# A high-performance AC excitation and measuring system for on-line estimation of Lithium-ion battery cells' internal impedance

Ioannis Botas, Olga Sokolaki, Emmanuel Tatakis

## ► To cite this version:

Ioannis Botas, Olga Sokolaki, Emmanuel Tatakis. A high-performance AC excitation and measuring system for on-line estimation of Lithium-ion battery cells' internal impedance. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0069. hal-05081838

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https://utc.hal.science/hal-05081838v1

Submitted on 23 May 2025

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# A high-performance AC excitation and measuring system for on-line estimation of Lithium-ion battery cells' internal impedance

Ioannis Botas, Olga Sokolaki, and Emmanuel Tatakis

UNIVERSITY OF PATRAS

Department of Electrical and Computer Engineering

University Campus of Patras, GR 26504, Rion-Patras, Greece

Tel.: +30 / 2610 - 996412

E-Mail: up1066629@ac.upatras.gr, osokolaki@ac.upatras.gr, and e.c.tatakis@ece.upatras.gr

URL: http://www.lemec.ece.upatras.gr

## Acknowledgements

This work is part of a Research Program, financially supported by «SUNLIGHT GROUP ENERGY STORAGE SYSTEMS». The Conference participation was supported by the IPCEI on ME/CT program of Infineon Technologies Austria AG.

## Keywords

«Lithium-ion battery», «Battery impedance measurement», «EIS», «Bi-directional converters», «Current loop»

## Abstract

Nowadays, there is a rapid increase in the use of Lithium-Ion batteries for many different applications, due to their superiority in terms of energy density, efficiency, and self-discharge rate. In this paper, a high-performance system for estimation of Lithium-ion battery cells' internal impedance is presented. The main concept lies in the excitation of cells through a multi-sine controlled current signal, instead of using the time-consuming sequential harmonic injection. The proposed technique was implemented for a LiFePO4, 3.2V/30Ah battery cell through a bi-directional power converter, for integrating both charging and discharging capability. A laboratory prototype of this converter was designed and implemented, to investigate theoretical assumptions. The experimental results are eventually analyzed, to evaluate the influence of the various experimental conditions on the cell internal impedance.

## I. Introduction

Lithium-ion batteries are ubiquitous in modern technology, powering a broad set of devices and systems. The most popular of their key

applications are the Energy Storage Systems (ESS) and the Electric Vehicles (EVs), while they are the basic candidate in Consumer Electronics and gain more ground in Aerospace and Medicine. This ever-wider usage has brought a more urgent need for monitoring of their internal state, to enhance the systems' safety and reliability, as well as to optimize their performance and prolong their lifespan.

A promising and increasingly popular method, which can provide insights to the internal state of a battery cell, is the Electrochemical Impedance Spectroscopy (EIS) [1], [2]. This technique is based on the AC signal injection in a frequency range and the measurement of the corresponding battery cell response. Concerning EIS application, most works use commercial testers off-line, to excite the cells and take the required measurements [3], [4]. On the other hand, there are very few works that apply the EIS technique on-line with a custom-made system, based either on a switching power converter [5]-[7], or a linear regulator [8], [9].

However, the systems used in the aforementioned works have shortcomings and limitations, mainly in means of the signal type and injection procedure, or the power handling capability.

