Experimental Investigation of Heat Dissipation of Lithium–Ion Cells and Its Correlation with Internal Resistance

Abstract

Power loss is a limiting factor for batteries and individual cells. The resulting heat generation due to the power loss leads to reduced battery performance and, thus, lower efficiency. These losses are largely due to the internal resistance of the cells. Therefore, it is important to accurately determine the value of the internal resistance of lithium–ion cells. From the literature, it was found that there are three widely used internal resistance-measurement methods (current step method, direct-energy-loss method, and calorimeter measurement), with negligible research on their comparison demonstrating the most efficient method. Henceforth, to find the most optimal method, this research adopts all three methods on a variety of cell chemistries, including Lithium-ion Manganese Oxide (LMO), Lithium Iron Phosphate (LFP), Nickel Manganese Cobalt (NMC), and Lithium Titanium-Oxide (LTO) for different c-rates (1 C, 2 C, and 3 C), with a wide temperature range (from 0 °C to 40 °C).

1. Introduction

Since the past decade, lithium–ion batteries (LiBs) have become the preferred technology in the field of rechargeable batteries. This technology is dominant from small-scale products such as mobile phones to large scale such as electric vehicles (EVs) [1]. The trend toward the electrification of private and public transport is seen as necessary to achieve a reduction in carbon dioxide emissions [2]. The performance of LiBs is significantly dependent on factors such as capacity, self-discharge, and internal resistance [3,4]. Out of all these factors, internal resistance is one of the crucial factors to keep an eye on because it helps to identify and compare different types of cells, calculate energy efficiency, and determine the cooling system along with the power estimation [5]. Moreover, it can be said that the internal resistance is also a signal for battery ageing and, therefore, impacts the performance of the battery i.e., improvement in the battery can be done if the current internal resistance values are identified, and this consequently leads to the safety of the battery (decomposition of the active materials leads to increase in internal resistance and temperature within the battery) [6].

The research conducted by Schweiger et al. compared different internal-resistance-measurement methods such as current step methods, alternating current (AC) methods, and energy-loss methods. These measurements resulted in similar results for the current step method without change of charge and two energy’loss methods (Watt hour counting and calorimetry) for 60% state of charge (SoC) at 25 °C. The calorimeter measurements showed that with specific measurement cycles, the dissipated heat of a cell during a charging/discharging cycle can be used to calculate the internal resistance [7]. Several researchers used calorimetric measurements to determine the heat generation of Lithium-ion cells and batteries [8,9]. Calorimeter systems for calorimetric measurements with different approaches were researched by Madani et al. [10]. The outcome of this research showed that calorimetric measurements play an important role in understanding the thermal effects in lithium–ion cells.

Therefore, the goal of this experimental investigation is to measure the internal resistance (R_i) of lithium–ion cells with different cell chemistries, i.e., Lithium-ion Manganese Oxide (LMO), Lithium Iron Phosphate (LFP), Nickel Manganese Cobalt (NMC), and Lithium Titanium-Oxide (LTO), at different temperatures (from 0 °C to 40 °C) and C-rates (from 1 C to 3 C), along with the comparison of the results obtained from a variety of measurement methods, i.e., the current step method, direct-energy-loss method, and the calorimeter-measurement method. Please note that for the better comparison, all three methods were applied with a single measurement cycle simultaneously.

This article contains detailed information about the measurement setup and the measuring process. The data analysis is shown, and the key findings are presented and described (Figure 1).

2. Background

A simplified battery model can be developed by considering the internal resistance (often called ohmic resistance) and the voltage source [11]. The magnitude of resistance is primarily identified by the processes at the interface between active materials and electrolytes. During the first charging/discharging cycle, the solid electrolyte inter-phase (SEI) is formed at the negative electrode i.e., the anode [3]. This layer is permeable to lithium ions and protects the electrode from further decomposition. It, thus, significantly increases the service life of a cell [12]. However, the SEI also increases the battery’s internal resistance of the battery, as now the ions need to pass through an additional layer [3]. Henceforth, the heat dissipated due to internal resistance, which not only limits the power of the battery but also increases the cooling demand of the battery system.

