# Electrochemical Impedance Spectroscopy as an Analytical Tool for the Prediction of the Dynamic Charge Acceptance of Lead-Acid Batteries

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{1}, the parameter indicating the size of the first/high-frequency semicircle, is smaller for cells with higher DCA. According to the literature, this semicircle represents the charge transfer reaction, thus confirming that current-enhancing additives may decrease the pore diameter of the negative electrode.

## 1. Introduction

## 2. Experimental

#### 2.1. Laboratory Test Cells

#### 2.2. Charge Acceptance Tests

_{c}) at 80% SoC, discharge history (I

_{d}) at 90% SoC and during real start-stop micro cycles (I

_{r}) at 80% SoC. Further explanations of the test usage at the cell level are given in [16], along with a quantitative comparison with other charge acceptance tests.

#### 2.3. Electrochemical Impedance Spectroscopy

_{20}/2. A superposed charging or discharging current, I

_{DC}, was used to avoid a change of polarity during the measurement and to force only one reaction direction upon the battery. Thereby, the maximum SoC change during one spectrum was 2% SoC. The red marks in Figure 1 indicate when the EIS measurements were carried out. Moreover, the most important measurement parameters are summarized in Table 2. This micro cycling approach was first introduced by Karden et.al. [17]. The micro cycles were repeated three times, and only the last impedance spectrum with superimposed charging current was used for the respective investigations to evaluate the charging processes.

## 3. Results

#### 3.1. Dynamic Charge Acceptance

_{c}and the I

_{r}parts of the DCA (EN 50342-6:2015) test are carried out. However, the I

_{d}part of the DCA (EN) test is taking place at 90% SoC. For cells with lower plate count the acid density is lower at 90% SoC, resulting in higher charge acceptance [18,19]. Another effect is that flooded batteries rapidly suffer from acid stratification during DCA tests [20]. The decreased acid stratification due to the excess acid in cells with low plate count increases the normalized DCA [9,21,22].

#### 3.2. Electrochemical Impedance Spectroscopy

_{0}. For all cell layouts, the Type 1 EFB − C cells exhibit the biggest first semicircle under this condition, followed by the Type 1 EFB + C and the smallest first semicircle for Type 2 EFB + C. This trend can be visualized for all cell layouts. Comparing the results of the charging currents (shown in Figure 2) with the EIS results (shown in Figure 3), the conclusion can be drawn that the higher the DCA, the smaller the first semicircle of the EIS.

#### 3.3. Kramers-Kronig

#### 3.4. Distribution of Relaxation Times

_{τ}is a multiple of the measured frequencies N

_{f}in the impedance spectra N

_{τ}= c·N

_{f}[27] with c ϵ {1,2,3} to obtain a smoother distribution function [26]. The matrix A needs to be solved for all frequencies and predefined time constants for the real and the imaginary parts:

^{2}to the optimization function [28]. λ is the regularization parameter.

_{0}≈ 0.003 s, the peak at τ

_{1}≈ 0.07 s, a smaller peak around τ

_{2}≈ 1 s and the last peak at τ

_{3}≈ 10 s. Each peak represents a single process. Figure 5b shows the DRT of the EIS at 80% SoC of the middle size cell, 3P2N, while Figure 5c shows the DRT of the EIS at 80% SoC of the small size cell, 2P1N. The time constants identified by the DRT are given within Table 4. For these cell layouts, the peak at the small time constant is not as distinct anymore. The peak at τ

_{1}≈ 0.07 s is visible for middle and small size cells for all types as well. For the middle and small size cell, the last peak at τ

_{3}≈ 10 s is also visible, surrounded by several smaller peaks, which might be present due to the time constant τ

_{2}. These findings are used to evaluate the most important processes, to choose the ECM elements and the starting parameters of the fitting algorithm.

^{2}, or the width, correlates with the frequency range at which the process occurs. A process with a sharp peak occurs at a small range of frequencies, while wide peaks point out more distributed processes.

