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Citation: 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
Academic Editor: Torsten Brezesinski
Received: 31 May 2022
Accepted: 28 June 2022
Published: 5 July 2022
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batteries
Article
Electrochemical Impedance Spectroscopy as an Analytical Tool
for the Prediction of the Dynamic Charge Acceptance of
Lead-Acid Batteries
Sophia Bauknecht 1,* , Julia Kowal 1,* , Begüm Bozkaya 2, Jochen Settelein 3and Eckhard Karden 4
1Electrical Energy Storage Technology, Technische Universität Berlin, 10587 Berlin, Germany
2Consortium for Battery Innovation, 120 New Cavendish Street, London W1W 6XX, UK;
[email protected]
3Fraunhofer Institute for Silicate Research ISC, Neunerplatz 2, 97082 Würzburg, Germany;
[email protected].de
4Advanced Power Supply & Energy Management, Ford Research & Innovation Center Aachen,
52072 Aachen, Germany; ekarden@ford.com
*Correspondence: [email protected] (S.B.); [email protected] (J.K.)
Abstract:
The subject of this study is test cells extracted from industrially manufactured automotive
batteries. Each test cell either had a full set of plates or a reduced, negative-limited set of plates.
With these
test cells the predictability of the dynamic charge acceptance (DCA) by using electrochem-
ical impedance spectroscopy (EIS) is investigated. Thereby, the DCA was performed according to EN
50342-6:2015 standard. The micro cycling approach was used for the EIS measurements to disregard
any influencing factors from previous usage. During the evaluation, Kramers-Kronig (K-K) was
used to avoid systematic errors caused by violations of the stationarity, time-invariance or linearity.
Furthermore, the analysis of the distribution of relaxation times (DRT) was used to identify a usable
equivalent circuit model (ECM) and starting values for the parameter prediction. For all cell types
and layouts, the resistance R
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.
Keywords:
dynamic charge acceptance; electrochemical impedance spectroscopy; equivalent circuit
model; lead-acid batteries; Kramers–Kronig; distribution of relaxation times
1. Introduction
Electrochemical impedance spectroscopy (EIS) is used to identify electrochemical
processes and to characterize materials or devices [
1
]. The first studies on battery impedance
were made in the 1940s with a limited frequency range [
2
] since only analog measurement
technologies were used at the time. The AC resistance is still used to estimate the state
of charge (SoC) and state of health (SoH) of lead-acid batteries [
3
,
4
]. By increasing the
frequency range, the capacitive behavior of the impedance could also be used to estimate
SoC [5], SoH [6].
Until the end of the last century, no monitoring of starting lighting and ignition (SLI)
batteries existed. Indeed, it was not necessary, as the battery was considered only as an
energy buffer used for starting the vehicle. However, there has been an increase in both
the number of loads installed and the maximum power required, an automatic start/stop
function and a brake energy regeneration system have been implemented, and many safety-
related functionalities have been introduced [
7
]. Currently, the enhanced flooded battery
(EFB) is used as the standard for SLI in automotive applications. To meet continuously
growing demands, a significantly improved dynamic charge acceptance (DCA) of up to
2 A/Ah
over the complete battery life is unavoidable [
8
]. In this context the DCA has been
Batteries 2022,8, 66. https://doi.org/10.3390/batteries8070066 https://www.mdpi.com/journal/batteries
Batteries 2022,8, 66 2 of 15
defined as the ability of a battery or cell to absorb high-rate charge pulses during micro
cycling operation within a partial state-of-charge (PSoC). However, automotive batteries
are operated at high SoCs, with the DCA of EFBs only accepting 1 A/Ah in new condition,
and often less than 0.5 A/Ah after several months in service [
9
,
10
]. It has been shown that
innovative negative electrode technologies, such as carbon additives or organic expanders
in the negative active material (NAM), allow for up to three times more stabilized DCA
after several months in operation [
10
]. Additives substantially influence the structure of
porous active materials [
11
]. Moreover, they influence the processing of the electrodes.
Within the context of this work, the symbol EFB
−
C is used to designate a classic EFB with
normal DCA, whereas the symbol EFB + C is only used for EFBs that include additional
current-enhancing additives.
Screening active materials with different additive combinations is a time-consuming
task and is mainly performed in small test cells that allow for a high sample throughput.
For this task, the EN 50342-6:2015, attempting to forecast the DCA in real applications
with reasonable but not perfect correlation [
12
], can be used. However, several weeks of
cell testing for material screening is challenging. The processes that limit DCA, as well
as the mechanisms by which various additives achieve substantially higher DCA, are not
fully understood yet. However, they might be visualized via EIS. Using EIS with a wide
frequency range for non-destructive cell-level monitoring can show the electrochemical
processes and serve as a basis for dynamic battery models. Thus, EIS has been identified as
one of the most promising alternative tools for predicting battery behavior and condition,
e.g., for SoC and SoH prediction [5,6,13,14].
Within this work, the experimental implementation of the EIS and its evaluation, using
Kramers-Kronig (K-K) transformation and the distribution of relaxation times (DRT) is
presented to determine the starting parameters of the fitting with an equivalent circuit
model (ECM). The fitting results and the parameters obtained from the fitting are compared
with the DCA results to find a conclusive correlation. This way, EIS measurements can be
used as a technique to predict a high or low DCA for test cells.
2. Experimental
2.1. Laboratory Test Cells
Within this work, commercial EFB 12 V, 70 Ah batteries were used to separate three
test cells out of one battery [
15
]. Therefore, the plastic case of every other cell needed to
be cut. The cost on straps of the current collectors of the sacrificed cells were kept intact
to use them as terminals for the retained test cells. Two of the test cells needed to be
opened to reduce the number of plates (3P2N and 2P1N), while one cell was left complete.