Firstly, in works [5], [8] the EIS technique is applied with a high AC/DC ratio in the injected current signals, for the sake of accuracy. Nevertheless, this approach comes at the cost of higher SoC variations, because of the micro-cycling effect caused by the superimposed AC component [10]. For results validity, the system should be linear and invariant during the injection and measurement process. To meet this requirement, the SoC and internal temperature variation limit is often set as 1% and 1°C respectively, as a rule of thumb throughout the process. Therefore, a high AC amplitude introduces an extra limitation to the system linearity region, by changing the maximum
acceptable product of current flow time and
amplitude, compared to the case of pure DC
current. In general, it is good practice to keep the
total measurement time the shortest possible, to
minimize system state variations. This is also an
issue for the measuring methods presented in [5],
[9], where the AC signal injection is performed
by sweeping the desired frequency spectrum, so
the different frequency components are applied
one after the other. This sequential injection is
time-consuming and mainly limits either the DC
current flowing into the battery cell, or the
minimum frequency, or the maximum number of
frequency components that can be injected. This
issue is addressed in [7], where the injection of a
square-wave excitation current is proposed as a
solution. Through this signal, the simultaneous
injection of multiple harmonics is achieved, but
the shortcoming of this choice lies in the
harmonic amplitude drop with frequency raise.
This fact leads to a corresponding accuracy drop,
additional to the one caused by the fewer samples
in higher harmonics, given an upper limit of the
sampling rate. This limit is determined by the
converter switching frequency, from the
perspective that this is the highest possible data
variation rate.

Secondly, the works [8], [9] are focused on the
EIS technique application in low-capacity battery
cells, in the order of hundreds mAh to some units
of Ah. Hence, the linear regulator used fits the
case for the simplicity it offers, given the low
power that must be handled. In our case though,
the high cell capacity rules out this choice,
because of high conduction losses.

To address the above issues and bridge the
mentioned gaps, we developed a system based on
a DC/DC converter, with both charging and
discharging capability, as well as an EIS
technique application feature. The converter can
operate bidirectionally under controlled DC
current, up to the C-rate of a LiFePO4, 3.2/30Ah
cell. This current rating was selected, so that the
full investigation of the cell behavior under
loading conditions can be realized through the
developed system. Regarding EIS, the signal
injection can be performed in a quite lower
AC/DC ratio than this mentioned in the former
relevant works, with adequately high resolution in
measurements of the cell current and voltage, as
will be presented further on. This AC ratio was
thoroughly selected through an extracted
equation, in contrast to the arbitrary selection
made in the existing literature.

II. The proposed system

The developed system concerns a laboratory
prototype with capability of EIS technique
application in high-capacity battery cells. EIS
excitation can be applied both in on-line (DC
charging/discharging current) and off-line (no DC
charging/discharging current) mode, while the
option to store the cell current and voltage signals
is provided. The AC excitation signals of EIS are
of major interest here, as they can lead to useful
inferences regarding the cell internal condition
and State-of-Health. In the following subsections
the developed system is presented and analyzed
as two main parts, the power converter (excitation
system) and the measuring system.

A. Power converter topology and control

The high-power part of the developed system,
shown in Fig. 1, is responsible for
charging/discharging and exciting the cell. This
part is based on a synchronous DC/DC converter,
which was designed to operate for applying a
controllable AC current, on the top of a DC
charging (Buck mode) or discharging (Boost
mode) current. This DC current can be controlled
up to the C-rate of the cell in both directions. It is
worth mentioning that this work is an attempt to
develop a more affordable and higher-current EIS
tester, compared to the commercial bench-top
instruments. These devices cannot usually apply
DC current higher than 5A, and if they may, add-
ons are required at the price of increasing weight,
volume and cost.

<figure><img src="image_2.png" /><figcaption>Fig. 1: Controlled charging/discharging and EIS excitation system</figcaption></figure>

The proposed converter was designed to operate
deply in CCM to minimize switching ripple
current, so that the resulting distortion in the
injected AC signals can be neglected. The system
control was focused on the regulation of the
current applied to the battery cell. This was
achieved through a PI compensator, designed so that the system closed-loop transfer function has constant gain in the whole desired frequency range for EIS application. At this point, it is stated that only Buck operation is considered in this paper. Due to the form of the Buck converter transfer function in CCM, the converter acts as a low-pass filter, so the control unit needs to compensate the filter attenuation in the frequency range of interest.

