Experimental Analysis on Li-ion Batteries for Sensorless Temperature Estimation using Electrochemical Impedance Spectroscopy

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D

IPARTIMENTO DI

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NGEGNERIA DELL’

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NFORMAZIONE

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L

M

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E

A

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2015/2016

“Experimental Analysis on Li-ion Batteries for Sensorless

Temperature Estimation using Electrochemical Impedance

Spectroscopy”

Relatori: Candidato:

Roberto Saletti Natalia Bernardini

Federico Baronti

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Abstract

Li-ion batteries represent the most efficient way to supply portable and mobile electronic devices with electrical energy. However, these batteries imply safety and stability problems. Exceeding of temperature and voltage safety ranges may cause battery degradation, permanent damages and explosions. Electrochemical Impedance Spectroscopy (EIS) is a widely used experimental method to gain deeper information on electrochemical systems. It allows analyzing some battery characteristics, such as state of charge (SOC), state of health (SOH) and internal temperature, via impedance by applying a sinusoidal signal and by measuring the response. Experimental analyses are carried out on two different tablet batteries, a LiFePO4 ANR26650 cylindrical cell and a

Panasonic 18650 cylindrical cell. Changes of the battery impedance’s spectrum with respect to temperature and to state of charge are widely investigated. This work confirms the existence of an intrinsic relationship between battery internal temperature and three different battery parameters: module and phase of the impedance and the frequency at which the imaginary part is zero. The three methods are exhaustively analyzed and compared in terms of accuracy and sensitivity. It is demonstrated that it is possible to monitor battery internal temperature by means of impedance measurements without the need of external sensors or inserted thermocouples.

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Contents

1.

2.

2.1. Li-ion batteries overview ... 11

2.1.1. Basic characteristic ... 11

2.1.2. Safety Issues... 18

2.2. Battery Management System ... 21

2.2.1. Main functions ... 21

2.3. Benefits of sensorless temperature estimation... 25

2.4. Electrochemical Impedance Spectroscopy overview ... 26

2.4.1. Introduction ... 26

2.4.2. Measurement technique ... 27

3.

3.1. Overview of the literature ... 32

3.2. Contribution to the state-of-the-art ... 47

4.

4.1. Initial problems and solution ... 50

4.2. Measurement setup ... 52 4.2.1. Description ... 52 4.2.2. Setup validation ... 61

5.

5.2.1. State of charge variation ... 68

5.2.2. Temperature variation ... 74

5.3. Protections effect ... 78

6.

6.1. Intercept frequency ... 82

6.1.1. Temperature dependency ... 82

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6.1.3. Measurement errors and sensitivity ... 85

6.2. New methods ... 92

6.3. Comparison ... 102

6.4. Application ... 105

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List of figures

Figure 1 Li-ion battery ... 11

Figure 2 Relative energy density of secondary cell chemistries [16] ... 14

Figure 3 Li-ion battery representation ... 15

Figure 4 Li-ion cell operating window [19] ... 19

Figure 5 BMS ... 23

Figure 6 Block diagram ... 28

Figure 7 Impedance vector representation ... 29

Figure 8 Current vs voltage ... 30

Figure 9 Randle’s Circuit for a Li-ion battery ... 33

Figure 10 Ideal impedance spectrum of a Li-ion cell (I) [2] ... 36

Figure 11 Voltage response of a Li-ion cell after a current pulse [1] ... 37

Figure 12 Ideal impedance spectrum of a Li-ion cell (II) [1] ... 38

Figure 13 Nyquist plots of Ni-MH w.r.t. ageing [3] ... 39

Figure 14 Nyquist plots of a Li-ion cell at three SOCs% [4] ... 40

Figure 15 Impedance spectra of LiFePO 4 [6] ... 41

Figure 16 Simplified schematic representation of Li-ion battery [7]. ... 42

Figure 17(a) Module vs temperature [6] ... 44

Figure 17(b) Phase shift vs temperature [6] ... 44

Figure 18 Experimental setup ... 52

Figure 19 Front of Keysight E5061B ENA Series Network Analyzer ... 55

Figure 20 EIS Experiment Setup: Gain-Phase Mode schematic view ... 56

Figure 21 EIS Experiment Setup: GP-Shunt Mode schematic view ... 58

Figure 22 EIS Experiment Setup: GP-Shunt Mode equivalent circuit ... 60

Figure 23 Impedance spectra of a tablet battery with and without calibration ... 62

Figure 24 Modules of known resistance values ... 63

Figure 25(a) Batteries tested ... 64

Figure 25(b) Connectors on the ANR26650 Li-ion cell ... 64

Figure 26 OCV vs SOC for tablet batteries ... 65

Figure 27 OCV vs SOC for ANR26650 battery ... 66

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Figure 29 Impedance spectra of tablet half-cell at 25% SOC: SOCwaiting_time ... 69

Figure 30 ANR26650 impedance spectra at various SOCs% (25°C) ... 70

Figure 31 18650 impedance spectra at various SOCs% (25°C) ... 71

Figure 32 Tablet single cell impedance spectra at various SOCs% (25°C) ... 73

Figure 33 Tablet half battery impedance spectra at various SOCs% (25°C) ... 73

Figure 34 Impedance spectra of tablet single cell at 20°C: Twaiting_time ... 75

Figure 35(a) Impedance spectra of ANR26650 cell ... 76

Figure 35(b) Zoom around the x-axis crossing ... 76

Figure 36 Impedance spectra of tablet single cell ... 77

Figure 37 Impedance spectra of tablet half battery ... 77

Figure 38 Tablet half battery with and without fuse at various temperatures ... 79

Figure 39(a) Tablet single cell with and without protection chip (10°C) ... 81

Figure 39(b) Tablet single cell with and without protection chip (25°C, 40ºC) ... 81

Figure 40 Tablet single cell: Intercept frequency vs temperature (90%SOC)... 83

Figure 41 ANR26650: Intercept frequency vs temperature at various SOCs ... 86

Figure 42(a) Comparison of the results: literature and GP-Shunt mode ... 87

Figure 42(b) Comparison of the results: literature,GP-Shunt mode and T/R mode ... 87

Figure 43 Tablet single cell: intercept frequency vs temperature (I) ... 88

Figure 44 Tablet single cell: intercept frequency vs temperature (II) ... 89

Figure 45 Tablet half battery: intercept frequency vs temperature ... 91

Figure 46 Tablet single cell: module vs temperature at 90%SOC... 92

Figure 47 Tablet single cell: module vs temperature at 5Hz ... 93

Figure 48 Tablet single cell: Bode plot of the impedance module ... 95

Figure 49 Tablet single cell: Bode plot of the impedance module ... 95

Figure 50 Tablet single cell: module vs temperature at 356Hz ... 97

Figure 51 Tablet single cell: Bode plot of the impedance phase ... 98

Figure 52 Tablet single cell: Bode plot of the impedance phase ... 98

Figure 53 Tablet single cell: phase vs temperature at 6kHz ... 99

Figure 54 Bode plots of module and phase of the tablet half battery impedance ... 100