The internal resistance of the battery is also affected by various parameters. The results presented by Chen et al. show that the internal resistance increases with decreasing SoC, and the temperature also affects the internal resistance, i.e., because higher temperatures favour chemical reactions inside the cell [13]. Degradation and aging also increase internal resistance. This occurs in the form of passivation and SEI formation, especially at the negative electrode [3].

2.1. Current-Step-Method

In the current step method, the battery cell is considered as a voltage source with an ohmic resistance. No voltage will be dropped across the internal resistance if no current flows. To determine the internal resistance, a discharging pulse, followed by a charging pulse, is applied to the cell, consequently leading to a visible voltage jump at the beginning of each step. The ohmic resistance causes this [14,15]. In this method, choosing a measuring current that is as high as possible is important. Step-amplitude reduction leads to a higher measurement error [7]. By measuring the voltage difference ∆U, the internal resistance R_i can be determined (see Equation (1)). The charging and discharging steps can be used to determine the voltage difference.

$$R_i=\left|{\displaystyle \frac{∆U}{∆I}}\right|$$

(1)

The current step methods differ for the cells that are with and without charge change. After the initial step, the duration of the current pulse leads to a change in the cell’s SoC. Therefore, measuring the voltage difference with the change in charge leads to a falsified result [7]. However, according to [7,16], the voltage difference of 100 ms after the initial/final step the timespan is quite short, so the influence of the change in charge on the result is negligible [7].

2.2. Direct-Energy-Loss Method

Chemical effects within the cell must be considered as they can influence heat release. One is the reversible heat effect. It is caused by the intercalation of lithium ions in the lattice [3]. At the beginning of a charging cycle, it is endothermic and becomes exothermic during most of the charging cycle. During discharge, this effect is the opposite. If the cell is charged and discharged symmetrically, the reversible heat effect of the cell is cancelled [7]. To measure the internal resistance with the direct-energy-loss method, a charge-neutral profile is applied to the cell/battery. The charged energy E_chg and the discharged energy E_dchg were determined with the measured values of the charging voltage U_chg and discharging voltage U_dchg, as well as the charging current I_chg and the discharging current I_dchg. The energy loss for a single charge/discharge step results from Equations (2) and (3).

$$Q_{loss}=E_{chg}-E_{dchg}$$

(2)

$$Q_{loss}=\left(U_{chg}{I}_{chg}{∆t}_{chg}\right)-\left(U_{dchg}{I}_{dchg}{∆t}_{dchg}\right)$$

(3)

With the energy loss, the internal resistance R_i can be calculated, i.e., dividing the energy loss Q_loss through the integrated current that was applied during a measurement cycle (see Equation (4)),

$$R_i={\displaystyle \frac{Q_{loss}}{{\int }_{t2}^{t1}{\left(I\right(t\left)\right)}^{2}dt}}$$

(4)

The cells’s SoC must not change during the measurement. Otherwise, the values of the charged energy and the withdrawn energy cannot be compared.

2.3. Calorimeter Measurement

The temperature rise caused by Joule’s heat can be measured under certain conditions. The generated heat should not be affected by the environment. Therefore, an adiabatic measurement system is required, for which a calorimeter is used.

A calorimeter consists of a vacuum flask and a plug. A calibration heater and a thermometer were placed inside the flask in which a liquid surrounds both the devices and the measurement probe. The liquid serves as a medium to absorb the heat. The magnetic stirrer within the flasks ensures an even and fast temperature distribution within the liquid. The connections of the cell, the calibration heater, and the thermometer are led through a plug as shown in Figure 2, i.e., the structure of the calorimeter.

Chemical effects within the cell, like the reversible heat effect, need to be considered within this method as it influences the heat release.