The excess
space within the 3P2N and 2P1N test cell was refilled with spacers to avoid acid
overage. The goal of this cutting procedure was to enable the investigation of size and
symmetry effects without the eventuality of additional cost effects of handmade plates.
A complete cell contains either 8 positive and 9 negative plates (8P9N) or 8 plates each
(8P8N). The complete cell were compared with smaller cell setups, such as 3P2N and 2P1N,
as they would typically be used in material investigation. Since the focus of this work is to
investigate the processes influenced by additives in the NAM, the negative plates will be
the limiting element in the cell setup.
Two different battery types were evaluated: For the first type, one version with a
conventional NAM recipe (Type 1 EFB
−
C) was tested as a reference in comparison to a
second version with enhanced DCA, but otherwise identical components (e.g., positive
electrodes), manufactured the same day on the same equipment (Type 1 EFB + C). For the
other battery design, only versions with enhanced DCA were studied (Type 2 EFB + C).
The plate count and the nominal capacity for the three cell sizes of both test cell types are
given in Table 1.
Batteries 2022,8, 66 3 of 15
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 Cn
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
2.2. Charge Acceptance Tests
The charge acceptance test used were according to EN 50342-6:2015 where the test is
defined for batteries. Therefore, small changes, such as voltage and current adjustments had
to be executed. Otherwise, the test was conducted as defined within the norm.
The DCA
is
the result of three parts: namely, after charge (I
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
The galvanostatic EIS is used within this work. The frequency range is between
10 mHz
and 6.5 kHz for each spectrum. For each frequency, 3 periods were measured, and
8 frequencies per decade were evaluated, which resulted in a test duration of
≈15 min
for one spectrum. The main demand of EIS is that the system under inspection has to
be (quasi-) stationary, linear or at least able to be linearized with respect to the signal
amplitudes in its working point and causal.
To ensure a (quasi-) steady state during an EIS measurement the temperature, SoC
and SoH changes should be kept as small as possible. To ensure the same SoH of all test
cells, they all underwent the same pre-tests (e.g., C20 and DCA (EN) test) before executing
the EIS measurements. Moreover, all EIS measurements were performed within a climate
cabin at a temperature of 25 ◦C.
Starting from 80% SoC, the cell was cycled by approximately 4% SoC around the
targeted SOC. Meanwhile, one impedance spectrum was recorded during each discharging
and charging period, always starting at 80% SoC with a superimposed DC current of
±
I
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 1indicate 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.
Batteries 2022, 8, x FOR PEER REVIEW 3 of 15
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 Cn
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
2.2. Charge Acceptance Tests
The charge acceptance test used were according to EN 50342-6:2015 where the test is
defined for batteries. Therefore, small changes, such as voltage and current adjustments
had to be executed. Otherwise, the test was conducted as defined within the norm. The
DCA is the result of three parts: namely, after charge (Ic) at 80% SoC, discharge history (Id)
at 90% SoC and during real start-stop micro cycles (Ir) 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
The galvanostatic EIS is used within this work. The frequency range is between 10
mHz and 6.5 kHz for each spectrum. For each frequency, 3 periods were measured, and 8
frequencies per decade were evaluated, which resulted in a test duration of ≈15 min for
one spectrum. The main demand of EIS is that the system under inspection has to be
(quasi-) stationary, linear or at least able to be linearized with respect to the signal ampli-
tudes in its working point and causal.
To ensure a (quasi-) steady state during an EIS measurement the temperature, SoC
and SoH changes should be kept as small as possible. To ensure the same SoH of all test
cells, they all underwent the same pre-tests (e.g., C20 and DCA (EN) test) before executing
the EIS measurements. Moreover, all EIS measurements were performed within a climate
cabin at a temperature of 25 °C.
Starting from 80% SoC, the cell was cycled by approximately 4% SoC around the
targeted SOC. Meanwhile, one impedance spectrum was recorded during each discharg-
ing and charging period, always starting at 80% SoC with a superimposed DC current of
±I20/2. A superposed charging or discharging current, IDC, was used to avoid a change of
polarity during the measurement and to force only one reaction direction upon the bat-
tery. 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 ap-
proach 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.
Figure 1. Micro cycles for superimposed current during electrochemical impedance spectrum.
Figure 1. Micro cycles for superimposed current during electrochemical impedance spectrum.
Batteries 2022,8, 66 4 of 15
Table 2. EIS test parameters.
Parameter Value
IDC I20/2
IAC,max 0.5 A
fmin 10 mHz
fmax 6.5 kHz
T 25 ◦C
SoC 80%
∆SoC 2%
The dynamic behavior of each electrode was investigated using a hydrogen reference
electrode Gaskatel 88010 form Gaskatel Gesellschaft für Gassysteme durch Katalyse und
Elektrochemie mbH, Kassel, Deutschland. Since the location of a reference electrode
becomes a crucial factor, its position was kept the same in each test cell.
3. Results
3.1. Dynamic Charge Acceptance
The DCA test results according to EN 50342-6:2015 for Type 1 EFB
−
C, Type 1
EFB + C and Type 2 EFB + C are shown in Figure 2a–c comparing the complete cell,
the middle size cell and the small size cell. The DCA results have already been shown
and discussed in [
16
] and will be compared with the EIS measurements within the next
sections. The charge currents according to EN 50342-6:2015 are increasing in the following
order: Type 1
EFB −C<Type1EFB+C<Type2EFB+C
. This trend is valid for all cell
layouts. In the case of the impact of plate count on charge currents, DCA of single cells
are quantitatively higher for lower plate count. Even though, the acid amount within the
small and middle size test cells was limited via spacers, a higher acid-to-mass ratio was
found within these cells [
15
]. The amount of electrolyte per Ah has been found to be the
most relevant influencing factor for the DCA [
15
]. The acid density within the test cell
was adjusted to 80% SoC where the I
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].