Figure 55 Tablet half battery: module vs temperature at 7Hz ... 101

Figure 56 Tablet half battery: phase vs temperature at 88Hz ... 101

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List of tables

Table 1 Summary of batteries electrical parameters ... 67

Table 2 ANR26650: Intercept frequency at various SOCs ... 84

Table 3 18650: Intercept frequency at various SOCs ... 84

Table 4 Tablet single-cell: Intercept frequency at various SOCs ... 84

Table 5 Tablet half-battery: Intercept frequency at various SOCs ... 84

Table 6 Intercept frequency: errors in the estimated temperature ... 90

Table 7 Impedance Module: errors in the estimated temperature... 94

Table 8 Accuracy and sensitivity for the tablet single cell ... 104

1. INTRODUCTION

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Chapter 1

Introduction

Nowadays Li-ion batteries represent the most efficient way to store and accumulate energy. Electrochemical batteries are extensively used for energy storage and power supply in industrial, telecommunications, medical, electric utility, consumer, and portable electronics applications. Due to this massive expansion, business and research in this field have dramatically growth in the last few years. Battery energy storage systems play a key role in such applications since they can significantly impact performance, life, cost, reliability, and safety of these systems. Indeed the main reasons of rechargeable Li-ions batteries success lies in their high energy and power density, the lack of memory effect and the low self-discharge rate. Although rapid progress has been made in terms of system storage technologies, there are still some issues due to battery dangerousness.

Security problems occur if the battery operates out of its Safe Operating Area. Boundaries are imposed in terms of terminal voltage and battery temperature. Security violation can reduce battery performances and cause permanently damages or even cell explosion compromising user’s safety. For these reasons a Battery Management System has to be implemented.

BMS is mandatory to ensure that the battery parameters stay into safety ranges. Focusing on temperature, modern BMS uses thermal sensor mounted on the case surface of the battery. Despite thermistors are simple and robust system design and easily applicable, they suffer from heat delay and do not provide information on internal temperature. Nowadays, methods that tend to estimate temperature without external sensors are proposed in literature. This kind of estimations uses Electrochemical

1. INTRODUCTION

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Impedance Spectroscopy, a powerful technique to gain a deeper insight on temperature, state of charge (SOC) and ageing effects by investigating the internal electrochemical impedance of Li-ion battery. The measurement is based on the study of the response of a system undergoing sinusoidal voltage/current stimuli at various frequencies. The ratio is calculated in a wide frequency range and a large part of the spectrum of the internal battery impedance is obtained. The aim of this thesis is to measure and analyze temperature effects on Li-ion batteries impedance by means of Electrochemical Impedance Spectroscopy. The evidence of a clear relationship between battery temperature and frequency at which the imaginary part of the impedance is null has been proven. This frequency is referred to as the intercept frequency. Since this point remains relatively constant with state of charge variations the main benefit provided is the possibility to accurately monitor, and consequently control, the internal temperature without external sensors and thermistors.

The thesis has been developed among Dialog Semiconductor S.R.L. in Livorno, Italy. The characterized batteries are two different tablet Li-ion cells, an ANR26650 LiFePO4 and Panasonic 18650 LiCoO2. Firstly, an overview on Li-ion battery has been

carried on and an introduction to electrochemical impedance spectroscopy has been provided. This theoretical background is reported in chapter 2. Recent works and articles have been studied and chapter 3 presents an exhaustive panoramic of EIS state-of-the-art for sensorless temperature estimation for Li-ion batteries applications.

Subsequently, the core of the thesis has been conducted in laboratory, testing the three different kinds of rechargeable batteries in various conditions.

These measurements have been found to be very thorny: the low value of the impedance makes easy to fall in measurement errors. Many setups have been tested before finding a valid setup, accurate enough to guarantee a significant reduction of cables effects. All the steps done to reach this aim and the final equipment are described in details in

chapter 4. Chapter 5 explains how the experiments have been carried out in the

laboratory environment. Firstly batteries have been tested at various states of charge and at a fixed temperature of 25°C. Then the state of charges has been fixed and batteries have been measured at various temperatures [10°C, 40°C]. In this way temperature and SOC effects have been evaluated separately. Once data have been collected, data extrapolation phase has been performed to evaluate temperature contribution and also to

1. INTRODUCTION

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analyze SOC effects. Some battery parameters show a great dependence on temperature and here comes the proposal of a sensorless measurement method. By measuring the value of the battery impedance, information on core temperature can be gained. Chapter

6 shows these results with particular attention to the intercept frequency which is

strongly dependent from temperature and hardly from the battery state of charge. The existence of an intrinsic relationship between the cell’s internal temperature and both the impedance module and imaginary part has been evaluated separately. A correlation with the state-of-the-art is also proposed in order to verify the correctness of the data found. The conducted investigation shows the way forward to future applications for battery temperature monitoring without application of external sensors. Possible future works and integration, for example in battery chargers, are also proposed. Finally chapter 7 presents the conclusions of this work.

2. BACKGROUND

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Chapter 2

Background

2.1. Li-ion batteries overview

2.1.1. Basic characteristic

Li-ion batteries represent the best technology to supply portable and mobile devices with electrical energy. Essentially a battery converts chemical energy into electric energy. If the battery belongs to the primary type, non-rechargeable ones, this process is irreversible and the battery can provide energy until its end of discharge. Instead the process is reversible in secondary or rechargeable batteries. Lithium metal, lithium-ion and lithium-ion polymer are the three main categories of lithium batteries. Lithium metal batteries are primary batteries, while both lithium-ion and lithium-ion polymer batteries are rechargeable.