The converter ideal small-signal transfer function in open loop, G<sub>id</sub>, can be described by Eq. (1), derived according to the procedure presented in [6]:

$$ G_{id}(s) = \frac{I_{cell}(s)}{V_{supply}} = \frac{s^3 LL_f C_o + s^2 LC_o Z(s) + s(L+L_f) + Z(s)}{s^3 LL_f C_o + s^2 LC_o Z(s) + s(L+L_f) + Z(s)} \quad (1) $$

with d(s) representing the converter duty cycle and Z(s) the battery cell internal impedance, the equivalent circuit model (ECM) of which is given in Fig. 2. This ECM was selected based on a trade-off between accuracy and simplicity, which is of concern especially in real-time applications.

<figure><img src="image_3.png" /><figcaption>Fig. 2: Typical ECM of a battery cell internal impedance (2<sup>nd</sup>-order model)</figcaption></figure>

Regarding the circuit elements, the series resistance R<sub>o</sub> mostly describes the electrolyte permeability, and it also represents the contact resistances of the cell terminals with the active electrode material [11]. The other two resistive-capacitive blocks simulate the evolution of two major intrinsic electrochemical processes. The R<sub>CT</sub>//C<sub>DL</sub> pair describes the ion transfer between the electrodes as current is flowing, and the R<sub>Diff</sub>//C<sub>Diff</sub> pair corresponds to the ion diffusion in the electrolyte bulk, as a result of their oriented movement.

Focusing now on the system control and based on Eq. (1), the PI compensator gains, K<sub>P</sub> and K<sub>I</sub>, were calculated according to the desired gain and phase margins, as well as the desired closed-loop bandwidth [12]. The latter is of critical importance in our application, because the controller must be capable of controlling current

in a frequency range instead of a pure DC component, as is usually the case. The bandwidth is directly related to the compensated system crossover frequency f<sub>C</sub>, which can be selected in a range, according to a rule of thumb for its limits [6]:

$$ f_{FLT} < f_C < 0.1 f_{SW} \quad (2) $$

where f<sub>FLT</sub> is the output filter dominant cut-off frequency and f<sub>SW</sub> the converter switching frequency.

Of course, the theoretical controller gains may need further adjustment on the experimental setup, as the system parasitic behavior and the battery cell internal impedance can only be estimated and not exactly known a priori. At this point it should be mentioned that the impedance estimation was based on bibliographic data [5]. Finally, it is worth highlighting the critical role of the output filter capacitor parasitic ESR in the filter response, as it damps the undesired oscillations in the LCL filter resonant frequency.

## B. Measuring system

The low-power part of the developed system concerns the measurements required for the battery cell impedance curve extraction, after the application of the EIS technique. Given the requirement for measurements under various DC current values, up to the cell C-rate, additional impedance estimation error is introduced due to the presence of the high DC current. This error results mainly because the AC and DC current components are sensed as a total signal through the same sensor, while AC being considerably lower than DC. Therefore, the clear separation of this superimposed AC part requires special attention, both in the schematic and the PCB design of the unit. In Fig. 3 the developed measurement circuit diagram is depicted, with “signal” representing either voltage or current.

<figure><img src="image_4.png" /><figcaption>Fig. 3: Measurement unit circuit diagram</figcaption></figure>

The measured signal is led to the microcontroller ADC unit, which has a resolution of 12 bits and a
sensing voltage reference equal to the supply voltage of the unit (3.3V). This results in a minimum sensing level of 0.8mV, meaning a theoretical error of 8% in the typical 10mV AC voltage response due to EIS current injection. A similar issue also exists in the current sensing, as we aim to an AC current that is significantly lower than the DC. Therefore, the signal DC component is first extracted from the total signal (1st stage), and then it is subtracted from it, to obtain the AC component (2nd stage). At this stage, the AC component is amplified with a high gain and finally it is added to an offset, to be observable by the ADC unit (3rd stage). The amplification gain $R_f/R$ was chosen for the maximum possible accuracy, achieved when the full measurement range of the ADC unit can be utilized. Hence, the full-scale value of the measured signal should be equal to the sensing voltage reference of the unit. In this way, the theoretical error can be significantly reduced, moving from 8% to 0.07%, for the case that the unit can sense up to 1.5 times the typical peak value of the battery cell AC voltage response.