Batteries 2022, 8, x FOR PEER REVIEW 4 of 15
Table 2. EIS test parameters.
Parameter Value
IDC I20/2
IAC,max 0.5 A
fmin 10 mHz
fmax 6.5 kHz
T 25 °C
SoC 80%
Δ SoC 2%
The dynamic behavior of each electrode was investigated using a hydrogen reference
electrode Gaskatel 88010 form Gaskatel Gesellschaft für Gassysteme durch Katalyse und
Elektrochemie mbH, Kassel, Deutschland. Since the location of a reference electrode be-
comes a crucial factor, its position was kept the same in each test cell.
3. Results
3.1. Dynamic Charge Acceptance
The DCA test results according to EN 50342-6:2015 for Type 1 EFB − C, Type 1 EFB +
C and Type 2 EFB + C are shown in Figure 2a–c comparing the complete cell, the middle
size cell and the small size cell. The DCA results have already been shown and discussed
in [16] and will be compared with the EIS measurements within the next sections. The
charge currents according to EN 50342-6:2015 are increasing in the following order: Type
1 EFB − C < Type 1 EFB + C < Type 2 EFB + C. This trend is valid for all cell layouts. In the
case of the impact of plate count on charge currents, DCA of single cells are quantitatively
higher for lower plate count. Even though, the acid amount within the small and middle
size test cells was limited via spacers, a higher acid-to-mass ratio was found within these
cells [15]. The amount of electrolyte per Ah has been found to be the most relevant influ-
encing factor for the DCA [15]. The acid density within the test cell was adjusted to 80%
SoC where the Ic and the Ir parts of the DCA (EN 50342-6:2015) test are carried out. How-
ever, the Id 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].
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]).
3.2. Electrochemical Impedance Spectroscopy
A close-up of the first semicircle of the negative half-cell EIS at 80% SoC for Type 1
EFB − C, Type 1 EFB + C and Type 2 EFB + C is shown in Figure 3a–c, comparing the
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]).
3.2. Electrochemical Impedance Spectroscopy
A close-up of the first semicircle of the negative half-cell EIS at 80% SoC for Type 1
EFB
−
C, Type 1 EFB + C and Type 2 EFB + C is shown in Figure 3a–c, comparing the
negative half-cell spectra of the complete cells, the middle size cells and the small size
cells. In Figure A1 the complete spectra of the negative half-cell EIS at 80% SoC are shown.
Batteries 2022,8, 66 5 of 15
The EIS measurements are conducted at 80% SoC, because the DCA (EN) test is executed
mostly at 80% SoC. Therefore, the processes effecting the DCA are expected to be most
visible in EIS measurements at this SoC. For this reason, the scope of the EIS measurements
was limited to 80% SoC. To improve the comparability, all spectra are shifted along the real
axis by the smallest real part of the impedance Re{
Z
} for all cell types and layouts. This
value is referred to as the internal resistance, R
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.
Batteries 2022, 8, x FOR PEER REVIEW 5 of 15
negative half-cell spectra of the complete cells, the middle size cells and the small size
cells. In Figure A1 the complete spectra of the negative half-cell EIS at 80% SoC are shown.
The EIS measurements are conducted at 80% SoC, because the DCA (EN) test is executed
mostly at 80% SoC. Therefore, the processes effecting the DCA are expected to be most
visible in EIS measurements at this SoC. For this reason, the scope of the EIS measure-
ments was limited to 80% SoC. To improve the comparability, all spectra are shifted along
the real axis by the smallest real part of the impedance Re{Z} for all cell types and layouts.
This value is referred to as the internal resistance, R0. 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.
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.
To enhance the comparability between the cell sizes, the impedance shown in Figures
3 and 4 is scaled by the number of active half plates. Since the original cell layout of the
complete cell differs between the battery types, where Type 1 test cells obtain 8P8N—
resulting in 15 active negative half plates out of 16 and the Type 2 test cell obtain 8P9N—
resulting in 16 active negative half plates out of 18. Within the middle size cell, four active
negative half plates are left; and, in the small size cells, only two active negative half plates
are left. The resulting scaling factor is given in Table 3.
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
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.
To enhance the comparability between the cell sizes, the impedance shown in Figures 3
and 4is scaled by the number of active half plates. Since the original cell layout of the com-
plete cell differs between the battery types, where Type 1 test cells obtain
8P8N—resulting
in 15 active negative half plates out of 16 and the Type 2 test cell obtain 8P9N—resulting in
16 active negative half plates out of 18. Within the middle size cell, four active negative
half plates are left; and, in the small size cells, only two active negative half plates are left.
The resulting scaling factor is given in Table 3.
Batteries 2022, 8, x FOR PEER REVIEW 6 of 15
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 4 shows the negative half-cell EIS at 80% SoC for Type 1 EFB − C, Type 1 EFB
+ C and Type 2 EFB + C in a Bode plot, where Figure 4a shows the absolute value of the
impedance of the complete cells, Figure 4b shows the phase shift of the complete cells,
Figure 4c shows the absolute value of the impedance of the middle size cells, Figure 4d
shows the phase shift of the middle size cells, Figure 4e shows the absolute value of the
impedance of the small size cells and Figure 4f shows the phase shift of the small size cells.