To determine the energy loss, firstly, the heat capacity of the system must be determined, for which a calibration procedure is performed. This procedure uses a heating resistor or so-called calibration heater to supply a known amount of energy to the system. The amount of heat Q_calibration supplied can be calculated from Equation (5),

$$Q_{calibration}=Pt=UIt$$

(5)

P is the electrical power of the resistor and t in s is the duration of the power supply where current flows through the resistor, i.e., the value of the resistor is known. Together with the set voltage and Ohm’s law, the amount of heat can be calculated based on Equation (6),

$$Q_{calibration}={\displaystyle \frac{U^2}{R_{resistor}}}t$$

(6)

The supplied energy leads to a temperature increase within the calorimeter. This temperature-difference ∆T calibration is measured with a thermometer. Based on this temperature difference and the amount of heat Q_calibration, the heat capacity C_system of the system can be determined according to Equation (7),

$$C_{system}={\displaystyle \frac{Q_{calibration}}{{∆T}_{calibration}}}$$

(7)

After the calibration, the cell is supplied with a current profile. Due to the cell’s internal resistance, heat is generated, which is absorbed by the calorimeter liquid. The resulting temperature rise ∆T_measurement is measured with the thermometer. With the heat capacity C_system and the temperature difference ∆T_measurement, the energy loss Q_loss can be calculated (Equation (8)),

$$Q_{loss}=C_{system}{∆T}_{measurement}$$

(8)

Then, the cell’s internal resistance can be determined with Equation (4).

3. Methodology

The cells used in the experiment (

Table 1. Cell specifications.

	Efest IMR18650	HETER HTCF18650	Samsung SDI INR18650-29E	LTO Battery Co. LTO18650
Nominal capacity [mAh]	3100 mAh	1600 mAh	2900 mAh	1300 mAh
Nominal voltage [V]	3.7 V	3.2 V	3.6 V	2.4 V
Charge cut-off voltage [V]	4.2 V	3.7 V	4.2 V	2.8 V
Discharge cut-off voltage [V]	2.5 V	2.5 V	2.5 V	1.5 V
Discharge current [A]	10 A	4.80 A	8.25 A	13 A
Temperature range (charging) [°C]	0 °C to 45 °C	0 °C to 45 °C	0 °C to 45 °C	0 °C to 45 °C

) were Efest (Shenzhen, China) IMR18650 LMO cells with a nominal capacity of 3100 mAh [17], HETER ELECTRONICS GROUP (Shenzhen, China) HTCF18650 LFP cells with a nominal capacity of 1600 mAh [18], Samsung SDI (Cheonan-si, Republic of Korea) INR18650-29E lithium NMC oxide cells with a nominal capacity of 2900 mAh [19] and LTO Battery Co. (Shenzhen, China) LTO18650 LTO cells with a nominal capacity of 1300 mAh [20]. A nickel stripe of 45 mm × 10 mm × 0.15 mm was welded on both sides of the cell poles.

A WIKA CTR-2000—precision thermometer from WIKA company (Klingenberg am Main, Germany) was used for temperature measurement and calibration inside the calorimeter [21]. Two of these thermometers were used with two PT-100 probes i.e., temperature sensors. The CTP5000-250 [22] reference thermometer from WIKA company consists of components such as a Resistance Temperature Detector (RTD), lead wire, sheath, and a connection. Both devices were supported by LabView software (v2.13.16) and had an RS232 interface. The CTR-2000 thermometer has an accuracy of 0.01 °K.

The HI190M magnetic stirrer from Hanna Instruments company (Woonsocket, RI, USA) [23] was used to ensure an even temperature distribution inside the calorimeter flask. The stirrer bar inside the calorimeter rotated at 250 rotations per minute (rpm).

The stainless-steel 0.5 L flask of Dewar GSS/DSS 500 model from KGW Isotherm company (Karlsruhe, Germany) was used for the experiment. The plug to cover the flask was made of polyurethane for good thermal isolation [24]. They formed a thermal insulator to obtain a quasi-adiabatic calorimeter. The reason for such insulation was to ensure that negligible heat energy could escape from the calorimeter. Moreover, to support the experiment for low temperatures, the calorimeter liquid was a mixture of 50% water and 50% glycol with a freezing point around −33 °C. This ensured that the water inside the calorimeter did not freeze at 0 °C and −10 °C testing temperatures. In total, 300 mL of liquid were used per calorimeter. Four flasks were procured for the experiments.

