# Multi-Mode Lithium-Ion Battery Balancing Circuit Based on Forward Converter with Resonant Reset
## 2 Balancing Circuit
### Circuit Analysis
\begin{array}{l}i_{Lm1}(t)=-\frac{\int_{0}^{t}tU_{L}dt}{L_{m1}}\\ i_{Lm2}(t)=\frac{\int_{0}^{t}tU_{H}dt}{L_{m2}}\end{array} \right. \tag{8}\]  
According to (6) to (8), at t\({}_{0}\), i\({}_{1}\) and i\({}_{2}\) in the two primary circuits can be deduced, respectively, as:  
\[\left\{ \begin{array}{l}i_{1}(t)=i_{Lm1}(t)+i_{T1a}=\frac{(t-t_{0})|U_{L}}{L _{m1}}-\frac{N^{2}(U_{H}-U_{L})}{2(N^{2}R_{T_{2}}+R_{T_{2}}+R_{o})}\\ i_{2}(t)=i_{Lm2}(t)+i_{T2a}=\frac{(t-t_{0})|U_{H}}{L_{m2}}+\frac{N^{2}(U_{H}-U _{L})}{2(N^{2}R_{T_{2}}+R_{T_{2}}+R_{o})}\end{array} \right. \tag{9}\]  
At t\({}_{1}\), the excitation currents i\({}_{Lm1}\) and i\({}_{Lm2}\) of the forward converters can be calculated by (10):  
\[\left\{ \begin{array}{l}i_{Lm1}(t_{1})=\frac{(t_{1}-t_{0})|U_{L}}{L_{m1}}= \frac{Dt_{S}U_{L}}{L_{m1}}\\ i_{Lm2}(t_{1})=\frac{(t_{1}-t_{0})|U_{H}}{L_{m2}}=\frac{Dt_{S}^{2}U_{H}}{L_{m2 }}\end{array} \right. \tag{10}\]  
According to (9) and (10), at t\({}_{1}\), i\({}_{1}\) and i\({}_{2}\) in the two primary circuits can be deduced, respectively, as:  
\[\left\{ \begin{array}{l}i_{1}(t_{1})=i_{Lm1}(t_{1})+i_{T1a}=\frac{2 DT_{S}U_ {L}(N^{2}R_{T_{2}}+R_{T_{2}}+R_{o})-N^{2}L_{m1}(U_{H}-U_{L})}{2L_{m1}(N^{2}R_ {T_{2}}+R_{T_{2}}+R_{o})}\\ i_{2}(t_{1})=i_{Lm2}(t_{1})+i_{T2a}=\frac{2 DT_{S}U_{H}(N^{2}R_{T_{2}}+R_{T_{2}} +R_{o})-N^{2}L_{m2}(U_{H}-U_{L})}{2L_{m2}(N^{2}R_{T_{2}}+R_{T_{2}}+R_{o})} \end{array} \right. \tag{11}\]  
Stage 2 [t\({}_{1}\)-t\({}_{2}\)]: At t\({}_{1}\), MOSFETs are turned off, L\({}_{m}\) and C\({}_{r}\) begin to resonate, and the circuit enters the resonant state. Because the voltage added to the excitation inductor is still positive at this time, the excitation currents i\({}_{Lm1}\) and i\({}_{Lm2}\) will continue to increase. Charges are accumulated on the resonant capacitors C\({}_{r1}\) and C\({}_{r2}\), respectively, and the resonant capacitors are charged. At t\({}_{2}\), the resonant capacitors C\({}_{r1}\) and C\({}_{r2}\) are charged to the voltage of cells \(U_{H}\) and \(U_{L}\), differently, and the voltage drops across the excitation inductors are zero. \(i_{Lm1}\) and i\({}_{Lm2}\) reach the maximum values of i\({}_{Lm1\text{ (max)}}\) and i\({}_{Lm2\text{ (max)}}\), respectively. Thus, the relationship between resonant current and resonant voltage in the balancing circuit at t\({}_{1}\) can be expressed by (12) and (13):  
\[\left\{ \begin{array}{l}U_{L}=L_{m1}\frac{di_{l,m1}(t)}{dt}+U_{Cr1}(t)\\ U_{H}=L_{m2}\frac{di_{l,m2}(t)}{dt}+U_{Cr2}(t)\end{array} \right. \tag{12}\]  
\[\left\{ \begin{array}{l}i_{Lm1}(t)=C_{r1}\frac{dU_{Cr1}(t)}{dt}\\ i_{Lm2}(t)=C_{r2}\frac{dU_{Cr2}(t)}{dt}\end{array} \right. \tag{13}\]  
From (12) and (13), resonant current and resonant voltage are respectively calculated using:  
\[\left\{ \begin{array}{l}U_{Cr1}(t)=U_{L}-U_{L}cos\frac{t-t_{1}}{\sqrt{L_{m1} \zeta_{r1}}}+i_{Lm1}(t_{1})\sqrt{\frac{t_{2m1}}{\zeta_{r1}}}sin\frac{t-t_{1}}{ \sqrt{L_{m1}\zeta_{r1}}}\\ U_{Cr2}(t)=U_{H}-U_{H}cos\frac{t-t_{1}}{\sqrt{L_{m2}\zeta_{r2}}}+i_{Lm2}(t_{1} )\sqrt{\frac{t_{2m}}{\zeta_{r2}}}sin\frac{t-t_{1}}{\sqrt{L_{m2}\zeta_{r2}}} \end{array} \right. \tag{14}\]  
\[\left\{ \begin{array}{l}i_{Lm1}(t)=i_{Lm1}(t_{1})cos\frac{t-t_{1}}{\sqrt{L_{m1 }\zeta_{r1}}}+U_{L}\sqrt{\frac{\zeta_{r1}}{L_{m1}}}sin\frac{t-t_{1}}{\sqrt{L_ {m1}\zeta_{r1}}}\\ i_{Lm2}(t)=i_{Lm2}(t_{1})cos\frac{t-t_{1}}{\sqrt{L_{m2}\zeta_{r2}}}+U_{H}\sqrt {\frac{\zeta_{r2}}{L_{m2}}}sin\frac{t-t_{1}}{\sqrt{L_{m2}\zeta_{r2}}}\end{array} \right. \tag{15}\]  
Stage 3 [t\({}_{2}\)-\({}_{3}\)]: At t\({}_{2}\), Cr1 and Cr2 continue to charge by L\({}_{m1}\) and L\({}_{m2}\). At the same time, U\({}_{Cr1}\) and U\({}_{Cr2}\) reach their maximum values U\({}_{Cr1}\) (max) and U\({}_{Cr2}\) (max) at t\({}_{3}\), while \(i_{Lm1}\) and \(i_{Lm2}\) decrease to zero to achieve a magnetic reset. Resonant current and resonant voltage at t\({}_{2}\) can be expressed as:  
\[\left\{