{T_{2}}+R_{T_{2}}+R_{o})}\\ i_{T2a}=NI_{TRANS}=\frac{N^{2}(U_{H}-U_{L})}{2(N^{2}R_{T_{2}}+R_{T_{2}}+R_{o})} \end{array} \right. \tag{6}\]  
and the secondary side currents \(i_{T1b}\), \(i_{T2b}\) are expressed as:  
\[\left\{ \begin{array}{l}i_{T1b}=-I_{TRANS}=-\frac{N(U_{H}-U_{L})}{2(N^{2}R_{ T_{2}}+R_{T_{2}}+R_{o})}\\ i_{T2b}=I_{TRANS}=\frac{N(U_{H}-U_{L})}{2(N^{2}R_{T_{2}}+R_{T_{2}}+R_{o})} \end{array} \right. \tag{7}\]  
At t\({}_{0}\), the excitation currents \(i_{Lm1}\) and \(i_{Lm2}\) of the forward converters can be calculated by (8):  
\[\left\{