A calibration heater was used to calculate the heat capacity of the setup. The calibration heater acted as a heating resistor. One axially wired resistor with a ceramic housing from Hi Tech Resistors HTR (Nagpur, India) was used for the experiments for each flask. The model has a resistance value of 6.8 Ω with a maximum load capacity of 11 W [25]. It was found that these resistors heat up the temperature of the calorimeter liquid by approx. 1.1 °C with a voltage supply of 5 V.

The BTS4000-5V6A model from Neware company (Shenzhen, China) was used for the pre-cycling of the cells to generate the current profile for the measurement cycle and as the power supply for the calibration heater. For each channel, it uses a four-wire measurement. The tester contained a control unit and several racks with eight channels each so that cells can pre-cycle quickly, with an aim to measure four cells at the same time [26]. To accelerate the temperature equalization, the calorimeter liquid was brought to the required test temperature using the F32 thermostat from JULABO (Seelbach, Germany) thermostat before it was filled into the calorimeter [27]. By using a VTS 4150-5 model of VÖTSCH (Balingen, Germany) climate chamber, it was possible to generate a stable ambient temperature for the calorimeter during the experiment [28].

The measurements, with the help of a calorimeter and magnetic stirrer, occurred in the climate chamber, while the temperature-measuring devices and the battery tester were located outside. The flask was placed on the magnetic stirrer with the stirring bar inside. The cables for the cell and the calibration heater ran through the plug of the flask, as well as the temperature sensors (CTP-5000), as shown in Figure 3.

Figure 4 provides an overview of a cell connection with its measurement devices. One channel of the battery tester can handle 6 A. To have sufficient current, two channels were connected in parallel. The calibration heater was connected to the third channel of a rack.

Before the measurements, each new cell was cycled 10 times from charge cut-off voltage to discharge cut-off voltage with the Neware Battery Tester. After that, the cells were set to 50% SoC again with the Neware Battery Tester.

Measuring the internal resistance of four different cells started with placing the four calorimeters inside the climate chamber a day before the measurement. The internal resistance is strongly temperature-dependent. To measure it at a certain temperature, the heating through the calibration and the heating through the measurement cycle must be considered. The target temperature (measurement temperature) has been selected so that it is in the middle of the temperature rise that is passed through during the measurement cycle. Previous measurements showed that the calibration of the system with the calibration heater leads to a temperature increase of about 1 K inside the calorimeter. This was considered when tempering the calorimeter liquid as well. The climate chamber was set to the same temperature as the calorimeter liquid to minimize temperature exchange between the calorimeter and the environment. When the liquid reached the desired temperature inside the thermostat, it was filled into calorimeters with a syringe. The climate chamber was then closed, and the temperature inside the calorimeters was recorded. Once the temperature was stable, measurements were undertaken.

The measurements started with a first resting period of 5400 s (90 min). This was necessary to determine the temperature differences between the calibration and measurement cycle. For calibration, the calibration heater was powered with 5 V for 420 s to bring a defined amount of energy into the system, followed by the second resting period of 5400 s (90 min). Thereafter, the measurement cycle was applied to the cells and the time was recorded by the Neware battery tester with a resolution of 1 Hz. After the measurement cycle, there was a final resting period of 5400 s (90 min). Figure 5 provides an overview of the whole measurement procedure.

The constant-defined measurement cycle (as shown in Figure 5) was used for all three methods: the current step method, the direct-energy-loss method, and the calorimeter method. It was noticed that continuous charging and discharging would exceed the safety limit of the cells, especially with high C-rates. Therefore, the cycles were defined with shorter charge/discharge steps. As can be seen in Figure 5, the cycle was set for the 40 s that contains charging, discharging, and rest periods. Then, this was repeated 45 times, which equates to 30 min in total for each cell.

It is vital to note here that the cycle shown in Figure 5 was created for each cell chemistry based on the respective cell specifications. Moreover, separate cycles were developed for each applied C-rate (1 C, 2 C and 3 C) according to the specific capacity of each cell. The reason for the chosen range of the C-rate is that C-rates below 1 C do not generate enough heat inside the cells to achieve meaningful results in terms of measurement accuracy. Parameters such as the voltage limits of the cells were considered as the safety limits in the cycle. The Neware BTS recorded the charge energy (E_chg) and the discharge energy (E_dchg). This allowed the determination of loss of energy. The current profile resulted in the cell heat-up, and this was measured by temperature sensors. With the temperature difference and the heat capacity of the setup, again the energy loss can be calculated to determine the internal resistance for the calorimeter method.