The DCA test according to EN 50342-6:2015 contains very short charging pulses (10 s),
where the largest part of the accepted current is taken in within the first second. That
might be the reason why the processes influencing the DCA can be visualized within the
first semicircle, between 1.5 and 365 Hz. This part of the spectra can also be seen within
the Bode plot, where the absolute value of the impedance at 2 Hz follows this trend: Type
1 EFB − C > Type 1 EFB + C > Type 2 EFB + C. This trend is the same for all cell layouts.
3.3. Kramers-Kronig
Before identifying an ECM for the EIS data obtained, they were preprocessed to avoid
systematic errors by using inaccurate measurement data due to violations of the station-
arity and time-invariance. For example, large residuals may appear at low frequencies
when the system was non-stationary during the measurement. A reliable method of de-
tecting irregularities in the measurement data is the K-K transformation.
The K-K transformation verifies the EIS measurement points that are stable and
causal if the real and the imaginary parts are interdependent [23,24]. The K-K transfor-
mation can be used to reconstruct the spectrum using the K-K test [25]. For this test, an
ECM consisting of a series connection of a single resistance and several RC circuits with
linear parameters is used. Each single RC circuit fulfills the K-K condition, and, therefore,
the entire model does as well. To minimize the residuals at the ends of the frequency dis-
persion, the range of the time constant investigated should be extended by some decades.
If the spectrum can be reconstructed with the K-K compliant RC circuits and the residuals
are evenly distributed, the measured spectrum can be considered valid. The data points
not fulfilling the K-K transformation are not evaluated any further.
3.4. Distribution of Relaxation Times
Finding characteristic points within an electrochemical impedance spectrum as well
as the determination of the ECM for representing the spectrum requires experience and
deep knowledge about the electrochemical processes at hand. The DRT features a more
reproducible and model-free approach to identifying the single processes [26]. It enhances
the impedance spectroscopy by transferring it from the frequency into the time domain
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.
Batteries 2022,8, 66 6 of 15
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
Figure 4shows the negative half-cell EIS at 80% SoC for Type 1 EFB
−
C, Type 1 EFB
+ C and Type 2 EFB + C in a Bode plot, where Figure 4a shows the absolute value of the
impedance of the complete cells, Figure 4b shows the phase shift of the complete cells,
Figure 4c shows the absolute value of the impedance of the middle size cells, Figure 4d
shows the phase shift of the middle size cells, Figure 4e shows the absolute value of the
impedance of the small size cells and Figure 4f shows the phase shift of the small size
cells. The DCA test according to EN 50342-6:2015 contains very short charging pulses (
10 s)
,
where the largest part of the accepted current is taken in within the first second. That might
be the reason why the processes influencing the DCA can be visualized within the first
semicircle, between 1.5 and 365 Hz. This part of the spectra can also be seen within the
Bode plot, where the absolute value of the impedance at 2 Hz follows this trend: Type 1
EFB −C > Type 1 EFB + C > Type 2 EFB + C. This trend is the same for all cell layouts.
3.3. Kramers-Kronig
Before identifying an ECM for the EIS data obtained, they were preprocessed to avoid
systematic errors by using inaccurate measurement data due to violations of the stationarity
and time-invariance. For example, large residuals may appear at low frequencies when
the system was non-stationary during the measurement. A reliable method of detecting
irregularities in the measurement data is the K-K transformation.
The K-K transformation verifies the EIS measurement points that are stable and causal
if the real and the imaginary parts are interdependent [
23
,
24
]. The K-K transformation can
be used to reconstruct the spectrum using the K-K test [
25
]. For this test, an ECM consisting
of a series connection of a single resistance and several RC circuits with linear parameters is
used. Each single RC circuit fulfills the K-K condition, and, therefore, the entire model does
as well. To minimize the residuals at the ends of the frequency dispersion, the range of the
time constant investigated should be extended by some decades. If the spectrum can be
reconstructed with the K-K compliant RC circuits and the residuals are evenly distributed,
the measured spectrum can be considered valid. The data points not fulfilling the K-K
transformation are not evaluated any further.
3.4. Distribution of Relaxation Times
Finding characteristic points within an electrochemical impedance spectrum as well
as the determination of the ECM for representing the spectrum requires experience and
deep knowledge about the electrochemical processes at hand. The DRT features a more
reproducible and model-free approach to identifying the single processes [
26
]. It enhances
the impedance spectroscopy by transferring it from the frequency into the time domain
using Fourier transformation [
27
]. Even when overlapping in the frequency domain, the
separation of each single polarization contribution is possible in the time domain, without
the need for any prior assumptions [
27
]. As a result, the major and minor polarization
processes involved are identified, even in complex impedance spectra.
To generate the DRT from the measured impedance spectra, the following optimization
problem needs to be solved
minn||A·x−b||2o(1)
By solving the algorithm, only non-negative values are allowed, since only positive
resistance values have a physical meaning. A common way to solve this problem is
the shift and cut-off approach: All measurements containing a positive imaginary part
are repudiated, and the ohmic offset is subtracted from all measurements, leaving only
Batteries 2022,8, 66 7 of 15
RC elements. Danzer introduced the generalized DRT analysis for EIS ohmic, inductive,
capacitive, resistive-capacitive and resistive-inductive effects [
27
]. Thus, the entire spectrum
can be reproduced.