It is worth mentioning that the overall experimental sensing accuracy is also affected by the tolerances and drifts of the various circuit components used, and therefore, the selection of these components is of critical importance. A countermeasure to this is the accurate calibration of the measuring unit, to consider its non-ideal functional behavior and minimize the stray effects.

# III. EIS technique approaches

## A. Frequency spectrum sweep mode

The EIS technique is usually applied by sweeping the desired frequency spectrum [5], [9]. An example of this approach is visualized in Fig. 4, where the sequential injection of five different harmonics is qualitatively illustrated, with each harmonic being applied for a two-period time.

<figure><img src="image_5.png" /><figcaption>Fig. 4: Qualitative representation of AC excitation signal (frequency spectrum sweep mode)</figcaption></figure>

The different harmonics are injected one after the other, and during this process, the system invariance must be ensured. Therefore, the minimum frequency should be selected appropriately, in combination with the DC current. It is worth mentioning that each frequency component should be applied for at least two periods, with the first one(s) serving as the “initialization” phase, and the last one as the time for measurements. This requirement results from the definite response time of the battery cell. During this, the charge-transfer process is still in transient stage, so the cell response to the application of any signal will not have reached the steady state. This implies that more than one signal period may be necessary for the “initialization” phase, when frequency increases beyond a point.

## B. Multi-sine mode

This approach comes as an optimization of the frequency sweep technique, mainly on the total time needed for injection. In Fig. 5, the simultaneous injection of the same harmonics as in the example of Fig. 4 is qualitatively illustrated.

<figure><img src="image_6.png" /><figcaption>Fig. 5: Qualitative representation of AC excitation signal (multi-sine mode)</figcaption></figure>

As the different harmonics are injected simultaneously, this process takes significantly less time than the sequential approach. Specifically, the time requirement here is at least two periods of the minimum-frequency signal, instead of at least two periods of every injected signal. Due to this, the minimum frequency that can be injected, $f_{min}$, is lower with the same DC and AC current components, given the same measurement time to meet the requirement for the SoC variation limit. This shift in $f_{min}$ depends on the number of injected harmonics and it is more intense as this number increases, to achieve a better depiction of the battery cell internal impedance in the characteristic spectrum. Alternatively, if only a specific part of the spectrum is of interest regarding the impedance, the DC rate can be increased in multi-sine
compared to the frequency sweep injection. This allows the cell characterization in a wider range of operating conditions, thus giving better insights into its dynamic functional behavior. In this paper, the application of the multi-sine technique was performed in four bands, one per decade inside the desired frequency spectrum, with each of the bands containing five harmonics of the same amplitude. The number of bands and the number of harmonics in each band were chosen based on the trade-off between sufficient amount of data points, for depiction accuracy, and an overall injection time shorter than this of the sweep mode, while keeping the total AC amplitude at the level of the sweep mode. Moreover, the sampling rate for each band was selected in such a way as to keep the number of samples in an acceptable range for every harmonic of the band, while having the converter switching frequency as its upper limit. This fact is an aftermath of the main reason for the spectrum partition, because otherwise the result would be either an immense number of samples in the fundamental period or a very poor accuracy in high frequencies. Finally, a representation requirement for FFT was that the frequency resolution, resulting from the sampling rate and the number of samples in the fundamental period, must allow the depiction of all the harmonics in each band.

Given that the complex signal fundamental period is longer than the battery cell response time, it is evident that the minimum time required for the multi-sine injection is about half of the corresponding time needed for the frequency sweep in this case. As aforementioned, this time reduction is greater as the number of harmonics in the multi-sine signal increases, or the respective frequencies are mostly concentrated close to the fundamental.