2. BACKGROUND

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Let’s describe the general characteristics and the typical behavior of Li-ion batteries. First of all international industry standards differentiate between the word cell and the word battery. The first is an elementary electrochemical unit that contains the basic components, such as electrodes, separator, and electrolyte as described later in detail. Cells can be composed in series to increase the battery voltage or in parallel to increase the capacity. The battery or battery pack is the collection of more cells or cell assemblies which are ready for use, as it contains an appropriate housing, electrical interconnections, and possibly electronics to control and protect the cells from failure. In this regard, the simplest "battery" is a single cell with a small electronic circuit for protection. The differentiation should be done when dealing with specific applications, for example, battery electric vehicles, where battery may indicate a high voltage system of 400V, and not a single cell. However in this thesis the two words are used indistinctly. The amount of charge that can be stored in a battery indicates its capacity. The rated capacity is represented by the amount of charge that a battery can deliver to a load from full charge in a well-defined condition, specified by the battery manufacturer. It indicates the amount of ampere that the battery can supply continuously in one hour. The available capacity of a battery depends upon the rate at which it is discharged. If a battery is discharged at a relatively high rate, the available capacity will be lower than the rated capacity. The electric potential between the battery external terminals is defined as terminal voltage. The terminal voltage varies with the operating conditions of the battery. The nominal voltage, sometime also referred to as Mid-Point Voltage (MID), is the voltage that is measured when the battery has discharged 50% of its total energy. The end-of-discharge (EOD) and the end-of-charge (EOC) conditions represent respectively the voltages when discharge and charge are considered complete. They are also called cut-off voltages. Let’s observe that the value of the real capacity of a battery depends on the definitions of the cut-off voltages. In fact, if the definitions of the EOD and EOC are different from those specified by the battery manufacturer, then also the battery maximum capacity will be different from the rated capacity. Once EOD and EOC voltages have been specified, it is possible to introduce the definition of the state of charge.

Knowing the amount of energy left in a battery compared with the energy it had when it was full gives the user an indication of how much longer a battery will continue

2. BACKGROUND

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to perform before it needs recharging. State of Charge (SOC) is the ratio of the amount of charge left in a battery to its maximum capacity. It is a measure of the short term capability of the battery and it is usually expressed in percentage. Its values range is [0%, 100%], where SOC=0% corresponds to the full-discharge condition of the battery and SOC=100% corresponds to full-charge condition. Note that the preferred SOC reference should be the rated capacity of a new cell rather than the current capacity of the cell. This is because the cell capacity gradually reduces as the cell ages. For example, towards the end of the cell's life its actual capacity will be approaching only 80% of its rated capacity and in this case, even if the cell was fully charged, its SOC would only be 80% of its rated capacity. In general after performing lots of cycles (the number depends from operating condition and battery chemistry) the battery remains with about 75% to 85% of the original capacity. When used in notebook computers or cellular phones, this rate of deterioration means that after three to five years the battery will have capacities too low to be still usable. Temperature and discharge rate effects reduce the effective capacity even further. This difference in reference points is important if the user is depending on the SOC estimation as he would for example in a real gas gauge application in a car. Unfortunately the SOC measurement reference is often defined as the current capacity of the cell instead of the rated capacity. In this case a fully charged cell, nearing the end of its life, could have an SOC of 100% but it would only have an effective capacity of 80% of its rated capacity and adjustment factors would have to be applied to the estimated capacity to compare it to its rated new one. Using the current capacity rather than the rated capacity is usually a design shortcut or compromise to avoid the complexity of determining and allowing for the age related capacity adjustments which are conveniently ignored.

State of health (SOH) is a figure of merit of the condition of the battery compared to

its ideal conditions. During the lifetime of a battery, its performance or "health" tends to deteriorate gradually due to irreversible physical and chemical changes that take place with usage and with age until eventually the battery is no longer usable or dead. The SOH is an indication of the point which has been reached in the life cycle of the battery and a measure of its condition relative to a fresh battery. The units of SOH are percent points and 100% indicates that the battery's conditions match the battery's specifications.

2. BACKGROUND

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The open-circuit voltage (OCV) represents the voltage at the terminals of the battery when there is no load applied. The open-circuit voltage depends, in first approximation, on the battery state of charge and on its internal temperature.

Figure 2 Relative energy density of secondary cell chemistries [16]

Li-ion batteries are mostly diffused in the markets of power portable devices, power tools and electric vehicle. The reasons of their success are many: very high energy density, low weight, very low self-discharge rate, no memory effect and possibility of fast charge. The energy density is a measure of the amount of energy per unit weight or per unit volume which can be stored in a battery. Thus for a given weight or volume a higher energy density cell chemistry will store more energy or alternatively for a given storage capacity a higher energy density cell will be smaller and lighter. Figure 2 shows some examples of relative energy densities for some secondary cell chemistries. Higher energy densities are obtained by using more reactive chemicals. However more reactive chemicals tend to be unstable and can require safety precautions. These will be deeply explained in the next paragraph. Batteries from different manufacturers with similar chemistries and similar construction may have different energy densities and discharge performances since the energy density is also dependent on the quality of the active materials used in cell. Impurities may limit the cell capacities that can be achieved. Lastly the energy density does not usually refer to the chemicals alone but to the whole cell, taking into account the cell casing materials and the connections. This is why there is often a difference between cylindrical and prismatic cells.

2. BACKGROUND

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To go back to the reasons of Li-ion batteries success, other advantages have to be described. They can be deep cycled, losing 20% of total capacity against 50% of Lead-acid batteries and thanks to their high cell voltage fewer cells are needed for high voltage batteries (one Li-ion cell can replace three metal-hydride or nickel-cadmium batteries of only 1.2V). The Li-ion batteries are considered expensive; however their price continues to fall as the technology gains more acceptances. Another advantage is the small size available (solid state chemistry can be printed on to ceramic or flexible substrates to form thin film batteries with unique properties) and the different shapes available. Prismatic cell are easy to be packaged and occupies low space. Cylindrical cells are small too and used in portable computers. Pouch cells have flat bodies and since no rigid container is present their weight is significantly reduced.

Despite the chemistries, the main battery principle is the conversion of electric energy in chemical energy during charging and vice versa during discharging. Let’s have a look on the composition of a Li-ion cell.

The cell is formed by a positive and a negative electrode. To avoid that the two conductors would touch together and generate an internal dangerous shortcut, they are kept isolated by an electrolyte separator (about 20-25μm of thickness) which has to be a conductor for ions and an insulator for the electrons.

Figure 3 proposes the elementary behavior of rechargeable batteries in charging and

discharging conditions. In the first scenario, the positive electrode oxides and it releases CHARGE DISCHARGE Li+ Lithium Metal Oxygen Graphite Li+ ANODE CATHODE

2. BACKGROUND

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electrons. The ions Li+ are attracted by the anode, as illustrated by the green arrow. Although, during discharge the anode (which is the negative electrode) oxides, the ions Li+ move towards the cathode (red arrow) and the electrons released by the anode flow through the external circuit resulting in the current that provides power to the load. In particular when the battery is fully charged there is a surplus of electrons on the anode giving it a negative charge and a deficit on the cathode giving it a positive charge resulting in a potential difference across the cell. When the circuit is completed the surplus electrons flow in the external circuit from the negatively charged anode which loses all its charge to the positively charged cathode which accepts it, neutralizing its positive charge. This action reduces the potential difference across the cell to zero. The circuit is completed or balanced by the flow of positive ions in the electrolyte from the anode to the cathode. Note that since the electrons are negatively charged the electrical current they represent flows in the opposite direction, from the cathode (positive terminal) to the anode (negative terminal).