#### 3.5. Fitting with an Equivalent Circuit Model

_{0}, an inductivity, and several semicircles, each representing a separate process, e.g., charge transfer, chemical reactions, mass transport or adsorption processes. In Figure 5, the DRT of the EIS at 80% SoC is shown. Since τ

_{0}is only distinct for complete cells, only τ

_{1}, τ

_{2}and τ

_{3}are used as parameters within the ECM. Each of the time constants is dedicated to one semicircle representing one process. A semicircle can be modelled by using a parallel connection between a resistor and a capacitor. If the measurement results in a flattened semicircle, which can be explained by the porosity of the electrodes, a constant phase element (CPE) in parallel to a resistor can be used instead. The formula for a CPE in parallel to a resistance, which is also referred to as ZARC element, is:

#### 3.6. Parameterization of the Equivalent Circuit Model

_{0}is equal to the removed offset of the smallest real part of the impedance Re{Z}. However, for some spectra, it is not possible to take this minimum in a frequency range where the phase of the impedance crosses zero. As a result, the upper limit is extended. The time constants identified by the DRT are used as fixed parameters for the fitting. The evaluation of the fitting had previously been executed with variable values for all ξ. However, for the purposes of comparability, the average value for each ξ

_{1}, ξ

_{2}and ξ

_{3}are taken, and the fit is repeated with fixed values.

_{1}, in Figure 9a for R

_{1}and in Figure 10a for C

_{1}. The parameters of the middle size cells are shown in Figure 8, Figure 9 and Figure 10b, and the fitting results of the small size cells are presented in Figure 8, Figure 9 and Figure 10c.

## 4. Discussion

^{2+}ions. The polarization of the negative electrode is stronger compared to the positive electrode due to the double-layer capacitance of the positive electrode being approximately 10 times higher [8]. Therefore, the DCA during dynamic operation is known to be restricted by the negative electrode [32,33,34].

_{4}crystal size, thus increasing its dissolution and decreasing the distance for Pb

^{2+}ion transport. These effects should be visible within the EIS measurements.

^{2+}ions takes place in a two-step electron transfer reaction [48,49]. Huck compared the spectra resulting from two charge transfer reaction steps with two additional adsorbates and different intermediate steps [50]. Using this two-step electron transfer reaction, the spectra of cells with current-enhancing additives would influence not only the high-frequency semicircle, but also the low-frequency semicircle. Even though the impact was the highest in the first semicircle, the measurement results indicate that the additives also have an effect on the low frequency semicircle as well.

## 5. Conclusions

_{1}is around 0.07 s

^{ξ}. The resistance R

_{1}, the parameter indicating the size of the high-frequency semicircle, increases for larger semicircles. R

_{1}is the highest considering Type 1 EFB − C for all test cell layouts. This is followed by the resistance R

_{1}for the Type 1 EFB + C cells, which is the second highest for each cell layout. The smallest R

_{1}for all cell layouts is that of the Type 2 EFB + C cells. Furthermore, it has been found that, disregarding the cells’ layout, the DCA is the biggest for Type 2 EFB + C cells, followed by Type 1 EFB + C cells, while the Type 1 EFB − C cell has the lowest DCA. Consequently, the inverse of the resistance R

_{1}, and thus the conductivity, shows the highest correlation to the DCA. On the other hand, there is no clear relationship between the capacitance C

_{1}and the DCA, and no correlation between the parameters of the second semicircle and the DCA.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Complete, unprocessed negative half-cell EIS at 80% SoC for (

**a**) complete cell: 8P8N/8P9N, (

**b**) middle size cell: 3P2N and (

**c**) small size cell: 2P1N visualized in a Nyquist diagram.

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**Figure 2.**Normalized charge current of laboratory test cells obtained from the DCA test according to EN 50342-6:2015 for (

**a**) complete cell: 8P8N/8P9N, (

**b**) middle size cell: 3P2N and (

**c**) small size cell: 2P1N (adapted from [16]).

**Figure 3.**First semicircle of the negative half-cell obtained from EIS at 80% SoC of the laboratory test cells visualized in a Nyquist diagram for (