As an example, to briefly describe the determination of the DRT, the reconstruction of
a series of RC elements is described. For an RC element, the imaginary and real parts are
described with
Im{ZRC}=ω·τ·R
1+(ω·τ)2(2)
and
Re{ZRC}=R
1+(ω·τ)2(3)
where
ω
is the angular frequency,
τ
is the time constant of the RC element and Ris the
resistance from the respective RC element. For the transformation procedure, both the
imaginary and real parts of the measured spectrum are used for the optimization function.
b=Re{ZRC}
Im{ZRC}(4)
The unknown distribution function of the DRT is
x=[h1. . . hk. . . hNτ]T(5)
where N
τ
is a multiple of the measured frequencies N
f
in the impedance spectra
Nτ=c·Nf[27]
with c
{1,2,3} to obtain a smoother distribution function [
26
]. The matrix Aneeds to be
solved for all frequencies and predefined time constants for the real and the
imaginary parts
:
A=
Re{1
1+j·ω1·τ1}. . . Re{1
1+j·ω1·τNτ}
.
.
.....
.
.
Re{1
1+j·ωN f ·τ1}. . . Re{1
1+j·ωN f ·τNτ}
Im{1
1+j·ω1·τ1}. . . Im{1
1+j·ω1·τNτ}
.
.
.....
.
.
Im{1
1+j·ωN f ·τ1}. . . Im{1
1+j·ωN f ·τNτ}
(6)
where the number of the RC elements used to display the spectra with an ECM must usually
be chosen to be higher than the number of measurements. The use of a regularization
term is one way to solve this ill-posed optimization problem. For example, Tikhonov
regularization can be used to calculate the DRT, adding the term |
λχ
|
2
to the optimization
function [28]. λis the regularization parameter.
By including the regularization term in the optimization problem, the matrix Aand
vector bare complemented:
AReg =A
λ·IeRnxn(7)
and
bReg =b
0eRn(8)
A further analysis of the number and location of peaks of the obtained distribution
function reveals the number of involved processes and even allows their quantification.
In Figure 5a, the DRT of the EIS at 80% SoC of the complete cells is shown. For all three
types of complete cells, four peaks are visible within the DRT, meaning that four processes
are involved: the peak at the smallest time constant
τ0≈
0.003 s, the peak at
τ1≈0.07 s
, a
smaller peak around
τ2≈
1 s and the last peak at
τ3≈
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,
Batteries 2022,8, 66 8 of 15
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
τ1≈
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
τ3≈
10 s is also visible, surrounded by several smaller peaks, which might be
present due to the time constant
τ2
. These findings are used to evaluate the most important
processes, to choose the ECM elements and the starting parameters of the fitting algorithm.
Batteries 2022, 8, x FOR PEER REVIEW 8 of 15
By including the regularization term in the optimization problem, the matrix A and
vector b are complemented:
𝐴
 =󰇣
𝐴
𝜆·𝐼 𝜖
ℝ
󰇤 (7)
and
𝑏 =󰇣 𝑏
0 𝜖
ℝ
󰇤 (8)
A further analysis of the number and location of peaks of the obtained distribution
function reveals the number of involved processes and even allows their quantification.
In Figure 5a, the DRT of the EIS at 80% SoC of the complete cells is shown. For all three
types of complete cells, four peaks are visible within the DRT, meaning that four processes
are involved: the peak at the smallest time constant τ0 ≈ 0.003 s, the peak at τ1 ≈ 0.07 s, a
smaller peak around τ2 ≈ 1 s and the last peak at τ3 ≈ 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 τ1 ≈ 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 τ3 ≈ 10 s is also visible, surrounded by several smaller peaks, which might be pre-
sent due to the time constant τ2. These findings are used to evaluate the most important
processes, to choose the ECM elements and the starting parameters of the fitting algo-
rithm.
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.
Table 4. Time constants determined with DRT of the EIS at 80% SoC of the laboratory test cells.
Type 1 EFB − C Type 1 EFB + C Type 2 EFB + C
Time Con-
stant
Complete
Cell
Middle Size
Cell
Small Size
Cell
Complete
Cell
Middle Size
Cell
Small Size
Cell
Complete
Cell
Middle Size
Cell
Small Size
Cell
τ0/sξ 0.003 - - 0.002 - - 0.003 - -
τ1/sξ 0.072 0.08 0.10 0.06 0.054 0.044 0.073 0.056 0.056
τ2/sξ 2.359 2.32 1.094 1.094 2.816 0.568 2.984 1.436 0.451
τ3/sξ 13.495 11.19 5.838 7.415 15.9 3.629 19.025 19.025 2.283
Assuming a Gaussian distribution within the DRT, the occurring peaks can be as-
signed to a single Gaussian bell shape, respectively. As a result, the location of the peak
correlates with the characteristic time constant of the process. The area that a bell shape
covers is equal to the polarization resistance of the correlating process. The variance σ2, or
the width, correlates with the frequency range at which the process occurs. A process with
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.
Table 4. Time constants determined with DRT of the EIS at 80% SoC of the laboratory test cells.
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
τ0/sξ0.003 - - 0.002 - - 0.003 - -
τ1/sξ0.072 0.08 0.10 0.06 0.054 0.044 0.073 0.056 0.056
τ2/sξ2.359 2.32 1.094 1.094 2.816 0.568 2.984 1.436 0.451
τ3/sξ13.495 11.19 5.838 7.415 15.9 3.629 19.025 19.025 2.283
Assuming a Gaussian distribution within the DRT, the occurring peaks can be assigned
to a single Gaussian bell shape, respectively. As a result, the location of the peak correlates
with the characteristic time constant of the process. The area that a bell shape covers is
equal to the polarization resistance of the correlating process. The variance
σ2
, 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
The measurement data can be evaluated within the frequency domain using the ECM,
consisting of simple passive electrical components such as resistors, capacitors and induc-
tors. These components are interconnected in networks and can represent electrochemical
processes. The modeling approach is mainly based on Randles’ circuit [
29
], which has
been employed with different levels of modification but always with the fundamental
structure. The illustration and implementation of the model structure are very simple [
30
].
