# Advanced Electrochemical Impedance Spectroscopy of Industrial Ni-Cd Batteries

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}, reflecting high intrinsic rates of the redox electron transfer processes in Ni-Cd cells.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Ni-Cd Cells and Block Arrangements

#### 2.2. Electrochemical Impedance Spectroscopy (EIS) and Calibration Method

_{sig}is swept within a defined frequency range and the signals are then transformed into the frequency domain by a discrete Fourier transform (DFT) process. The resulting complex signals I(ω) and V(ω) are then used to compute the complex raw impedance Z(ω)

_{raw}= V(ω)/I(ω).

_{ser}and the gain error term G

_{e}. The measured impedance Z

_{M}is given by the following equation:

_{T}is the true impedance, and Z

_{ser}and G

_{e}are the error coefficients. The two error coefficients Z

_{ser}and G

_{e}are determined in a calibration process where at least two different known calibration standards are measured. Here, we use one short and one 10 mΩ shunt standard. Once the error coefficients are known, the true impedance can be calculated by transforming Equation (1) into the following form:

_{sol}and the charge transfer resistance R

_{ct}. The double-layer capacitance C

_{dl}is derived from the constant phase reactance CPE

_{dl}as follows [21,22]:

_{diff}is obtained from the Warburg impedance Z

_{diff}by the following relation:

^{2}/D, L is the length of the diffusion, D is the diffusion coefficient, and the fractional-order P is set as 0.5.

#### 2.3. Electromagnetic Finite Element Method (FEM) Modeling

## 3. Results and Discussion

#### 3.1. EIS Calibration Method

_{T}) is obtained by Equation (2).

#### 3.2. Calibrated EIS versus SoC

#### 3.3. Equivalent Electric Circuit Model and Fit Parameters

_{L}], a solution resistance element R

_{sol}, a ZARC element [R

_{ct}/CPE

_{dl}], and a Warburg diffusion element (W

_{diff}). The constant phase element (CPE) is used because of the porous nature of the electrodes and adsorption capacitances [31,32], while the inductive element is included to model the physical arrangement of the cell plates and connectors. From this Randles circuit, the following equation is used to obtain the impedance of the “battery under test” (BUT), Z

_{BUT}:

_{c}is the characteristic angular frequency of the CPE, α is the fractional order of the CPE, L is the inductance, and Z

_{diff}is the impedance of the Warburg diffusion.

_{sol}as identified by the intersection point of the impedance curve with the x-axis. At intermediate frequencies, the charge transfer resistance and the double-layer capacitance are shown as a depressed semi-circle. At low frequencies, the diffusion tail is obtained on the right part of the Nyquist plot.

_{sol}shows very similar values for the three cells, with an average of R

_{sol}= 1.433 mΩ. Due to the depressed semi-circle and the corresponding fit errors, the charge transfer resistance R

_{ct}values differ significantly. An average value of R

_{ct}= 0.042 mΩ is obtained, with a significantly higher value for Cell 3, which is sourced from Block 2, while Cell 1 and Cell 2 are from Block 1. Similarly, the diffusion resistances R

_{diff}of Cell 1 and Cell 2 are very similar, while R

_{diff}of Cell 3 is higher, which can be, for instance, due to small differences in the SoC leading to faster electrode kinetics [33]. Lastly, the inductance values L are very similar for all three cells because the inductance is related to the geometry of the cell and electrode plates, which is very similar for Block 1 and Block 2. Thereby, it is noted that the imaginary part of the impedance is dominated by the diffusion resistance, whereas the internal resistance is dominated by the resistance of the solution resistance, which agrees with literature observations [13].