**a**) complete cell: 8P8N/8P9N, (

**b**) middle size cell: 3P2N and (

**c**) small size cell: 2P1N.

**Figure 4.**Negative half-cell EIS at 80% SoC of the laboratory test cells for (

**a**) absolute value Z of the complete cell: 8P8N/8P9N, (

**b**) phase shift of the complete cell: 8P8N/8P9N, (

**c**) absolute value Z of the middle size cell: 3P2N, (

**d**) phase shift of the middle size cell: 3P2N, (

**e**) absolute value Z of the small size cell: 2P1N and (

**f**) phase shift of the small size cell: 2P1N visualized in a Bode plot.

**Figure 5.**DRT of the EIS at 80% SoC of the laboratory test cells for (

**a**) complete cell: 8P8N/8P9N, (

**b**) middle size cell: 3P2N and (

**c**) small size cell: 2P1N.

**Figure 7.**Negative half-cell EIS at 80% SoC and fit with ECM of the laboratory test cells for (

**a**) complete cell: 8P8N/8P9N, (

**b**) middle size cell: 3P2N and (

**c**) small size cell: 2P1N visualized in a Nyquist diagram.

**Figure 8.**Fitting result τ

_{1}of the laboratory test cells for (

**a**) complete cell: 8P8N/8P9N, (

**b**) middle size cell: 3P2N and (

**c**) small size cell: 2P1N.

**Figure 9.**Fitting result R

_{1}of the laboratory test cells for (

**a**) complete cell: 8P8N/8P9N, (

**b**) middle size cell: 3P2N and (

**c**) small size cell: 2P1N.

**Figure 10.**Fitting result C

_{1}of the laboratory test cells for (

**a**) complete cell: 8P8N/8P9N, (

**b**) middle size cell: 3P2N and (

**c**) small size cell: 2P1N.

**Table 1.**Test cell setup and its nominal capacity, where P is the number of positive plates, and N is the number of negative plates (Reprinted from Ref. [16]).

Cell Size | Plate Count Type 1/Type 2 | Nominal Capacity C_{n}Type 1/Type 2 |
---|---|---|

Complete cell | 8P8N/8P9N | 70 Ah |

Middle size cells | 3P2N | 17.50 Ah/18.70 Ah |

Small size cells | 2P1N | 8.75 Ah/9.30 Ah |

Parameter | Value |
---|---|

I_{DC} | I_{20}/2 |

I_{AC,max} | 0.5 A |

f_{min} | 10 mHz |

f_{max} | 6.5 kHz |

T | 25 °C |

SoC | 80% |

ΔSoC | 2% |

**Table 3.**Scaling factors according to the number of active plates within a cell (Reprinted from Ref. [16]).

Original Plate Count | Type 1: 8P8N | Type 2: 8P9N |
---|---|---|

Complete cell | 15/16 | 16/18 |

Middle size cells | 4/16 | 4/18 |

Small size cells | 2/16 | 2/18 |

Type 1 EFB − C | Type 1 EFB + C | Type 2 EFB + C | |||||||
---|---|---|---|---|---|---|---|---|---|

Time Constant | Complete Cell | Middle Size Cell | Small Size Cell | Complete Cell | Middle Size Cell | Small Size Cell | Complete Cell | Middle Size Cell | Small Size Cell |

τ_{0}/s^{ξ} | 0.003 | - | - | 0.002 | - | - | 0.003 | - | - |

τ_{1}/s^{ξ} | 0.072 | 0.08 | 0.10 | 0.06 | 0.054 | 0.044 | 0.073 | 0.056 | 0.056 |

τ_{2}/s^{ξ} | 2.359 | 2.32 | 1.094 | 1.094 | 2.816 | 0.568 | 2.984 | 1.436 | 0.451 |

τ_{3}/s^{ξ} | 13.495 | 11.19 | 5.838 | 7.415 | 15.9 | 3.629 | 19.025 | 19.025 | 2.283 |