Compared to other models, the ECM offers a higher computing speed and can there-
fore be used to simulate complex systems or to implement even small simulation step
sizes [
30
].
This makes
it possible to even simulate dynamic battery operations. However,
parameterization measurements must be executed, and ECMs have a restricted validity
range [30].
Batteries 2022,8, 66 9 of 15
Most ECMs consist of an internal resistance R
0
, an inductivity, and several semicir-
cles, 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
τ0
is only distinct for complete cells, only
τ1
,
τ2
and
τ3
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:
ZARC =1
1
R+A·(jω)ξ(9)
The CPE element fits the spectra much better and still contains a capacitive charac-
teristic, which is used for evaluation. In Figure 6, the chosen ECM for the electrochemical
impedance spectra is given. For each time constant identified with the DRT, one ZARC
element is used.
𝐴
𝑗
𝑗
𝑗
𝑗
𝐴
Figure 6. ECM for the EIS data.
The formula of the chosen ECM is
Z=R0+L·(jω)γL+R1
1+τ1·(jω)ξ1+R2
1+τ2·(jω)ξ2+R3
1+τ3·(jω)ξ3(10)
with
τ=R·A(11)
3.6. Parameterization of the Equivalent Circuit Model
The first semicircle of the negative half-cell EIS at 80% SoC for Type 1 EFB
−
C, Type 1
EFB + C and Type 2 EFB + C, with the corresponding fit, are shown in Figure 7a–c for the
complete cells, for the middle size cells and for the small size cells, correspondingly. Even
though the fitting result is not exactly the same as the measurement, the first semicircle is
well represented. To minimize the deviation between the modelled data and the EIS, the
complex least-squares fitting algorithm was employed [
31
]. The starting parameters, lower
and upper limits for the ECM fit are given in Table 5. The resulting fitting parameters used
are stated in Table 6.
The starting parameter for the internal resistance R0is 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 ξ1,ξ2and ξ3are taken, and the fit is repeated with fixed values.
The fitting results of the complete laboratory test cells are shown in Figure 8a for
τ1
,
in Figure 9a for R
1
and in Figure 10a for C
1
. The parameters of the middle size cells are
shown in Figures 8,9and 10b, and the fitting results of the small size cells are presented in
Figures 8,9and 10c.
Batteries 2022,8, 66 10 of 15
Batteries 2022, 8, x FOR PEER REVIEW 10 of 15
semicircle is well represented. To minimize the deviation between the modelled data and
the EIS, the complex least-squares fitting algorithm was employed [31]. The starting pa-
rameters, lower and upper limits for the ECM fit are given in Table 5. The resulting fitting
parameters used are stated in Table 6.
The starting parameter for the internal resistance R0 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 exe-
cuted with variable values for all ξ. However, for the purposes of comparability, the av-
erage value for each ξ1, ξ2 and ξ3 are taken, and the fit is repeated with fixed values.
Table 5. Starting parameters, lower and upper limits for the ECM fit.
Parameter Lower Limit Starting Value Upper Limit
R0/Ω R0 R0 R0 + 0.05
L/μH 0 200 10,000
λL 0 0.4 1
R1/Ω 0 0.3 1
τ1/sξ τ1 τ1 τ1
ξ1 0.849 0.849 0.849
R2/Ω 0 0.4 1
τ2/sξ τ2 τ2 τ2
ξ2 0.664 0.664 0.664
R3/Ω 0 0.5 2
τ3/sξ τ3 τ3 τ3
ξ3 0.75 0.75 0.75
Figure 7. Negative half-cell EIS at 80% SoC and fit with ECM of the laboratory test cells for (a) com-
plete cell: 8P8N/8P9N, (b) middle size cell: 3P2N and (c) small size cell: 2P1N visualized in a Nyquist
diagram.
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.
Table 5. Starting parameters, lower and upper limits for the ECM fit.
Parameter Lower Limit Starting Value Upper Limit
R0/ΩR0R0R0+ 0.05
L/µH 0 200 10,000
λL0 0.4 1
R1/Ω0 0.3 1
τ1/sξτ1τ1τ1
ξ10.849 0.849 0.849
R2/Ω0 0.4 1
τ2/sξτ2τ2τ2
ξ20.664 0.664 0.664
R3/Ω0 0.5 2
τ3/sξτ3τ3τ3
ξ30.75 0.75 0.75
Table 6. Parameters for the ECM fit and error.
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
R0/Ω0 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
λL0.94 0.98 0.97 0.94 0.94 0.95 0.95 1 0.18
R1/Ω0.4 0.42 0.52 0.34 0.303 0.2 0.309 0.28 0.16
τ1/sξ0.072 0.08 0.10 0.06 0.054 0.044 0.073 0.056 0.056
ξ10.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85
R2/Ω0.534 0.533 0.3 0.3 0.6 0.3 0.384 0.3 0.3
τ2/sξ2.359 2.32 1.094 1.094 2.816 0.568 2.984 1.436 0.451
ξ20.664 0.664 0.664 0.664 0.664 0.664 0.664 0.664 0.664
R3/Ω0.218 0.62 1.16 0.41 1.452 0.366 0.37 0.101 0.1
τ3/sξ13.495 11.19 5.838 7.415 15.9 3.629 19.025 19.025 2.283
ξ30.75 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
Batteries 2022,8, 66 11 of 15
Batteries 2022, 8, x FOR PEER REVIEW 11 of 15
Table 6. Parameters for the ECM fit and error.