### C. AC current amplitude selection

One of the concerns around the EIS technique is the selection of the superimposed AC component amplitude. This should be high enough to be sensed with sufficient accuracy over switching noise and be clearly distinguished from the quite higher DC component. Concurrently, it should be kept below a limit, to eliminate micro-cycling effect, as mentioned earlier. In current literature there is a gap regarding this point, as the relevant works make this choice arbitrarily. To give a better insight, this paper includes an attempt to derive a generalized rule for a more thorough and methodic selection of the AC amplitude. The

main concept, on which the rule is based, is the charge amount being transferred by each of the AC and DC components. More specifically, the charge transfer $Q_{AC}$ during a period $T_{AC}$ of the alternating current, can be expressed as:

$$Q_{AC} = Q_{total} - Q_{DC} = (I_{RMS} - I_{DC})T \quad (3)$$

The $I_{RMS}$ term represents the total current that affects SoC and its value corresponds to the magnitude of the vector resulting from the sum of the $I_{AC}$ and $I_{DC}$ vectors:

$$I_{RMS} = \sqrt{I_{AC}^2 + I_{DC}^2} \quad (4)$$

Now, considering a linear relation between the amplitude of AC and DC components:

$$\hat{I}_{AC} = aI_{DC}, \quad (5)$$

Eq. (4) becomes:

$$I_{RMS} = I_{DC} \sqrt{1 + \frac{a^2}{2}}, \quad (6)$$

and consequently, Eq. (3) becomes:

$$Q_{AC} = Q_{DC} \left( \sqrt{1 + \frac{a^2}{2}} - 1 \right) \quad (7)$$

Therefore, the upper limit of the AC amplitude can be determined according to the maximum acceptable SoC error because of micro-cycling ($\Delta SoC_{AC}$), as following:

$$a_{max} = \sqrt{2[(1 + \Delta SoC_{AC})^2 - 1]} \quad (8)$$

Regarding the lower limit of the AC amplitude, the selection criterion derives from the minimum desired SNR, so the worst-case AC signal-to-noise ratio. Regarding this, a rough estimation can be made, considering that noise results from the switching steady-state ripple and transient phenomena.

### IV. Results and Discussion

This section deals with the illustration of the comparative results, obtained after performing the EIS tests as described previously. The tests were conducted both in frequency spectrum sweep and multi-sine mode under different current rates, during charging of a LiFePO4, 3.2V/30Ah battery cell. For the cell voltage measurement, the 4-wire technique was used, so that the voltage can be taken directly across the cell terminals and not on the converter output. In
this way, the voltage drop on the cables in between due to the current flow was avoided, thus allowing the true cell voltage depiction. The laboratory test setup is shown in Fig. 6, where the volume scale of the developed system can be observed.

<figure><img src="image_7.png" /><figcaption>Fig. 6: Experimental setup of the developed laboratory prototype</figcaption></figure>

The charging current was above 5A for all tests, while the selected AC current amplitude was kept under 1A based on Eq. (8), for a maximum of 1% micro-cycling error. This implies that while the peak of the harmonics in frequency sweep mode can reach up to 1A, the peak of the corresponding harmonics in multi-sine mode should be lower. Regarding data acquisition, the cell current and voltage data were collected during the last period of each injected AC signal and stored through a data-logger, developed by the authors. Following this, FFT analysis and further processing was performed off-line to extract the impedance values, based on which the Nyquist plots can be formed in every case. The expected form of these plots for the selected model of Fig. 2 is shown in Fig. 7.

<figure><img src="image_8.png" /><figcaption>Fig. 7: Nyquist plot typical form for 2^nd^-order battery cell impedance models [9]</figcaption></figure>

In Fig. 8, the cell current and voltage responses are presented for the case of EIS injection in multi-sine mode. The test was performed under 5A DC current and superimposed AC current of 1A peak, while the injected harmonics were in the range of 0.1 to 1000 Hz.