Parameter | Lower Limit | Starting Value | Upper Limit |
---|---|---|---|

R_{0}/Ω | R_{0} | R_{0} | R_{0} + 0.05 |

L/μH | 0 | 200 | 10,000 |

λ_{L} | 0 | 0.4 | 1 |

R_{1}/Ω | 0 | 0.3 | 1 |

τ_{1}/s^{ξ} | τ_{1} | τ_{1} | τ_{1} |

ξ_{1} | 0.849 | 0.849 | 0.849 |

R_{2}/Ω | 0 | 0.4 | 1 |

τ_{2}/s^{ξ} | τ_{2} | τ_{2} | τ_{2} |

ξ_{2} | 0.664 | 0.664 | 0.664 |

R_{3}/Ω | 0 | 0.5 | 2 |

τ_{3}/s^{ξ} | τ_{3} | τ_{3} | τ_{3} |

ξ_{3} | 0.75 | 0.75 | 0.75 |

Type 1 EFB − C | Type 1 EFB + C | Type 2 EFB + C | |||||||
---|---|---|---|---|---|---|---|---|---|

Time Constant | Complete Cell | Middle Size Cell | Small Size Cell | Complete Cell | Middle Size Cell | Small Size Cell | Complete Cell | Middle Size Cell | Small Size Cell |

R_{0}/Ω | 0 | 0.0062 | 0 | 0 | 0 | 0 | 0.0119 | 0.0121 | 0 |

L/μH | 420 | 108 | 11.8 | 249 | 46.8 | 13.1 | 257 | 277 | 2500 |

λ_{L} | 0.94 | 0.98 | 0.97 | 0.94 | 0.94 | 0.95 | 0.95 | 1 | 0.18 |

R_{1}/Ω | 0.4 | 0.42 | 0.52 | 0.34 | 0.303 | 0.2 | 0.309 | 0.28 | 0.16 |

τ_{1}/s^{ξ} | 0.072 | 0.08 | 0.10 | 0.06 | 0.054 | 0.044 | 0.073 | 0.056 | 0.056 |

ξ_{1} | 0.85 | 0.85 | 0.85 | 0.85 | 0.85 | 0.85 | 0.85 | 0.85 | 0.85 |

R_{2}/Ω | 0.534 | 0.533 | 0.3 | 0.3 | 0.6 | 0.3 | 0.384 | 0.3 | 0.3 |

τ_{2}/s^{ξ} | 2.359 | 2.32 | 1.094 | 1.094 | 2.816 | 0.568 | 2.984 | 1.436 | 0.451 |

ξ_{2} | 0.664 | 0.664 | 0.664 | 0.664 | 0.664 | 0.664 | 0.664 | 0.664 | 0.664 |

R_{3}/Ω | 0.218 | 0.62 | 1.16 | 0.41 | 1.452 | 0.366 | 0.37 | 0.101 | 0.1 |

τ_{3}/s^{ξ} | 13.495 | 11.19 | 5.838 | 7.415 | 15.9 | 3.629 | 19.025 | 19.025 | 2.283 |

ξ_{3} | 0.75 | 0.75 | 0.75 | 0.75 | 0.75 | 0.75 | 0.75 | 0.75 | 0.75 |

error | 0.042 | 0.029 | 0.066 | 0.051 | 0.047 | 0.034 | 0.036 | 0.023 | 0.051 |

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## Share and Cite

**MDPI and ACS Style**

Bauknecht, S.; Kowal, J.; Bozkaya, B.; Settelein, J.; Karden, E.
Electrochemical Impedance Spectroscopy as an Analytical Tool for the Prediction of the Dynamic Charge Acceptance of Lead-Acid Batteries. *Batteries* **2022**, *8*, 66.
https://doi.org/10.3390/batteries8070066

**AMA Style**

Bauknecht S, Kowal J, Bozkaya B, Settelein J, Karden E.
Electrochemical Impedance Spectroscopy as an Analytical Tool for the Prediction of the Dynamic Charge Acceptance of Lead-Acid Batteries. *Batteries*. 2022; 8(7):66.
https://doi.org/10.3390/batteries8070066

**Chicago/Turabian Style**

Bauknecht, Sophia, Julia Kowal, Begüm Bozkaya, Jochen Settelein, and Eckhard Karden.
2022. "Electrochemical Impedance Spectroscopy as an Analytical Tool for the Prediction of the Dynamic Charge Acceptance of Lead-Acid Batteries" *Batteries* 8, no. 7: 66.
https://doi.org/10.3390/batteries8070066