Type 1 EFB − C Type 1 EFB + C Type 2 EFB + C
Time Con-
stant
Complete
Cell
Middle Size
Cell
Small Size
Cell
Complete
Cell
Middle Size
Cell
Small Size
Cell
Complete
Cell
Middle Size
Cell
Small Size
Cell
R0/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
λL 0.94 0.98 0.97 0.94 0.94 0.95 0.95 1 0.18
R1/Ω 0.4 0.42 0.52 0.34 0.303 0.2 0.309 0.28 0.16
τ1/sξ 0.072 0.08 0.10 0.06 0.054 0.044 0.073 0.056 0.056
ξ1 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85
R2/Ω 0.534 0.533 0.3 0.3 0.6 0.3 0.384 0.3 0.3
τ2/sξ 2.359 2.32 1.094 1.094 2.816 0.568 2.984 1.436 0.451
ξ2 0.664 0.664 0.664 0.664 0.664 0.664 0.664 0.664 0.664
R3/Ω 0.218 0.62 1.16 0.41 1.452 0.366 0.37 0.101 0.1
τ3/sξ 13.495 11.19 5.838 7.415 15.9 3.629 19.025 19.025 2.283
ξ3 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
The fitting results of the complete laboratory test cells are shown in Figure 8a for τ1,
in Figure 9a for R1 and in Figure 10a for C1. The parameters of the middle size cells are
shown in Figures 8 till 10b, and the fitting results of the small size cells are presented in
Figures 8 till 10c.
Figure 8. Fitting result τ1 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 R1 of the laboratory test cells for (a) complete cell: 8P8N/8P9N, (b) middle
size cell: 3P2N and (c) small size cell: 2P1N.
Figure 8.
Fitting result
τ1
of the laboratory test cells for (
a
) complete cell: 8P8N/8P9N, (
b
) middle
size cell: 3P2N and (c) small size cell: 2P1N.
Batteries 2022, 8, x FOR PEER REVIEW 11 of 15
Table 6. Parameters for the ECM fit and error.
Type 1 EFB − C Type 1 EFB + C Type 2 EFB + C
Time Con-
stant
Complete
Cell
Middle Size
Cell
Small Size
Cell
Complete
Cell
Middle Size
Cell
Small Size
Cell
Complete
Cell
Middle Size
Cell
Small Size
Cell
R0/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
λL 0.94 0.98 0.97 0.94 0.94 0.95 0.95 1 0.18
R1/Ω 0.4 0.42 0.52 0.34 0.303 0.2 0.309 0.28 0.16
τ1/sξ 0.072 0.08 0.10 0.06 0.054 0.044 0.073 0.056 0.056
ξ1 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85
R2/Ω 0.534 0.533 0.3 0.3 0.6 0.3 0.384 0.3 0.3
τ2/sξ 2.359 2.32 1.094 1.094 2.816 0.568 2.984 1.436 0.451
ξ2 0.664 0.664 0.664 0.664 0.664 0.664 0.664 0.664 0.664
R3/Ω 0.218 0.62 1.16 0.41 1.452 0.366 0.37 0.101 0.1
τ3/sξ 13.495 11.19 5.838 7.415 15.9 3.629 19.025 19.025 2.283
ξ3 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
The fitting results of the complete laboratory test cells are shown in Figure 8a for τ1,
in Figure 9a for R1 and in Figure 10a for C1. The parameters of the middle size cells are
shown in Figures 8 till 10b, and the fitting results of the small size cells are presented in
Figures 8 till 10c.
Figure 8. Fitting result τ1 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 R1 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.
Batteries 2022, 8, x FOR PEER REVIEW 12 of 15
Figure 10. Fitting result C1 of the laboratory test cells for (a) complete cell: 8P8N/8P9N, (b) middle
size cell: 3P2N and (c) small size cell: 2P1N.
4. Discussion
It has been found that, disregarding the layout of the cells, the DCA is the largest for
Type 2 EFB + C cells, followed by Type 1 EFB + C cells, while the Type 1 EFB − C cell
exhibits the lowest DCA [16]. During the recuperation with voltage restriction, the charg-
ing current cannot be infinitely increased, since it is limited by the supply of Pb2+ 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 [32–34].
The mechanisms of how additives enable higher DCA are not yet fully understood.
One suggestion is that the current-enhancing additives may increase the porosity of the
negative electrode. This would increase the electrochemical active surface of the negative
electrode, decrease the maximum PbSO4 crystal size, thus increasing its dissolution and
decreasing the distance for Pb2+ ion transport. These effects should be visible within the
EIS measurements.
Interpreting EIS measurement is not straightforward, as the results found in the lit-
erature showed different shapes for battery, cell and half-cell spectra, mainly depending
on the measurement regime, e.g., SoC, superimposed DC current and frequency range.
For most EIS measurements, two capacitive semicircles can be identified [5,6,8,17,35–45].
It is mostly agreed that each semicircle relates to one separate process of the electrochem-
ical reaction, which could be charge transfer, chemical reactions, mass transport or ad-
sorption processes. However, the interpretation of the semicircles varies between the
sources. That said, it is mostly agreed that one of the found capacitive semicircles is most
likely related to the charge transfer reaction alongside the double-layer capacitance; where
both are related to the porosity of the electrodes and are thereby, associated with the elec-
trochemical reaction at the electrode-electrolyte interface [45]. According to the suggested
mechanisms regarding the influence of the additives, the charge transfer process might be
the semicircle where the impact of additives is noticeable.
