## 3 Results and Discussion
### Results of Extraction Directional Charge-Transfer Resistance in AC Nyquist Curve  
In Figure 13c,d, both magnitude and phase bode plots have obvious fitting errors at low frequencies. The errors are mainly due to \(C_{bulk}\)- \(z_{Warburg}\) is supposed to reduce these errors, but is not convenient for BMS design. The sums of resistances in \(FPLC_{ACequ}\) are shown in Equations (25)-(28), and their amplitudes in magnitude bode plots are marked in Figure 13a. Based on contacting resistance, each step is corresponding to one charge-transfer resistance, and 0.5\(R_{ct,nd}\), 0.5\(R_{ct,nc}\), 0.5\(R_{ct,pd}\) and 0.5\(R_{ct,pc}\) orderly add up. The sums of resistances in \(BHLAC_{ACequ}\) are shown in Equations (29)-(30), and \(R_{ct,p}\) \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Elements in First** & **Values** & **Error** & **Error\%** &
\begin{tabular}{c} **Elements in** \\ **Behavioral Circuit** \\ \end{tabular} & **Values** & **Error** & **Error\%** \\ \hline \(R_{CFP}\) & 604.4 \(\mu\Omega\) & 10.9 \(\mu\Omega\) & 1.8 & R\({}_{\text{CBH}}\) & 628.2 \(\mu\Omega\) & 8.1 \(\mu\Omega\) & 1.3 \\ \(L_{SFP}\) & 185.9 \(\mu\Omega\) & 4.9 \(\mu\Omega\) & 2.6 & L\({}_{\text{SBH}}\) & 182.5 \(\mu\Omega\) & 5.2 \(\mu\Omega\) & 2.9 \\ \(C_{bulkP}\) & 12,300.0 \(\mu\Omega\) & 754.9 \(\mu\Omega\) F & 6.1 & C\({}_{\text{ChulBH}}\) & 11,855.0 \(\mu\Omega\) & 73.7 \(\mu\Omega\) F & 6.2 \\ \(R_{ct,nd}\) & 116.2 \(\mu\Omega\) & 59.452 \(\mu\Omega\) & 51.1 & R\({}_{\text{ctn}}\) & 410.0 \(\mu\Omega\) & 24.6 \(\mu\Omega\) & 6.0 \\ \(R_{ct,nc}\) & 467.6 \(\mu\Omega\) & 147.8 \(\mu\Omega\) & 31.6 & R\({}_{\text{ct,p}}\) & 978.5 \(\mu\Omega\) & 42.1 \(\mu\Omega\) & 4.3 \\ \(R_{ct,pd}\) & 643.2 \(\mu\Omega\) & 153.6 \(\mu\Omega\) & 23.9 & C\({}_{\text{dl,n}}\) & 125.2 \(\mu\Omega\) & 10.8 \(\mu\Omega\) & 8.7 \\ \(R_{ct,pc}\) & 1711.6 \(\mu\Omega\) & 153.9 \(\mu\Omega\) & 9.0 & C\({}_{\text{dl,p}}\) & 1272.0 \(\mu\Omega\) & 111.8 \(\mu\Omega\) & 8.8 \\ \(C_{dl,nl}\) & 34.5 \(\mu\Omega\) & 19.2 \(\mu\Omega\) & 55.7 & & & \\ \(C_{dl,nc}\) & 83.9 \(\mu\Omega\) & 25.7 \(\mu\Omega\) & 30.7 & & & \\ \(C_{dl,pd}\) & 358.0 \(\mu\Omega\) & 196.9 \(\mu\Omega\) & 55.0 & & & \\ \(C_{dl,pc}\) & 1036.0 \(\mu\Omega\) & 227.0 \(\mu\Omega\) & 31.6 & & & \\ Chi-squared & 0.011 & & & Chi-squared & 0.013 & & \\ Weighted sum of & 0.98 & & & Weighted sum of & 1.27 & & \\ squares & & & & squares & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Values of elements in \(FPLC_{ACequ}\) and \(BHLAC_{ACequ}\) for E-08#. Figure 13: Linear circuits fitting for bode plots: (**a**) \(FPLC_{ACEqu}\) fitting in magnitude; (**b**) \(BHLAC_{ACEqu}\) fitting in magnitude; (**c**) \(FPLC_{ACEqu}\) fitting in phase; (**d**) \(BHLAC_{ACEqu}\) fitting in phase. orderly add up. They are also marked in Figure 13b. In magnitude bode plots, the asymptotes which roughly outline measured and fitting curves are drawn according to poles and zeros points. \[R_{1 FP}=R_{CFP}+0.5R_{ct,nd}, \tag{25}\]  
\[R_{2 FP}=R_{CFP}+0.5R_{ct,nd}+0.5R_{ct,nd}, \tag{26}\]  
\[R_{3 FP}=R_{CFP}+0.5R_{ct,nd}+0.5R_{ct,nd}+0.5R_{ct,pd}, \tag{27}\]  
\[R_{4 FP}=R_{CFP}+0.5R_{ct,nd}+0.5R_{ct,nd}+0.5R_{ct,pd}+0.5R_{ct,pc}, \tag{28}\]  
\[R_{1 BH}=R_{CBH}+R_{ct,nd}, \tag{29}\]  
\[R_{2 BH}=R_{CBH}+R_{ct,n}+R_{ct,p}, \tag{30}\]  
In experiments, batteries' capacities are almost full-charging; Therefore, charge-transfer resistances for discharging (\(R_{ct,pd}\)_or_\(R_{ct,nd}\)) are smaller than charge-transfer resistances for charging (\(R_{ct,pc}\)_or_\(R_{ct,nd}\)). Furthermore, if batteries' SoH is good, \(R_{ct,pd}\)_or_\(R_{ct,nd}\) for discharging are less than \(R_{ct,pc}\)_or_\(R_{ct,nd}\) for charging across the broad SoC range; otherwise, when batteries begin to discharge, \(R_{ct,pd}\)_and_\(R_{ct,nd}\) are used to increase significantly, and the batteries' terminal voltages drop obviously. Extracting the DC directional charge-transfer resistance from the AC Nyquist curve has engineering significance.