Additionally, as a battery is cycled, the formation of a new passivating film can be noticed, the Solid-Electrolyte Interface (SEI). Its growth results from irreversible electrochemical decomposition of the electrolyte. Due to the many different reactions that are involved in its formation, this film has not been completely understood yet. In the case of Li-ion batteries, SEI is formed at the negative electrode because typical electrolytes are not stable at the operating potential of this electrode during charging. The product of this decomposition forms a solid layer on the surface of the active material. The layer is composed of electrolyte-carbonate reduction products that serve both as ionic conductor and electronic insulator. As many works discover, this interface is favored by elevated temperature and remains consolidate up to 100 ˚C. Basically solid electrolytes and organic solvents easily decompose on the anodes during the charge, forming this new layer. The deposition of the SEI layer is an essential part of the formation process when the cells take their first charge. Initially, this formation protects the electrode against solvent decomposition at large negative voltage but it also increases the cell internal impedance and reduces the possible charge rates as well as the high and low temperature performance. Moreover the thickness of the SEI layer is not homogeneous and increases with age, increasing the cell internal impedance, reducing its capacity and hence its cycle life. However if the SEI growth was sustained at the

2. BACKGROUND

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formation rhythm during the whole battery operation, the latter would become unusable due to the continual loss of lithium. Instead the reason that batteries can continue to operate is that the SEI does not conduct electrons, and it is almost impenetrable to electrolyte particles. After the SEI formation, the inability of electrolyte molecules to travel through it and to reach the active material surface, where they could react with lithium ions and electrons, suppresses additional SEI growth. Intercalation is suppressed far less, because lithium ions can easily pass through the SEI through the exchange of ions between the solvent, SEI compounds and the lithium intercalated in the active material. Thus the battery is able to experience many charge-discharge cycles with little additional SEI build-up.

Unlike lead-acid, nickel metal hydride or nickel-cadmium batteries, there is no fixed chemistry for the lithium-ion cell, but it depends on the combination of anode, cathode and electrolyte materials. In general, the cathode is of metal oxide (ready to hold ions) while the anode consists of porous carbon. Depending on materials choices, the voltage, energy density, life, and safety of a Li-ion battery can change vividly.

LCO batteries consist of a cobalt oxide cathode and a graphite carbon anode; they present a high specific energy that makes it a popular choice for portable devices. The drawback of LCO chemistry is the relatively short life duration, low thermal stability and limited specific power. Instead Lithium Manganese Oxide (LMO) batteries have a high thermal stability and enhanced safety, but the cycle and calendar life are limited. One of the most successful Li-ion technologies is the nickel-manganese-cobalt oxide (NMC). The advantage of this type of batteries lies in the cathode chemistry which is a combination of nickel-manganese-cobalt. Nickel is known for its high specific energy but poor stability. Manganese manages to have a low internal resistance but offers a low specific energy. Combining the metals enhances each other strengths. A further development of NMC is lithium nickel-cobalt-aluminum technology. These batteries offer high specific energy and reasonably good specific power and long life duration. Adding aluminum gives the chemistry greater stability. High energy and power densities, as well as good life span, make NCA a candidate for EV field. High cost and marginal safety are negatives. Less good are safety and cost. Lithium Iron Phosphate (LFP) cells have high current rating and long cycle life, besides good thermal stability,

2. BACKGROUND

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enhanced safety and tolerance if abused. Li-phosphate has excellent safety and long life span but moderate specific energy and elevated self-discharge.

In all the chemistries above the anode is made of graphite, but also an alternative on battery anode exists. LTO batteries present lithium titanate anodes. The cathode is graphite and resembles the architecture of a typical lithium-metal battery. The cycle life should be higher than the one of a regular Li-ion. LTO batteries are safe and have excellent low-temperature performances.

To sum up, NCA enjoys the highest specific energy. However, manganese and phosphate are superior in terms of specific power and thermal stability. Li-titanate instead has the best life span. Consequently the orientation of manufacturers and researchers is to realize hybrid batteries with mixed cathode materials in order to obtain the maximum possible advantages.

2.1.2. Safety Issues

As repeated many times, Li-ion batteries are nowadays the technology of choice to supply portable and mobile electronic devices and electric vehicles with electrical energy. Their diffusion is due to the many advantages they provide: no memory effect, compact size, low weight, high power and energy density. Nevertheless Li-ion batteries can become unstable and even explode, so analyses have to be made and precautions have to be taken in order to guarantee full safety. Apart from the obvious inconvenience, the cost of replacing the battery can be prohibitive. This is particularly true for high voltage and high power automotive batteries which must operate in hostile environments and which at the same time are subject to abuse by the user. In the mid-2006 recall of 4.1 million Sony batteries used in Dell laptops was a consequence of internal contamination with metal particles. It wasn’t the first recall of its kind but the largest. During the past decade there have been numerous recalls of Li-ion batteries in cellular phones and laptops owing to overheating problems.

Critical conditions mainly regard voltage and temperature. Figure 4 reports typical operating ranges of Li-ion cells. The yellow area shows the safety operating area. The typical lower limits are 2V and 0°C and the typical upper limits are 4.2V and 60°C.

2. BACKGROUND

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Figure 4 Li-ion cell operating window [19]

First issue occurs while operating at high temperatures and it can lead to the destruction of the cell. Thermal condition of the battery depends from environment: if the ambient temperature is high the battery will gain heat from its surroundings. In this circumstance, the Arrhenius effect (see equation (15) in chapter 3) helps to get higher power out of the cell by speeding up the reaction rate. In addition higher currents give rise to higher I2R heat dissipation (Joule effect) and this even further increases temperature. Battery designers strive to keep the internal resistance of the cells as low as possible, but even with mΩ resistances the heating can be substantial. Joule effect can be the start of a positive temperature feedback and unless heat is removed faster than it is generated the result will be thermal runaway. Indeed in addition to I2R effect,

electrochemical reactions are exothermic and consequently reinforce the heat generated

by the current flow. In Li-ion batteries during charging the reaction is initially endothermic then becomes exothermic; the reverse during discharging. The first step for thermal runway is the breakdown of the passivating solid-electrolyte interface on the anode due to overheating (excessive current, over-charge or high external temperature). At about 80°C the electrolyte reacts with carbon anode uncontrollably. This reaction is exothermic and as the temperature continues to increase the breakdown of the organic solvents of the electrolyte occurs releasing flammable hydrocarbon gases but no

-50 0 50 100 150 200 250 300 350 0 2 4 6 8 10 T emper ature (° C) Cell Voltage(V) Lithium plating Cathode Breakdown Anode Copper dissolve SEI breakdown

Electrolyte exothermic breakdown Cathode Active Material Breakdown

Thermal Runaway Death and Law Suits

No fire yet Possible Venting

Battery Safety

2. BACKGROUND

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Oxygen. Pressure builds up into the cell but no fire occurs since there isn’t free Oxygen. Eventually heat from the electrolyte breakdown causes breakdown of the metal oxide cathode material releasing Oxygen which enables burning of both the electrolyte and the gases inside the cell. The breakdown of the cathode is also highly exothermic taking the temperature and pressure even higher (about 200°C).