After the measurement cycle, there was a third resting phase of 5400 s (90 min) again for the determination of the temperature differences. The temperature was recorded with the CTP-5000 temperature sensor with a time resolution of 1 Hz.

Data Evaluation

To calculate the internal resistance with the current step method, only the Excel file generated from the Neware BTS for the measurement cycle was necessary. After each resting step in the measurement cycle, the voltage difference ∆U and the current difference ∆I were determined. As mentioned in Section 2.1, the timespan of the voltage and current measurement was 100 ms.

After loading the values for the voltage/current and time into the Matlab workspace, the Matlab file selected the required data. As with the current step method, only the Excel file of the measurement cycle was needed. The Neware BTS determined the charge/discharge energy after each charging/discharging step. In the corresponding Matlab file, the values of all charging steps and the values for the discharging steps were summed up. The difference between the charging and the discharging energy was the energy loss Q_loss in mWh. The unit is transferred into Ws. This simplification is possible because the value of the current corresponds to a square wave signal and has either the value zero or the value of the selected C-rate. I_m is the magnitude of the current during the experiment. The duration of the current flow is tm. Due to the current profile, it is 900 s for each measurement. The corresponding Matlab file took the temperature and time values from the Excel document generated during the measurement by the LabView software. It also reads the current, voltage, and resulting energy values. These were taken from the Excel document generated by the Neware BTS during calibration. Afterwards, the time segments of the measurement were manually transferred to the Matlab file. The curve-fitting toolbox was then used to extrapolate the rest phases.

Subsequently, the starting point, before a temperature rise, and an ending point, after a temperature rise, were determined based on the last intersection of the temperature curve with the extrapolated straight lines. Then, the areas of the triangles above and below the temperature curve were calculated. In a while loop, a variable was generated to change the x-value (time value) of the extrapolated lines. The area of the lower triangle was made up of the temperature curve and the distance of the variable from the starting point. The area of the upper triangle was made up of the temperature curve and the distance of the variable to the endpoint. The areas were determined with the “trapz” command. Then, the lower area was divided by the upper area and the result was saved. Thereafter, the value of the variable was increased by one. At the point where both areas were approximately equal, the result was approximately equal to one. The temperature difference was then determined at this point.

Based on the temperature difference obtained from the calibration and the measurement, the internal resistance was determined. By using the energy Q_calibration taken from the calibration file and the temperature difference ∆T_calibration for Equation (7), the heat capacity of the system was calculated. By using the temperature difference ∆T_measurement in Equation (8), the energy loss Q_loss was calculated. Then, the internal resistance R_i was calculated with Equation (9).

For determining ∆T, the area-compensation method was used. The pre-period (resting phase before a temperature rise) is extrapolated forwards, and the post-period (the resting phase after a temperature rise) is extrapolated backwards until the two triangular areas on either side of the main period (time of temperature rise) are equal (Figure 6). The temperature rise found in this way is the temperature difference ∆T [10]. This method was used twice, once for the calibration and once for the measurement.

The red lines in Figure 6 are the extrapolated resting phases. Together with the blue line and the black temperature curve, they enclose the mentioned triangles. The blue line is drawn at the point where the area of the two respective triangles is equal. At this point, it represents the temperature difference ∆T.

4. Results and Discussion

This section presents the results of each cell chemistry in Figure 6, Figure 7, Figure 8 and Figure 9. The internal resistances determined by the three methods were each shown in the same color i.e., Blue—calorimeter method, Grey—current step method, and Orange—direct-energy-loss method.

During the charging process of the LMO cell, the maximum cell voltage of 4.2 V was exceeded. Therefore, the measurements for 2 C at 10 °C and 3 C at 0 °C and 10 °C were not recorded. This could have been avoided with smaller C-rates. However, using smaller C-rates would not generate enough heat inside the cell to determine the internal resistance with the calorimeter method as mentioned in the previous section.