#### 3.4. EIS Comparison of Cells and Blocks

#### 3.5. Electrochemical Interpretation

_{sol}, R

_{ct}, and C

_{dl}in relation to the principal components of a Ni-Cd cell. The electrochemical redox reactions (reduction and oxidation) include the discharge where Ni

^{3+}is reduced to Ni

^{2+}and the hydroxide ions (OH

^{−}) are transferred to the cadmium anode, where the oxidation occurs from Cd

^{0}to Cd

^{2+}. The main electrochemical model parameter describing the redox reaction is the charge transfer resistance R

_{ct}, reflecting the exchange between the ions and the electrons at the electrode interfaces. R

_{ct}is related to the reaction kinetics described by the Butler–Vomer equation for a charge-transfer controlled electrochemical reaction with an inverse relation to the exchange current density, R

_{ct}= R ×T/(n × F × i

_{0}), where i

_{0}is the exchange current density, R is the gas constant, T is the temperature, n is the number of electrons in the redox reaction, and F is the Faraday constant [34]. Considering Cell 3 with an R

_{ct}of 0.079 mΩ obtained at 296 K, and a two-electron charge transfer process, then the exchange current density is equal to i

_{0}= 0.23 A·cm

^{−2}, which corresponds to 162 A per cell with an electrode surface area of 700 cm

^{2}. The exchange current density reflects intrinsic rates of electron transfer between electrodes, and provides insights into the nature of the electrodes, their structure, as well as their physical parameters such as surface roughness [35]. The larger the exchange current density, the faster the redox reactions. Compared to Li-ion batteries which show typical i

_{0}values of around 5 mA·cm

^{−2}[36], the Ni-Cd current density is significantly larger. For instance, this is apparent by the fact that higher C-rates can be obtained from Ni-Cd batteries, which are typically required in uninterruptible power supply (UPS) applications. Such applications require mixed high and low discharge rates for electrical loads between 30 min and 3 h [37]. The highly conductive ionic solution in the Ni-Cd cell is represented in the model by a purely resistive element R

_{sol}. The solution resistance depends on the ionic concentrations, temperature, and geometry of the area in which the current is flowing. The conductivity k of the ionic solution, in units of [S/m], is obtained from the solution resistance R

_{sol}by the following relation, k = L

_{n}/(R

_{sol}× A), where A is the surface area, and L

_{n}is the length of the charge path. Considering Cell 3 with an R

_{sol}= 1.43 mΩ and based on the geometrical dimensions of the Ni-Cd cell, we obtain conductivity values around 1 S·cm

^{−1}, which compare very well with the conductivity of bulk KOH solution found in literature [38]. The deposition of ion charges at the electrode interface is countered by electronic charges at the electrode interface, creating a double-layer capacitance CPE

_{dl}. Whereas the ion diffusion process inside the electrode particles of the positive Ni active material is represented by a Warburg impedance, which is a solid-state physical diffusion process [39].

## 4. Conclusions

_{sol}, R

_{ct}, and C

_{dl}were described in relation to the principal components of a Ni-Cd cell and the electrochemical redox reactions. Based on the modelled parameters, intrinsic properties were estimated such as the solution conductivity and the exchange current density, reflecting high intrinsic rates of the redox electron transfer processes in Ni-Cd cells compared to LiBs.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) A sketch of the Ni-Cd block showing the two cells connected in series with a bar connector, and the respective cell electrode terminals. (

**b**) Two Ni-Cd blocks (Block 1, Block 2) consisting of two cells per block connected in series with a bar connector. (

**c**) EIS measurement setup consisting of a power analyzer containing two source measure units (SMUs), a PC with EIS control and visualization software, and the calibration standards and fixture. (

**d**) EIS measurement modelled by combining an ideal impedance meter with the series error impedance Z

_{ser}and a gain error term G

_{e}. The true impedance Z

_{T}is computed from the measured and error-affected impedance Z

_{M}.

**Figure 2.**(

**a**) The 3-D electromagnetic computer-aided design (CAD) model of the Ni-Cd battery block containing two cells connected in series and the cable fixture. A four-wire connection scheme for force and sense is considered, and a metallic cell connector is used to connect both cells. (

**b**) The 3-D EMPro simulation of the cell fixture and contacts, including a planar view of the magnetic H-filed distribution around the wires. (

**c**) Simulated real impedance (upper panel) and imaginary impedance (lower panel) across the frequency range of 10 Hz to 1 kHz. The geometry and model dimensions are based on the original Ni-Cd block (SAFT SBM112) and the EIS setup.