Secondly, safety is compromised in case of internal shortcut between the two electrodes. This event is encouraged by both under-voltage and over-voltage condition. By allowing the cell voltage to fall below its end-of-discharge by over-discharging or storage for extended period results in progressive breakdown of the electrode materials. The anode copper current collector starts to dissolve into the electrolyte and disperses through it. When the voltage is increased again the copper ions precipitates as metallic copper wherever they causally happen to be, not necessary back on the current collector foil. This can ultimately cause a short circuit between the electrodes.

In addition if the charging voltage is increased beyond the recommended upper voltage threshold excessive current flows into the battery and gives rise to overheating. Temperature increases and the problems described above occur. Moreover this over-voltage condition is the main cause of Lithium plating. Indeed it implies excessive current and the ions can’t be accommodated quickly enough between the intercalation layers of the carbon anode (see figure 3). As a result they accumulate on the surface of the anode where they are deposited as metallic Lithium. The consequence of this is a reduction of free Li-ion, hence an irreversible capacity loss and since the plating is not necessarily homogeneous, but dendritic in form, it can ultimately result in a short circuit between the electrodes. The metallic lithium accumulate can physically create a mass that creates a contact cathode and anode.

Less obvious, also operating at low temperatures can cause no minor matters: still according to Arrhenius Law in this range the reaction rate decreases, implying a reduction in the current carrying capacity of the cell both for charging and discharging. In other words its power handling capacity is reduced. Furthermore, the reduced reaction rate slows down, and makes more difficult, the insertion of the Li-ion into the intercalation spaces. As with over-voltage operation, when the electrodes can’t accommodate the current flow, the result is reduced power and Lithium plating.

2. BACKGROUND

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Moreover, usually a battery package is not built by only one cell. During a thermal runaway, the high heat of the failing cell can propagate to the next cell, causing it to become thermally unstable as well. In some cases, a chain reaction occurs in which each cell disintegrates at its own timetable. A pack can get destroyed within a few short seconds or linger on for several hours as each cell is consumed one-by-one. To increase safety, packs are fitted with dividers to protect the failing cell from spreading to neighboring cells. This underlines the necessity to control single cell in operating and storage conditions.

2.2. Battery Management System

2.2.1. Main functions

Since the 90s, Li-ion batteries have come to dominate small-scale applications such as portable phones and laptops. However, the breakthrough in large scale applications such as (Hybrid) Electrical Vehicles was laborious, which is mainly caused by the high initial cost and safety issues. To ensure full safety, the automotive industry has established requirements and regulations in terms of security and trustworthiness.

To guarantee the highest safety Battery Management System (BMS) is necessary. It can consist on Battery Monitoring, keeping a check on the key operational parameters during charging and discharging such as voltages and currents and the battery internal and ambient temperature. The monitoring circuits would normally provide inputs to protection devices which would generate alarms or disconnect the battery from the load or charger should any of the parameters become out of limits. Some systems encompass not only the monitoring and protection of the battery but also methods for keeping it ready to deliver full power when called upon and methods for prolonging its life. This includes everything from controlling the charging regime to planned maintenance. The main aims of BMS are the protection of the cells from damage, the prolongation of battery life and the maintenance of the battery in a state in which it can accomplish the functional requirements of the application for which it was specified.

2. BACKGROUND

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In the following, the specific functions, that BMS may incorporate to achieve these objectives, are reported. The protection of the battery from out of tolerance operating conditions is fundamental to all BMS applications. Also charging control is essential to get the charge into the battery, to optimize the charging rate and knowing when to stop.

SOC and SOH determination could be required in many applications may be simply for

providing the user with an indication of the capacity left in the battery or may be for ensuring optimum control of charging process. In multi-cell battery chains one function can also be the Cell Balancing: small differences between cells due to production tolerances or operating conditions tend to be magnified with each charge/discharge cycle. Weaker cells become overstressed during charging causing them to become even weaker, until they eventually fail causing premature failure of the battery. Cell balancing is a way of compensating for weaker cells by equalizing the charge on all the cells in the chain and thus extending battery life. Storing the battery's history is another possible function of the BMS. This is needed in order to estimate the State of Health of the battery, but also to determine whether it has been subject to abuse. Parameters such as number of cycles, maximum and minimum voltages and temperatures and maximum charging and discharging currents can be recorded for subsequent evaluation. This can be an important tool in assessing warranty claims. For critical battery applications, or with expensive batteries, authentication is often employed to prevent the use of unapproved batteries in the application. This may be to avoid compatibility problems with system protection and power management schemes or with the applicable software revision or it could be to avoid damage to the reputation of the product and the brand if inferior or unreliable cells would be used. Hence some BMS give the possibility to record information about the cell such as the manufacturer's type designation and the cell chemistry. The battery package can incorporate a unique method of identifying the cell such as a code written in a memory device. The application checks the code and if isn’t correct a switch inside the application will not allow the power to be connected to the rest of the circuit. Communications is another main function of BMS. The so called Intelligent Batteries require communications with other system devices or with external equipment. Some have links to other systems interfacing with the battery for monitoring its condition or its history. In addition sometime users need access to the battery for modifying the BMS control parameters or for diagnoses and tests and also in this case communication is necessary.

2. BACKGROUND

23

Figure 5 BMS

Figure 5 shows the block diagram of a typical BMS.

To sum up, BMS have to manage rechargeable batteries and ensure they operate in their Safe Operating Area, within voltage, pressure, current and temperature restrictions.

Let’s explore in detail how temperature operating limits impact the security. Safety for batteries is strongly related to the temperature. When the temperature becomes too high, the battery can catch fire or even explode. However while such incidents are rare, consequences include costly recalls and potential risks for human safety.

For the success of the electric vehicle market these problems would be detrimental, if not fatal. Search for newer cathode and anode materials that do not compromise safety while providing high-energy and high-power densities is ongoing. In spite of these wide-ranging options, it is thus of high importance to have a reliable on-board

thermal management system with proper temperature gauging.

Temperature management should have these design objectives: the protection of the cell from overheating usually simply monitoring the temperature and interrupting the current path if the temperature reaches the limit, the dissipation of surplus heat generated and a minimum addition to the weight to include forced cooling. In addition there could be localized hot spots within the battery pack exceeding temperature limit, hence uniform heat distribution is another design objective. A temperature gradient across the battery pack can seriously affect the life of the battery.