<figure><img src="image_9.png" /><figcaption>Fig. 8: Battery cell current and voltage response in multi-sine mode (I<sub>DC</sub>=5A, I<sub>AC</sub>=1A peak)</figcaption></figure>

As can be observed, the applied spectrum was divided into four bands, one per frequency decade. Each of the complex signals represented the sum of five harmonics, resulting in a total of twenty injected signals of different frequencies. Each signal was applied for at least two fundamental periods, so that the cell response to the excitation signal can reach the steady state. The minimum band duration was chosen as 5 seconds, so that the overall injection and measurement time is kept under the limit of 36 seconds, which results in 1% SoC variation under C-rate (30A). In our case, the whole EIS process required 35 seconds, given the fundamental frequencies of 0.1, 1, 10, and 100 Hz for the four bands respectively, and the application of two periods for the first one, and 5 seconds for the following three bands. For the sake of clarity, it is also stated that for each band, the 1^st^, 2^nd^, 4^th^, 5^th^, and 8^th^ order harmonics were added. These were selected so that each of the harmonics can have an integer number of samples in one period, given the sampling rate of the complex signal. The latter was determined based on the choice of 1000 samples in the fundamental period, balancing the trade-off between signal depiction clarity and low memory requirement.

At this point, it should be highlighted that the converter noise (100 kHz switching ripple and transients in MHz range) was observed on the scale of 300 mV peak, as depicted in Fig. 9.
<figure><img src="image_10.png" /><figcaption>Fig. 9: Battery cell voltage (blue) and charging current (pink)</figcaption></figure>

At the same time, the expected AC voltage response due to EIS (0.1-1000 Hz) is on the scale of 5mV, as also shown in Fig. 8, pointing out the challenge in clear extraction of the desired signals. Given this, it can be inferred that the unit of Fig. 3 filters out effectively the undesired harmonic content, as will become evident from the below results as well. Regarding signal depiction, therefore, this 5 over 300 mV ratio makes necessary the acquisition of the oscilloscope figures in high resolution mode.

After the multi-sine EIS application, the test was repeated in frequency spectrum sweep mode, for the sake of comparison. The injected harmonics were of 1A amplitude and the same frequencies as in the multi-sine case. It should be noted that for the sweep mode, each of the harmonics was applied for exactly two periods and not more. This was decided to keep the injection time the shortest possible, while avoiding the controller transient response during the first period. In Fig. 10, the battery cell current and voltage responses are shown.

<figure><img src="image_11.png" /><figcaption>Fig. 10: Battery cell current and voltage response in frequency spectrum sweep mode ($I_{\text{DC}}=5\text{A}$, $I_{\text{AC}}=1\text{A}$ peak)</figcaption></figure>

From the comparative study it was observed that, for the applied frequencies, the required injection time was significantly greater in the sweep mode compared to the multi-sine, leading to a SoC variation higher than the typical linearity limit of 1% under C-rate. This implies that the sequential approach is not recommended in cases when a battery cell is in high-current operation and also in cases of slow kinetics investigation during charging or discharging, because of the extended injection time needed.

Following the EIS excitation and measurements, the experimental points of impedance were extracted off-line, for each of the modes respectively. These are presented in a common graph for comparison (Fig. 11), along with the fitting curve on the points of the proposed method (multi-sine $5A_{\text{DC}}$, 1A peak).

<figure><img src="image_12.png" /><figcaption>Fig. 11: Experimental points, obtained in multi-sine (blue) and frequency spectrum sweep mode (red), with $I_{\text{DC}}=5\text{A}$ and superimposed $I_{\text{AC}}=1\text{A}$ peak, along with the fitting curve (green)</figcaption></figure>

As shown in Fig. 11, the experimental points are mainly among each other and follow the same trend for both cases, forming the expected shape of Nyquist plot. This fact indicates a high level of consistency in the results, regardless of the excitation current form, given the preserved system linearity and invariance during the measurements. However, the sequential technique has a drawback over multi-sine, which lies in the extended required injection time. Indicatively for the frequencies used in our case, the injection time was 47 seconds in sweep mode over 35 seconds in multi-sine, and this difference increases with the number of low-frequency harmonics in general.