For the LMO cells in Figure 7 (LMO cell at 1 C), the values of the calorimeter method and the direct energy loss method are almost the same at 112 mΩ, while the current step method value is 25% lower than the other two methods i.e., 85 mΩ (Figure 7). The smallest measurement value of 29 mΩ is recorded at the highest temperature (40 °C) for the current step and the direct-energy-loss method.

For the LMO cell at 2 C and 3 C, the direct-energy-loss method and the current step method yield same results within the measurement accuracy. Consistently higher results can be seen for the calorimeter method. An explanation for this is additional heating inside the flask of the calorimeter through to contact resistances from the connection of the cells to the Neware Battery Tester.

It is visible that there is a strong temperature dependence of the internal resistance. The measurement uncertainties depend on the current applied to the cells (higher current equals more heat equals less measurement uncertainties).

Like LMO cells, LFP cells also exceeded the safety limit while charging. The maximum charging voltage of 3.65 V was reached during the measurement cycle’s first charging step. Hence, the measurements for 0 °C at 2 C and 3 C and 10 °C at 3 C were not recorded.

For the LFP cells in Figure 8, measurement errors for the calorimeter and direct-energy-loss methods were higher than those of the LMO cells in Figure 7. The reason is the smaller current of 1.6 A per c-rate, which leads to smaller heating of the cell compared, for example, to the LMO cells. However, it can be noticed that the current step method presents consistently low internal resistance as compared to other values throughout the selected temperature range of 0 °C to 40 °C in absolute numbers.

Figure 8 shows that the calorimeter method always provides the highest value for the internal resistance, followed by direct-energy-loss and current step methods for the complete range of temperatures. In contrast to the LMO cell, none of the methods provide matching results.

For the NMC cells, the measurements for 0 °C at 2 C and 3 C, along with 10 °C at 3 C, were not recorded as they exceeded the maximum cell voltage.

Figure 9 (NMC cell at 1 C) shows that, at 0 °C, the results of the energy-loss methods are almost identical. The current step method is significantly lower. It has also been noticed that the higher the temperature during the measurement, the higher the results of the direct-energy-loss method and the current step method converge.

Comparing the graphs from the measurement at 2 C and at 3 C (Figure 9), it is visible that the direct-energy-loss methods provide almost identical results within the measurement accuracy for temperatures from 20 °C to 40 °C as it is visible for the LMO cell. The point that the calorimeter method provides higher results in absolute numbers than the other methods can be addressed here as well.

Figure 10 shows the internal resistance of the LTO cell at various temperatures. Due to the small current during the measurement, Figure 10 (LTO cell at 1 C) has some significant measurement uncertainties. The current at 1 C is 1.3 A, and the resulting heating of the cell is very low. This increases the influence of the accuracy of the temperature measurement.

The measurements with 2 C are shown in Figure 10 (LTO cell at 2 C). At 0 °C, the calorimeter method and the direct-energy-loss method are within the same measurement accuracy. The current step method is significantly lower, around 40% less than the calorimeter method. With increasing temperature, the internal resistance decreases due to the temperature dependence of the cell. This is visible for all methods. The current step method consistently yields the lowest results compared to the other methods.

The results from the calorimeter method for the LTO cells are between 3 mΩ and 5 mΩ higher than those of the direct-energy-loss method and the results of the current step as it is visible in Figure 10 (LTO cell at 2 C). The value of the internal resistance of the LTO cell differs from the other cell chemistries, as it is significantly lower.

At 3 C, the graph shows the same behavior as at 2 C, with the calorimeter method providing the highest value for internal resistance and the current step method providing the smallest values for each measured temperature.

It is noticeable that all the measurements have a wide range of internal resistance values, which are significantly different for each used method. The values obtained at 2 C and 3 C for the calorimeter method are always higher than the values (covering both parameter resistance and temperature) of the remaining two methods for each cell chemistry. When looking at the results obtained for the cells, it is noticeable that the calorimeter method is between 10% and 20% higher as compared to the direct-energy-loss method. A possible explanation for such behavior of the calorimeter method is due to additional heating within the calorimeter. This can be caused by contact resistance at the cable connections and the resistances of the cables themselves.