2. BACKGROUND

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Therefore quality of a thermal management system can be referred to as the ability to estimate the temperature distribution during operation. Existing thermal management systems typically use temperature sensor mounted to the surface of the cell and equate the measured temperature to the cell mean temperature. Devices such as thermocouples, thermal fuses, current interrupt device (CID), and resistance temperature detectors (RTDs) thermistors are well-established for temperature monitoring.

Excessive temperatures can be detected by protection circuit that incorporates a

thermal fuse which will permanently shut down the battery if its temperature exceeds

the limit. Furthermore resettable fuse can be used: it is triggered when the temperature threshold is reached, but then it will reset once the fault conditions have been removed and after it has cooled down again to its normal state. The temperature rise could be caused by self-heating of the thermistor due to the current passing through it or by conduction or convection from the ambient environment. Thus it is an over-current and over-temperature protection. Thermistors are circuit devices whose resistance varies with temperature and they are divided in two categories: PTC (Positive Temperature Coefficient) thermistor has a resistance which increases with temperature, while NTC (Negative Temperature Coefficient) thermistor has resistance that decreases as temperature rises. These devices provide a voltage analogue of temperature and can be used for example to disconnect the battery from the charger in an over-temperature condition.

Just to report a couple of examples, Forgez et al. [13] measured the internal battery temperature using a thermocouple to develop a thermal model and to determine model parameters. A thermocouple was inserted in the center of a commercial cell (LFP) by drilling a hole into the battery. As it can easily be thought, this is not a suitable approach for mass production such as for commercial and military applications. The thermocouple insertion requires destructive modification of active cells, and potentially introduces new locations where the structural integrity of the cell could be compromised if exposed to extreme conditions. Lee et al. [14] developed a flexible micro-temperature sensor to in-situ monitor the temperature; obviously, in this case the disadvantage is that a piece of hardware must be placed inside the battery.

Consequently, non-invasive diagnostic techniques for Li-ion batteries, such as Electrochemical Impedance Spectroscopy (EIS), have recently garnered considerable

2. BACKGROUND

25

attention. The idea is to move into a direction where no sensor is needed to evaluate battery temperature.

2.3. Benefits of sensorless temperature estimation

Regardless of the simple and robust sensor design and the easy packaging, thermistors imply some no minor problems. Surface mounted thermal sensors like thermistors or thermocouples suffer from heat transfer delay due to the thermal mass of batteries. Therefore, if the system is not in thermal equilibrium, the internal battery temperature differs from the external one. An improved estimate of cell internal temperature can be made by using a lumped-parameter thermal model. However, also these methods suffer from a number of drawbacks: firstly the difficult to estimate the required model parameters and secondly the eventuality that a rapid fluctuation in internal temperature may not be registered by surface mounted sensor. With this limitation thermal runaway may not be detected, since associated timescales are often shorter than timescales associated with heat conduction through the cell.

The main drawback of using thermistors is their placement on the outer surface of the battery case. When placing a thermistor on the outside of the cell, the accuracy and the dynamics of the cell temperature measurement, especially in peaks and overload detection, convey both the thermal coupling between sensor and cell and the thermal capacity of the electrode and housing. An abnormally high internal cell temperature is a nearly universal manifestation of something going on within the cell, presaging a potentially catastrophic event if corrective measures are not taken immediately. Additionally, a common problem is the wiring of these sensors, which directly impact the cost of the battery package.

One method to avoid these problems could be an embedded micro-temperature sensor within the cell. With this new approach it should be possible to directly obtain information of internal temperature, nevertheless the additional manufacturing and instrumentation requirements would significantly increase cost and complexity.

As already introduced in chapter 1, one approach to overcome all these problems is EIS. It is a valid possibility to investigate cell temperature. It studies the impedance of

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the battery and it has been demonstrated that it is intrinsically related to battery internal temperature. Thus measuring battery impedance will give information of the core temperature of the battery. Methods based on EIS do not suffer from the disadvantages, recorded above, of conventional BMS. They give information directly of the internal temperature without any heat delay. The thermal inertia of a cell, depending on size and design, doesn’t affect this method. This can be a benefit for battery chargers. The high precision and the speed of the temperature measurement allow the charger to drive a high current longer than it can do with conventional external sensors. This reduces the charging time. In addition EIS doesn’t require any insertion into the battery implying cost reduction. So let’s have a deep insight into Electrochemical Impedance Spectroscopy.

2.4. Electrochemical Impedance Spectroscopy overview

2.4.1. Introduction

Over the past two decades, Electrochemical Impedance Spectroscopy has emerged as the most powerful electrochemical technique for defining reaction mechanisms, for investigating corrosion processes, and for exploring distributed impedance systems. The establishment of EIS has been initiated in the late 19th century through the work on operational calculus by Oliver Heaviside. Emil Warburg apparently was the first to extend the concept of impedance to electrochemical systems at the turn of the 19th century, when he derived the impedance function for a diffusion process that still bears his name. EIS was employed extensively using reactive bridges to measure the capacitance of ideally polarizable electrodes, interfacial impedance, leading to the development of models for the electrified interface. The principal reason why EIS did not find extensive use in defining corrosion and electro dissolution reaction mechanisms during this period is that the lowest accessible frequency was too high to detect relaxations involving reaction intermediates, except for the fastest of mechanisms. Indeed in general these techniques were limited to frequencies above about 100 Hz. There was an exception, the Berberian-Cole Bridge, which, in principle, has no lower frequency limit, but which, in practice, proved to be so laborious to use that it was never

2. BACKGROUND

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adopted on a widespread basis. However, the reactive bridge techniques were used extensively to measure the double layer capacitance at solid and liquid metal electrodes, with the classical work of D. C. Grahame in the 1950s being seminal in nature. Still, it was the invention of the potentiostat in the 1940s and the result of work by Epelboin in Paris in the 1960s with the development of the Frequency Response Analyzer that pushed EIS into the forefront as an electrochemical and corrosion mechanism analytical tool. The discovery revolutionized this field, primarily because of its ability to probe electrochemical systems at very low frequencies. These inventions have led to an explosion in the use of EIS for exploring a wide range of systems and processes.

Nowadays, there are no doubts that EIS has become a powerful tool for the analysis of complex processes that are influenced by many variables with regard to electrolyte, materials and interfacial geometry. The reason of its success lies in four main aspects: linearity of the technique, discovery of all the information that can be gleaned from the system by linear electrical perturbation/response techniques, experimental efficiency and the fact that the validity of the data is readily determined using integral transform techniques (the Kramers–Kronig transforms) which are independent from the physical processes involved. Let’s scrutinize in details how this technique works.

2.4.2. Measurement technique

As said in the previous paragraph, EIS is a powerful measurement method to gain different kind of information about electrochemical systems. It allows studying impedance of an electronic device in a wide frequency range [µHz, MHz] and that’s good to investigate the electrochemical process dynamics, superficial phenomena and inner structure of electrochemical systems.