Regarding the fitting, the evaluation was performed based on the coefficient of determination ($R^2$), in combination with the relative Root Mean Square Error (rRMSE %) for
a more complete assessment. The latter is expressed as a percentage of the expected value, instead of the absolute error that is usually employed with RMSE, providing a better understanding of the error scale. According to a rule of thumb for the metrics, a high-quality fitting is achieved when R² is over 0.9 and rRMSE under 10% [13]. For the fitting curve extraction, the ECM of Fig. 2 was initially employed; however, a poor fitting quality was obtained. Therefore, this ECM was modified in terms of circuit element arrangement, type, and parameter values, based on the experimental data. Finally, it was found that a good fitting can be achieved from the ECM presented in Fig. 12. As shown, a series inductor is added for capturing the high-frequency cell response, and a Constant-Phase Element (CPE) is employed instead of a resistive-capacitive combination, for describing the diffusion effect.

<figure><img src="image_13.png" /><figcaption>Fig. 12: Revised internal impedance ECM for a LiFePO4, 3.2V/30Ah battery cell</figcaption></figure>

According to the literature [14], [15] the CPE is a promising candidate for describing the non-ideal capacitive behavior of the various electrochemical processes, due to the roughness of the electrode surfaces [16] and the resulting imperfect contact with the rest material. CPE represents a mathematical model that simulates an element with resistive-capacitive behavior, and its impedance varies with frequency keeping a constant phase. The CPE impedance is described by a fractional-order transfer function, as indicated by:

$$Z_{CPE} = \frac{1}{s^a Q}, a \in [0, 1] \quad (9)$$

Finally, the revised ECM parameter values were determined as R₀ = 2.5mΩ and L = 0.125uH, for the series resistance and inductance, R<sub>CT</sub> = 4.75mΩ and C<sub>DL</sub> = 1.5F, for the charge-transfer process, and Q<sub>Diff</sub> = 750F and a<sub>Diff</sub> = 0.45, for the ion diffusion process. The parametrization was performed following a trial-and-error fitting process, while keeping the evaluation metrics in high standards (R² > 0.9 and rRMSE < 10%). For

the diffusion parameters there was only a rough estimation, due to the lack of enough data in extra-low frequencies.

Further on and focusing solely on multi-sine excitation mode, the second comparison concerns the amplitude of the superimposed AC current and its effect on the measurements. In Fig. 13, the battery cell current and voltage responses are shown for multi-sine mode, with DC current of 5A and AC amplitude of 0.5A instead of 1A used previously.

<figure><img src="image_14.png" /><figcaption>Fig. 13: Battery cell current and voltage response in multi-sine mode (I<sub>DC</sub>=5A, I<sub>AC</sub>=0.5A peak)</figcaption></figure>

In Fig. 14, the experimental points are depicted in Nyquist plot form, for comparison of the results obtained by adjusting the AC excitation current amplitude.

<figure><img src="image_15.png" /><figcaption>Fig. 14: Experimental points, obtained in multi-sine mode under I<sub>DC</sub>=5A, with superimposed I<sub>AC</sub>=1A peak (blue) and I<sub>AC</sub>=0.5A peak (red), along with the fitting curve (green)</figcaption></figure>

It is obvious that the experimental points show a similar behavior in both cases, and the ones obtained using the lower excitation current are slightly more scattered. This scattering is expected because noise constitutes now a larger part of the total voltage response signal.
Nevertheless, the typical shape of the Nyquist plot is still well preserved, as indicated by the fitting evaluation metrics. This implies that the AC current and the resulting voltage response are effectively distinguished from noise in both cases. The third comparison is focused on the level of the battery cell working DC current and its effect on the measurements. In Fig. 15, the cell current and voltage responses are shown, corresponding to Fig. 8 and 13. In this test the excitation was applied also in multi-sine mode, with AC current amplitude of 1A and DC current of 15A instead of 5A used in Fig. 8.

