## 3 Results and Discussion
### Results of Steady-State AC Fundamental Voltage Responses on Battery Terminals  
Figure 11a is \(FPLC_{ACqu}\) on battery, and Figure 11b is \(BHL_{ACqu}\) on battery. \(FPLC_{ACqu}\) is shown in Equation (23) and \(BHL_{ACqu}\) is shown in Equation (24). The Nyquist curve from the EIS of the same 500 Ah battery (E-08#) is plotted with diagonal cross points in Figure 12. In Figure 12, the fitting curve of \(FPLC_{ACqu}\) is plotted with cross points, and the fitting curve of \(BHL_{ACqu}\) is plotted with dot  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & **Pulse Procedures** & **Timing (s)** & **Output (mV)** \\ \hline
23 & 240 s after 0.5 Hz/5 A injection close & 4520 & 654 \\
24 & 240 s after 0.5 Hz/5 A injection open & 4815 & 663 \\
25 & Before 1.0 Hz/5 A injection & 5970 & 663 \\
26 & 150 s after 1.0 Hz/5 A injection close & 6110 & 676 \\
27 & 150 s after 1.0 Hz/5 A injection open & 6340 & 668 \\
28 & Before 1.0 Hz/5 A injection & 6490 & 660 \\
29 & 150 s after 1.0 Hz/5 A injection open & 6720 & 668 \\
30 & 150 s after 1.0 Hz/5 A injection open & 6900 & 663 \\
31 & Before 1.0 Hz/5 A injection & 7130 & 674 \\
32 & Before 1.0 Hz/5 A injection & & \\
33 & 180 s after 1.0 Hz/5 A injection open & 7310 & 670 \\
34 & Before 1.0 Hz/5 A injection & 7535 & 665 \\
35 & 180 s after 1.0 Hz/5 A injection close & 7310 & 670 \\
36 & 180 s after 1.0 Hz/5 A injection open & 7535 & 665 \\
37 & Before 1.0 Hz/5 A injection & & \\
38 & 210 s after 1.0 Hz/5 A injection close & 7745 & 674 \\
39 & 210 s after 1.0 Hz/5 A injection open & 7990 & 669 \\
40 & Before 1.0 Hz/5 A injection & & \\
41 & 210 s after 1.0 Hz/5 A injection close & 8200 & 669 \\
42 & 210 s after 1.0 Hz/5 A injection open & 8450 & 661 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Measurement timings and results.  
Figure 10: Time series of outputs of detection circuit.  
points. The physical meanings of elements in the above equations are explained in Table A1 of the Appendix A.  
\[\begin{split} z_{FPLC_{ACqu}}&=R_{CFP}+j\omega L_{SFP}+ \frac{1}{j\omega C_{bulkFP}}+\frac{0.5R_{ct,pc}}{1+j\big{(}0.5R_{ct,pc}\big{)} \omega\big{(}2C_{dl,pc}\big{)}}+\frac{0.5R_{ct,nc}}{1+j(0.5R_{ct,nc})\omega \big{(}2C_{dl,nc}\big{)}}+\\ &\frac{0.5R_{ct,pd}}{1+j\big{(}0.5R_{ct,pd}\big{)}\omega\big{(}2C _{dl,pd}\big{)}}+\frac{0.5R_{ct,nd}}{1+j\big{(}0.5R_{ct,nd}\big{)}\omega\big{(}2 C_{dl,nd}\big{)}},\\ z_{BHLC_{ACqu}}&=R_{CBH}+j\omega L_{SBH}+\frac{1}{j \omega C_{bulkBH}}+\frac{R_{ct,p}}{1+j\mathrm{k}_{ct,p}\omega C_{dl,p}}+\frac{ R_{ct,n}}{1+jR_{ct,n}\omega C_{dl,n}},\\ \end{split} \tag{23}\]  
The Nyquist curve is fitted by \(FPLC_{ACqu}\) and \(BHLC_{ACqu}\) through software Zview, and element values in circuits are compared in Table 2. According to test conditions, SoC on batteries are nearly 100%; charge-transfer resistances (\(R_{ct,pc}\)_and_\(R_{ct,nc}\)) for charging of \(FPLC_{ACqu}\), are much higher than charge-transfer resistances (\(R_{ct,pd}\)_and_\(R_{ct,nd}\)) for discharging [3; 20]. Charge-transfer resistors and double-layer capacitors are directional different. In Figure 4, elements values for simulation are also from Table 2.  
Figure 11: Circuits fitting for EIS and Bode plots: (**a**) \(FPLC_{ACqu}\); (**b**) behavioral linear circuit (\(BHLC_{ACqu}\)).  
Figure 12: Experiment Nyquist curve is fitted by \(FPLC_{ACqu}\) and \(BHLC_{ACqu}\).