It is also called AC Impedance Measurement. It studies the electrochemical characteristics of the system by applying an AC sinusoidal signal as input and measuring its output, according to the traditional Transfer Function method. This function would be the impedance of the device under test (DUT).

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Figure 6 Block diagram

The analysis can be conducted in two ways: potentiostatic (PEIS) by applying a voltage signal and measuring the current or galvanostatic (GEIS) by stimulating the device with current and measuring the voltage. In most cases, the two modes are equivalent and result in the same impedance diagrams providing that a sine current amplitude is “equivalent” to the voltage sine amplitude. However, in certain conditions, typically when the system evolves during the measurement, results from the two techniques may be different. In literature there are conflicts about which one is the best but a final agreement has not been found yet.

For example if the galvanostatic method is considered, the perturbation applied to the system is a sinusoidal current 𝑖(𝑡) = 𝐼_𝑀𝑠𝑖𝑛(𝜔𝑡) of one pulsation 𝜔 and amplitude 𝐼𝑀. The resulting voltage is 𝑣(𝑡) = 𝑉𝑀𝑠𝑖𝑛(𝜔𝑡 + 𝜃) of still pulsation 𝜔 and

amplitude 𝑉_𝑀. Since current has phase null, 𝜃 represents the different phase between the input and the output signal. Consequently 𝑍(𝑡) is defined as the ratio between the response signal and the input one and is calculated as follow:

𝑍(𝑡) = 𝑣(𝑡) 𝑖(𝑡) = 𝑉_𝑀𝑠𝑖𝑛(𝜔𝑡 + 𝜃) 𝐼_𝑀𝑠𝑖𝑛(𝜔𝑡) = 𝑍𝑀 𝑠𝑖𝑛(𝜔𝑡 + 𝜃) 𝑠𝑖𝑛(𝜔𝑡) (1)

In this work case the current will be the IBAT, the current which flows through the

battery, and the voltage will be VBAT, the terminal voltage of the battery during

measurement conditions. With Euler’s law, the impedance can be represented as a complex number. The current 𝑖(𝑡) can be associated to the complex number 𝐼_𝐵𝐴𝑇 = 𝐼_𝑀𝑒(𝑗𝜔𝑡) _{and the voltage 𝑣(𝑡) to 𝑉}

𝐵𝐴𝑇 = 𝑉𝑀𝑒(𝑗𝜔𝑡+𝜃).

The equation below reports the impedance calculus.

System (DUT)

2. BACKGROUND

29 𝑍(𝜔) =𝑉𝑀𝑒(𝑗𝜔𝑡+𝜃)

𝐼_𝑀𝑒(𝑗𝜔𝑡) = 𝑍𝑀𝑒(𝑗𝜃) = 𝑍𝑀(𝑐𝑜𝑠(𝜃) + 𝑗𝑠𝑖𝑛(𝜃)) (2)

𝑍𝑀 and 𝜃 are respectively the module and the phase of the impedance. In this case the

real and the imaginary part of the impedance are well-separated. And in figure 7 a simple schematic data representation in the complex plane is proposed.

Figure 7 Impedance vector representation

EIS measurement consists in exploring the impedance value in a wide frequency range and one popular format for evaluating electrochemical impedance data consists in plotting the imaginary part of the impedance against the real part at each excitation frequency (Nyquist plot). This diagram is also known as a Cole-Cole plot. On the plot the impedance can be represented as a vector of length 𝑍_𝑀 and the angle between this vector and the x-axis is the phase 𝜃. As it will be presented in detail in chapter 3, this plot format makes it easy to see the effects of the Ohmic resistance (resistance of the electrolyte) thanks to the possibility of reach higher frequencies. In addition this plot option emphasizes circuit components that are in series. On the other hand in the Nyquist plot format frequency doesn’t appear explicitly.

Another possible format for evaluating electrochemical impedance data is the Bode plot consisting in representing module and phase of impedance against the logarithm of the frequency. With this format it’s easy to understand how impedance depends on frequency since it appears explicitly as one of the axes. However the shape of the curves can change if the circuit values change. Therefore, in this work both plots are reported to take a complete analysis of electrochemical impedance data.

θ

Z

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As introduced above, high efficiency and linearity are two of main advantages of EIS. However electrochemical cells are not linear system: doubling the current will not necessarily double the voltage, but they can be pseudo-linear. Figure 8 shows how electrochemical systems can be pseudo-linear. Looking at a small enough portion of a cell's current versus voltage curve, it appears to be linear.

Figure 8 Current vs voltage

Particularly important is the amplitude of the input signal, because it has to be small enough so that the response of the system is linear. In this kind of system, the current response to a sinusoidal potential will be a sinusoid at the same frequency but shifted in phase. If the excitation is too big harmonics are generated and EIS does not work. On the other hand if the input is too small the output could remain covered by noise perturbation and it could be undetectable. The right trade-off has to be found. Linearity is one of the main criteria for valid EIS as well as stability and causality. A stable system does not change with time and returns to its original state after the perturbation. The response of a causal system is due only to the applied perturbation.

Another main advantage of EIS is that applying small signal perturbation it is a non-destructive method. The measurement doesn’t damage device performances, EIS is non-invasive and for example it doesn’t require battery disassembly or physical insertion of thermocouples for temperature monitoring. Thanks to its extensive range of working EIS can help to analyze different electrochemical phenomena which have different time constants and hence they are visible in a wide frequency range. Moreover in the battery field this method can be used during all the stages of the development of new devices, from the half-cell to the packaged batteries and it is applicable to any kind of batteries independently from chemistry and size.

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One of the main drawbacks of this measurement method is the high cost instrumentation. Second problem may be given by the time necessary to complete EIS measurement. The system being measured must be at a steady state throughout the time required to measure the EIS spectrum. Results are not immediate because of the complex data analysis and sometime tests actually require long time to be executed (for example evaluating the State of Health of a Li-ion battery).

Despite the problems shown above the use of EIS for electro-analysis is spreading every year. Researchers are moving also towards online parameters estimation. The cell impedance is measured online, while the battery is operating, without the need to disconnect it from the system. In this way there is no need to add AC signal injection circuit and costly response measurement. However let’s have a look on EIS for Li-ion batteries applications.

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Chapter 3

State-of-the-art of Sensorless

Temperature Estimation

3.1. Overview of the literature

Temperature monitoring in Li-ion battery applications is mandatory to avoid safety issues. Many analyses in literature highlight the advantages in using EIS for Sensorless Temperature Estimation instead of BMSs which conventionally use external surface-mounted sensors. First of all it has to be explained why temperature estimation is possible with AC Impedance Measurement. Hence let’s have an overview on how recent works and articles deal with EIS applications for Li-ion batteries.