In several works, the high-frequency semicircle is attributed to the reactions inside
the porous structure of both positive and negative electrodes and thus the charge transfer
of the charge/discharge reaction [5,6,36–39,43,45–47]. The current [46,47] and SoC depend-
ency [5,43] of the high-frequency semicircle also suggests its coherence with the charge
transfer reaction. The highest correlation between DCA, the spectra and the fitting param-
eters is indeed obtained within the first semicircle. Therefore, the authors agree with the
statements found in the literature, that the high-frequency semicircle represents the
charge transfer reaction which can be differentiated by using different additives for en-
hancing the DCA.
However, Kowal et al. described the SoC dependency of all semicircles [40] and
showed a current dependency of the low-frequency semicircle [17,40]. Both SoC and cur-
rent are highly correlating to the charge transfer reaction. Instead of separating the two
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.
4. Discussion
It has been found that, disregarding the layout of the cells, the DCA is the largest
for Type 2 EFB + C cells, followed by Type 1 EFB + C cells, while the Type 1 EFB
−
C
cell exhibits the lowest DCA [
16
]. During the recuperation with voltage restriction, the
charging current cannot be infinitely increased, since it is limited by the supply of Pb
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 [32–34].
The mechanisms of how additives enable higher DCA are not yet fully understood.
One suggestion is that the current-enhancing additives may increase the porosity of the
negative electrode. This would increase the electrochemical active surface of the negative
electrode, decrease the maximum PbSO
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.
Batteries 2022,8, 66 12 of 15
Interpreting EIS measurement is not straightforward, as the results found in the
literature showed different shapes for battery, cell and half-cell spectra, mainly depending
on the measurement regime, e.g., SoC, superimposed DC current and frequency range.
For most
EIS measurements, two capacitive semicircles can be identified [
5
,
6
,
8
,
17
,
35
–
45
].
It is
mostly agreed that each semicircle relates to one separate process of the electrochemical
reaction, which could be charge transfer, chemical reactions, mass transport or adsorption
processes. However, the interpretation of the semicircles varies between the sources. That
said, it is mostly agreed that one of the found capacitive semicircles is most likely related
to the charge transfer reaction alongside the double-layer capacitance; where both are
related to the porosity of the electrodes and are thereby, associated with the electrochemical
reaction at the electrode-electrolyte interface [
45
]. According to the suggested mechanisms
regarding the influence of the additives, the charge transfer process might be the semicircle
where the impact of additives is noticeable.
In several works, the high-frequency semicircle is attributed to the reactions inside
the porous structure of both positive and negative electrodes and thus the charge trans-
fer of the charge/discharge reaction [
5
,
6
,
36
–
39
,
43
,
45
–
47
]. The current [
46
,
47
] and SoC
dependency [5,43]
of the high-frequency semicircle also suggests its coherence with the
charge transfer reaction. The highest correlation between DCA, the spectra and the fitting
parameters is indeed obtained within the first semicircle. Therefore, the authors agree
with the statements found in the literature, that the high-frequency semicircle represents
the charge transfer reaction which can be differentiated by using different additives for
enhancing the DCA.
However, Kowal et al. described the SoC dependency of all semicircles [
40
] and
showed a current dependency of the low-frequency semicircle [
17
,
40
]. Both SoC and
current are highly correlating to the charge transfer reaction. Instead of separating the
two semicircles
into two processes, others suggested that one reaction occurs in
two separate
semicircles, if it is executed in several consecutive or parallel steps. Hampson suggested
that the oxidation from lead to Pb
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
To obtain convenient EIS results, all influence factors need to be kept constant during
the whole measurement of each spectrum. To neglect influences from the previous usage
of the test cell, a micro cycling approach was used. However, during the evaluation of
electrochemical impedance spectra, enhanced care needs to be taken. A semicircle might
not be observable because it occurs at low frequencies that are not accessible under the
present conditions, or one semicircle might be masked by another. Therefore, the analysis
of the DRT can give an idea about the usable ECM and identify starting values for the
parameter prediction. This powerful method can be used to model the behavior of batteries
without requiring previous knowledge.
For all cell types and layouts, the time constant
τ1
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
Batteries 2022,8, 66 13 of 15
between the capacitance C
1
and the DCA, and no correlation between the parameters of
the second semicircle and the DCA.
As a conclusion, EIS measurements can be used as a technique to predict a high
or low DCA for cells. Further measurements need to be validated for more conclusive
results. Thereby, different additives need to be validated to confirm the correlation between
EIS parameters and DCA. Moreover, different superimposed DC currents and various
SoCs should be tested in future work to indicate the best suitable testing procedure to
forecast DCA.
Author Contributions:
Conceptualization, S.B.; methodology, S.B. and J.K.; software, S.B.; validation,
S.B., J.K. and E.K.; formal analysis, S.B.; investigation, S.B.; resources, S.B.; data curation, S.B.;
writing—original draft preparation, S.B.; writing—review and editing, S.B., J.K., B.B., J.S. and E.K.;
visualization, S.B.; supervision, J.K.; project administration, S.B.; funding acquisition, J.K. All authors
have read and agreed to the published version of the manuscript.
Funding:
This research was funded by Advanced Lead-Acid Battery Consortium, grant number
1618BST_CNP7_INTW.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement:
For any details regarding the data reported please contact S. Bauknecht
via mail: [email protected].
Acknowledgments:
The authors would like to thank Eberhard Meissner and Matt Raiford for their
continuous feedback and discussions. Moreover, we would like to thank Harald Koch and his team
at combatec GmbH for the tremendous work they did in disassembling the LABs to the cell level.
Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design
of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or
in the decision to publish the results.
Appendix A
Batteries 2022, 8, x FOR PEER REVIEW 14 of 15
Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the
design of the study; in the collection, analyses, or interpretation of data; in the writing of the manu-
script, or in the decision to publish the results.
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 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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