Comparing the cells with each other is rather difficult because of the different currents, per the c-rate. For LMO and NMC cells, which have very similar specifications, the direct-energy method and the current step method show nearly identical results at temperatures from 20 °C to 40 °C and at c-rates from 2 C to 3 C. These are the cells with the highest current per c-rate due to their higher capacity.

The heat generation inside the cell is important for the calorimeter method, as small heating leads to high measurement uncertainties. When using the calorimeter method, it must be ensured that the measurement current is set high enough.

No impact on temperature has been identified due to the methods used. Some generic observations made is that the temperature difference between the energy-loss method and the current step method is higher at low temperatures (between 0 °C and 10 °C) than at temperatures above 20 °C.

4.1. Estimation of the Measurement Uncertainty

This section describes the error measurement approach that was adopted for each type of selected method. The maximum error method was used to determine the measurement uncertainties. This method determines the maximum error of an indirect measurement from all measured variables. To determine this, the function is derived after each variable with a measurement uncertainty. The values of the derivatives are then added together.

4.1.1. Current Step Method Measurement Uncertainty

The value of the internal resistance R_i was determined by applying Equation (9) directly to the values measured by the Neware BTS,

$$\left|R_i\right|=\left|{\displaystyle \frac{∆U}{∆I}}\right|=\left|{\displaystyle \frac{U_1-U_2}{I_2-I_1}}\right|$$

(9)

For the absolute error, Equations (10) and (11) were used,

$$\left|{∆R}_i\right|=\left|{\displaystyle \frac{{\partial R}_i}{{\partial U}_1}}\right|{∆U}_1+\left|{\displaystyle \frac{{\partial R}_i}{{\partial U}_2}}\right|{∆U}_2+\left|{\displaystyle \frac{{\partial R}_i}{{\partial I}_2}}\right|{∆I}_2$$

(10)

$$\left|{∆R}_i\right|=\left|{\displaystyle \frac{-1}{I_2}}\right|{∆U}_1+\left|{\displaystyle \frac{1}{I_2}}\right|{∆U}_2+\left|{\displaystyle \frac{U_1-U_2}{{\left(I_2\right)}^2}}\right|{∆I}_2$$

(11)

The value for U₁ was the voltage before the current jump. The value for U₂ represents the voltage after the current jump. The current I₂ corresponds to the value of the current of the selected C-rate.

4.1.2. Direct-Energy-Loss Method Measurement Uncertainty

With ∆Q_loss found according to the formulas in Section 2.2, the measurement error for the direct-energy-loss method is calculated as follows (Equations (12) and (13)),

$$\left|{∆R}_i\right|=\left|{\displaystyle \frac{{\partial R}_i}{{\partial Q}_{loss}}}\right|{∆Q}_{loss}+\left|{\displaystyle \frac{{\partial R}_i}{\partial I}}\right|{∆I}_m$$

(12)

$$\begin{gathered} \left|{∆R}_i\right|=\left|{\displaystyle \frac{1}{I_m^2{t}_m}}\right|(\left|I_{chg}{t}_{chg}\right|{∆U}_{chg}+\left|U_{chg}{t}_{chg}\right|{∆I}_{chg}+ \\ \left|I_{dchg}{t}_{dchg}\right|{∆U}_{dchg}+\left|U_{dchg}{t}_{dchg}\right|{∆I}_{dchg})+\left|{\displaystyle \frac{{-2Q}_{loss}}{I_m^3{t}_m}}\right|{∆I}_m \end{gathered}$$

(13)

The time t_m stands for the duration of the current flow during measurement and is constantly 900s. The current I_m is the amount of current based on the selected C-rate. The duration of charge time t_chg and discharge time t_dchg are always equal. U_chg and I_chg represent the charging voltage and the charging current, while U_dchg and I_dchg represent the discharging voltage and the discharging current.

4.1.3. Calorimeter Method Measurement Uncertainty

In the calorimeter method, the size of the measurement error depends not only on the accuracy of the battery tester but also on the thermometer. Its accuracy influences both the calibration and the actual measurement [12].