It is clear for what explained in chapter 2, that there are several phenomena driving the behavior of a Li-ion battery. Looking back at figure 3, mass transport in the electrolyte, charge transfer reaction on the surface of the active electrode material and diffusion of lithium within the active electrode are the main reactions. The good point is that they exhibit time-dependent responses that are evidently detectable in the frequency domain, therefore EIS allows precisely this analysis. A sine wave voltage/current signal is applied to the battery and the respective current/voltage response is measured in a wide range of frequencies. After performing EIS, a great number of diagnostic plots can be obtained.

In practice each physical internal process can be modeled using electric circuit component such as resistances, capacitances, inductances etc. From Nyquist plots and data analysis it’s possible to extract the various components of the total impedance of

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the Li-ion battery. Although a lot of researches have already been carried on, the exact identification of which process is responsible for which part of the spectrum is not trivial. Several models have been proposed in literature.

Let’s focus on the Randle’s circuit which tends to model the main electrochemical phenomena of Li-ion batteries and is a good starting point for other more complex models.

Figure 9 shows the following elements:

• Electrolyte Resistance (RE) represents the migration of the lithium ions through

the electrolyte:

𝑅𝐸 = 𝜌 𝑙

𝐴 (3)

In this formula 𝜌 is the resistivity, 𝑙 and 𝐴 are respectively length and area of the two electrodes.

• Double-layer Capacitance (CDL) exists on the interface between an electrode

and its surrounding electrolyte. This double layer is formed as ions from the solution "stick on" the electrode surface. The charged electrode is separated from the charged ions. The separation is very small, often on the order of angstroms. Charges separated by an insulator form a capacitor. On a bare metal immersed in an electrolyte, it can be estimated that there will be from 20μF up to 60μF of capacitance for every 1cm2

of electrode area. The value of CDL depends

on many variables. Electrode potential, temperature, ionic concentrations, types of ions, oxide layers, electrode roughness, impurity adsorption, etc. are all

Figure 9 Randle’s Circuit for a Li-ion battery

Z

C

R

3. STATE-OF-THE-ART OF SENSORLESS TEMPERATURE ESTIMATION

34

factors. The equivalent electrical circuit for the redox reaction taking account of this double-layer capacitance is the parallel of CDL and RCT.

• Charge transfer Resistance (RCT) is generated by a single kinetic chemical

reaction at the electrode-solution interface and it represents the migration of ions from electrode to electrolyte.

𝑀𝑒⇒ 𝑀𝑒𝑥 𝑛+_{+ 𝑛𝑒}− ₍₄₎

In this equation electrons enter the metal and metal ions diffuse into the electrolyte. Charge is being transferred. This charge transfer has a certain velocity which depends on the type of reaction, temperature, product concentration and potential. The potential of the cell is related to the current by the following general equation:

𝑖 = 𝑖₀( 𝐶0 𝐶₀∗𝑒 𝛼𝑛𝐹𝜂 𝑅𝑇 −𝐶𝑅 𝐶_𝑅∗𝑒− −(1−𝛼)𝑛𝐹𝜂 𝑅𝑇 ) (5)

C0, CR, C0* and CR* are respectively the product and reactant concentrations at the

superficies and at the bulk, i0 is the exchange current density, η is the potential, F

and R are the Faraday and the gas constant, α is the reaction order and n is the number of electrons involved.

When the concentration in the bulk is the same as the electrode surface, C0=C0*

and CR=CR*, the Butler-Volmer Equation is obtained:

𝑖 = 𝑖₀( 𝑒𝛼𝑛𝐹𝜂𝑅𝑇 − 𝑒− −(1−𝛼)𝑛𝐹𝜂𝑅𝑇 ) (6)

Moreover when 𝜂 is very small the charge transfer resistance can be found as follow: 𝑅𝐶𝑇 = 𝜂 𝑖 = 𝑅𝑇 𝑛𝐹𝑖₀ (7)

• Warburg Impedance (ZW) characterizes ions diffusion in the electrode. It is

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Nyquist plot in the range of low frequencies as a diagonal straight line with a slope of 45 degrees. Indeed at high frequencies the Warburg impedance is small since diffusing reactants don’t have to move very far. For a “semi-infinite” distribution this impedance can be calculated just as follow (σ is the Warburg

coefficient): 𝑍𝑊 = 𝜎 √𝜔(1 − 𝑗) (8) 𝜎 = 𝑅𝑇 𝑛2_𝐹2_𝐴√2( 1 𝐶₀∗_√𝐷0+ 1 𝐶_𝑅∗_√𝐷𝑅) (9)

𝐷0 and 𝐷𝑅 are the diffusion coefficient of respectively oxidant and

reductant, 𝐴 the electrode area and 𝑛 the number of electrons involved. The equation (8) should be valid only if the thickness of the diffusion layer is infinite. The correct formula is more general and should consider these boundaries and change impedance behavior at lower frequencies:

𝑍_𝑊 = 𝜎

√𝜔(1 − 𝑗)𝑡𝑎𝑛 ℎ (√𝑗𝜔 𝛿

𝐷) (10)

𝛿 is the Nernst diffusion layer thickness and D is some average value of the diffusion coefficients of the diffusing species. Indeed if 𝛿 tends to infinite, or at higher frequencies (𝜔 ⇾ ∞), tanh(√𝑗𝜔𝛿_𝐷) would tend to the unit and the equation (10) would be simplified as the infinite diffusion equation (8). However in literature none of the works consider the second equation since the first one is a reasonable approximation to model the diffusion appeared from experimental results.

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According to the Randle’s circuit shown in figure 9, in figure 10 [2] is presented a corresponding qualitative Nyquist plot. In the graphic three different sections can be observed and this confirms that at various frequencies dissimilar processes contribute:

• At low frequencies only the diffusion effect is significant and indeed in the impedance spectrum only the single slope line caused by the Warburg impedance is visible;

• At medium frequencies the charge-transfer resistance is dominant

(Butler-Volmer’s equation) and the Nyquist plot shows a semicircle due to the “parallel”

between RCT and CDL. By extending the semicircle also in the low frequency

range, in other words if the semicircle is prolonged, it would reach the x-axis. Here the real part of the impedance is RE+RCT. So the diameter of the semicircle

indicates the value of charge transfer resistance. In addition the maximum of the semicircle is reached at the pulsation 𝜔𝑔. Knowing the value of the frequency

gives information on the value of CDL.

𝜔_𝑔 =_𝑅 1

𝐶𝑇𝐶𝐷𝐿 (11)

• At high frequencies the Ohmic effect is highlight. Here the x-axis cross occurs and the battery impedance is purely real and only the value of RE is relevant. The

frequency at which the imaginary part is null is called intercept frequency and it