**Figure 3.**EIS calibration (upper row) and correction process (lower row) using printed circuit board (PCB) shunt standard. The shunt standard is measured at the BUT measurement plane and the short is measured at the BUT negative terminal (upper right). The calibration raw data (upper left) are used together with the standard definition data to obtain the error coefficients which are forwarded to the correction process. The raw BUT data (lower left) and the calibrated data (lower right) are shown for a Ni-Cd single cell with 112 Ah measured at 23 °C.

**Figure 4.**(

**a**) Calibrated EIS results shown as the real impedance (resistance) and imaginary impedance (reactance), across the measurement frequency of 10 mHz to 500 Hz. The results are shown for Ni-Cd Cell 1 at six SoC charging levels, from 0% to 90%, and measured at 23 °C. The high, intermediate, and low SoC curve transitions are indicated in the graphs. (

**b**) The corresponding EIS results shown in a Nyquist plot.

**Figure 5.**Equivalent electric circuit model of a Ni-Cd cell. (

**a**) Randles model with four elements. (

**b**) Comparison of experimental impedance spectra and simulation for Cell 1, (

**c**) Cell 2, and (

**d**) Cell 3 at 10% SoC and 23 °C. (

**e**) Interpretation of the electrochemical processes in the Nyquist plot at different frequencies from 100 mHz to 500 Hz.

**Figure 6.**Calibrated EIS spectra of three cells and the extracted electrochemical parameters. (

**a**) Nyquist plot of the impedance spectra of the three cells. (

**b**) Extracted electrochemical parameters from the fitted EIS results for Cell 1, Cell 2, and Cell 3, simulated using Z-view software. The spectra were obtained in the frequency range of 100 mHz to 500 Hz.

**Figure 7.**Nyquist plot of the impedance spectra of the single cells, Cell 1, Cell 2, Cell 3, and Cell 4, compared to the block impedance for Block 1 (

**a**) and Block 2 (

**b**). The cells are at 20% SoC and the EIS measurements are performed using 3 A excitation current across the frequency range of 10 mHz to 2 kHz at 23 °C.

**Figure 8.**Electrochemical model parameters and principal components of a Ni-Cd cell. During discharge, the cathode (Ni

^{3+}O

^{2−}-OH

^{−}) reacts with water to produce Ni

^{2+}(OH)

_{2}and hydroxide ions (OH

^{−}) are transferred to the cadmium anode. The main electrochemical processes during discharge are described, including equivalent circuit model elements, such as R

_{ct}, R

_{sol}, and C

_{dl}(left and right panels).

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**MDPI and ACS Style**

Al-Zubaidi R-Smith, N.; Kasper, M.; Kumar, P.; Nilsson, D.; Mårlid, B.; Kienberger, F.
Advanced Electrochemical Impedance Spectroscopy of Industrial Ni-Cd Batteries. *Batteries* **2022**, *8*, 50.
https://doi.org/10.3390/batteries8060050

**AMA Style**

Al-Zubaidi R-Smith N, Kasper M, Kumar P, Nilsson D, Mårlid B, Kienberger F.
Advanced Electrochemical Impedance Spectroscopy of Industrial Ni-Cd Batteries. *Batteries*. 2022; 8(6):50.
https://doi.org/10.3390/batteries8060050

**Chicago/Turabian Style**

Al-Zubaidi R-Smith, Nawfal, Manuel Kasper, Peeyush Kumar, Daniel Nilsson, Björn Mårlid, and Ferry Kienberger.
2022. "Advanced Electrochemical Impedance Spectroscopy of Industrial Ni-Cd Batteries" *Batteries* 8, no. 6: 50.
https://doi.org/10.3390/batteries8060050