• Calibration

For calibration, a known amount of energy Q_calibration is fed into the system by using the calibration heater and the Neware BTS according to Equation (6). With the accuracy of the current and voltage measurement, the measurement error for Q_calibration is calculated according to Equation (14),

$$∆Q_{calibration}=\left|{\displaystyle \frac{∆U}{U}}\right|+\left|{\displaystyle \frac{∆I}{I}}\right|$$

(14)

For the accuracy of the heat capacity of the system, it results in Equations (15) and (16),

$$∆C_{system}=\left|{\displaystyle \frac{1}{∆T_{calibration}}}\right|∆Q_{calibration}+\left|-{\displaystyle \frac{Q_{calibrtion}}{∆T_{calibrtion}^2}}\right|∆T_{error}$$

(15)

$${∆C}_{system}=\left|{\displaystyle \frac{1}{{∆T}_{calibration}}}\right|\left(\left|{\displaystyle \frac{∆U}{U}}\right|+\left|{\displaystyle \frac{∆I}{I}}\right|\right)+\left|-{\displaystyle \frac{Q_{calibration}}{{∆T}_{calibration}^2}}\right|{∆T}_{error}$$

(16)

• Measurement

In the measurement, the heat capacity of the system C_system and the measured temperature change ∆T_measurement are used to determine the loss energy Q_loss of the cell according to Equation (17),

$$Q_{loss}=C_{system}∆T_{measurement}$$

(17)

With the determined measurement error for C_system and the accuracy of the temperature measurement, the result is Equation (18),

$$Q_{loss}=\left|∆T_{measurement}\right|∆C_{system}+\left|C_{system}\right|∆T_{error}$$

(18)

The further calculation was conducted analogue to the direct-energy method following Equation (12) for determining the internal resistance and Equation (18) for determining Q_loss.

$$\left|{∆R}_i\right|=\left|{\displaystyle \frac{1}{I_m^2{t}_m}}\right|(\left|∆T_{measurement}\right|{∆C}_{system}+\left|C_{system}\right|{∆T}_{error})+\left|{\displaystyle \frac{{-2Q}_{loss}}{I_m^3{t}_m}}\right|{∆I}_m$$

(19)

5. Conclusions

Three methods (the current step method without charge change, the direct-energy-loss method, and the calorimeter method) were implemented in this research to measure the internal resistance of four different lithium–ion cell chemistries. The cell chemistries used were LMO, LFP, NMC, and LTO.

The calorimeter method yields, consistently, the highest value for internal resistance in absolute values presumably caused by additional heating at contact resistances from the cell’s connection.

For the current step method, the most optimal results obtained are at temperatures of 20 °C and above at C-rates from 2 C. Here, the current step method delivered similar results for the LMO and NMC cells compared to the direct energy loss method. For LFP and LTO cells, the current step method is around 15% to 25% below the energy-loss methods. However, below 20 °C, the deviation of the current step method is even higher. A possible explanation cannot be derived from such results.

The utmost measurement accuracy is achieved at high C-rates (3 C). The calorimeter method showed a large measuring error. The reason for this is the low heating of the cells at small currents. In addition to the current and voltage measurement, the accuracy of the temperature measurement inflects the accuracy of the calorimeter method. The direct-energy-loss method and the current step method provide a similar accuracy of about ± 1 mΩ from 2 C, as they only depend on the accuracy of the current and voltage measurement of the battery tester.

Overall, this experimental research presents the three internal resistance measurement methods and compares them against each other based on four different cell chemistries, three different C-ratings, and with a temperature range of 0 °C to 40 °C. According to the authors’ knowledge, there is no such in-depth comparative analysis available in the literature. Therefore, this research widens the opportunity for future researchers to select the appropriate method for identifying the internal resistance of the cell.

For future work to compare, especially, the calorimetric measurements to the other two methods, additional factors inside the calorimeter such as contact resistances must be considered. Therefore, the measurements can be repeated with an improved calorimeter, and the contact resistances can be measured and included in the evaluation. Additionally, the latest version of the current step method can be adopted from the literature for future experiments. Lastly, there is a possibility to change the approach up to a certain extent. Rather than taking the fixed C-rates for all types of cells, the researchers can use the equal value of current during experiments to ensure that the appropriate influence of the current on cell heating.