<figure><img src="image_16.png" /><figcaption>Fig. 15: Battery cell current and voltage response in multi-sine mode ($I_{DC}=15\text{A}$, $I_{AC}=1\text{A}$ peak)</figcaption></figure>

In Fig. 16 the Nyquist plots of experimental points are illustrated, for comparison of the results obtained by adjusting the DC current.

<figure><img src="image_17.png" /><figcaption>Fig. 16: Experimental points, obtained in multi-sine mode under $I_{DC}=5\text{A}$ (blue) and $I_{DC}=15\text{A}$ (red), with superimposed $I_{AC}=1\text{A}$ peak, along with the fitting curve (green)</figcaption></figure>

As can be observed, the experimental points follow the typical Nyquist plot shape, as in Fig. 11 and 14, but here an interesting fact arises regarding the impedance. Specifically, with increasing current it is noticed that the series resistance $R_0$ is slightly increasing (right shift of

the real-axis crossing point), and the charge-transfer resistance $R_{CT}$ decreases (shrinking of the Nyquist plot semicircle). Regarding $R_0$, our observation requires a subtle analysis. Based on the literature, a common investigation approach is focused either on the behavior of the series resistance $R_0$ with temperature [3], or on the dependence of the total DC resistance with current [17]. However, both approaches lead to an ambiguous depiction of the current flow effect on this specific $R_0$ resistance. To address this, the duration of the tests performed for this analysis was kept short, meaning that the 35 seconds of EIS injection were added to a 3-minute interval for DC current settling. In this way, the temperature difference between the compared cases was maintained under the linearity limit and therefore, the temperature effect on the electrolyte permeability can be neglected. Due to the latter, the decoupling of the two facets of $R_0$ was realized and so the ohmic behavior of the contact resistances was considered as the dominant factor, leading to an overall rise of the resistance with higher current. Regarding $R_{CT}$, its decreasing trend derives from its nature. In fact, this resistance represents the difficulty in ion mobility during cell operation, which declines with higher current. This reduction is also indicated by the negative change in the Butler-Volmer curve gradient. Because of these variations, the fitted impedance model is less representative than in cases of Fig. 11 and 14, but still promising, as shown by the fitting quality metrics in Fig. 16.

## V. Conclusion

In this paper, the concept and implementation of a system for controlled charging/discharging and EIS excitation of a high-capacity Lithium-Ion cell was presented. The cell was excited with AC signals within a frequency spectrum, and its impedance was estimated based on the measured current and voltage during its charging process. The fast measurement was achieved through multi-sine EIS mode, as an improvement over the sequential harmonic injection. From the comparative tests performed it became clear that overall, the multi-sine approach excels the sequential, as they both result in similar impedance values, while the first outperforms the second in terms of time duration. Moreover, the impact of AC and DC current in the impedance results was investigated. Regarding the AC component, a slightly higher measurement precision was observed as the harmonic peak
current increased. As per the DC component, the
nature and physical interpretation of the battery
cell impedance became evident, thus giving better
insights into the functional behavior of the cell
across its operating range. In all the above cases,
the fitting curve of the experimental data from the
proposed method was extracted and the fitting
quality was evaluated comparatively for all the
cases, showing promising results.

Overall, this work was a first step, oriented to the
proof of the presented concept. In next attempts,
the battery cell discharging mode will be tested
and further on, the system extension from a single
battery cell to a battery cell stack will be
investigated. The latter is necessary to move in
real on-line conditions (e.g. operation on an
energy storage system in cooperation with RES).
Finally, the potential for embedding a real-time
impedance estimation technique will be
examined, for the live extraction of inferences
regarding the internal state of the cell or stack
under test.

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