Post on 17-Mar-2020
KIT – Universität des Landes Baden-Württemberg undnationales Forschungszentrum in der Helmholtz-Gemeinschaft www.iam.kit.edu/wet
Elektrochemische Modelle für Lithium -Ionen Batterien
André Weber
Institut für Angewandte Materialien - Werkstoffe der Elektrotechnik IAM-WETAdenauerring 20b, Geb. 50.40 (FZU), Raum 314
phone: 0721/608-7572, fax: 0721/608-7492andre.weber@kit.eduwww.iam.kit.edu/wet
Modellbildung elektrochemischer Systeme
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 2, 10.07.2018
www.iam.kit.edu/wet
Elektrochemisches ModellStand der Technik & Motivation
IAM-WET
Verhaltensmodell
geringe örtliche AuflösungechtzeitfähigParametrierung einfach
R1
UO
CV
R0 C1
Uba
tt
Ibatt
grobe Verhaltensprädiktion im vermessenen Parameterraum
Differentialgleichungen & FEM
hohe örtliche Auflösungnicht echtzeitfähigParametrierung aufwendig
Simulation von unbekannten Strukturen und MaterialienInter-/Extrapolation des Parameterraums
( )cDt
c ∇⋅∇=∂∂
( ) ( )
−
−−= …00
1exp UU
RT
zFjj
α
physikalisch motiviertes Ersatzschaltungsmodell
geringe örtliche AuflösungechtzeitfähigParametrierung mittel
Diagnose durch Simulation Fehler-/DegradationsfälleInter-/Extrapolation des Parameterraums
UO
CV
Uba
tt
Ibatt
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 3, 10.07.2018
www.iam.kit.edu/wet
Distribution of Relaxation Times as Basis for Time-Domain Models of Li-Ion Batteries
IAM-WET
Impedance Measurement
Physically MotivatedModel
Time-DomainSimulation
Model Targets:
Good Real-time Capabilities - model structure deployable on microcontroller
Scalability - accuracy and computational effort easy adaptable
Automization of Parameterization - model quality does not depend on human expertise
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 4, 10.07.2018
www.iam.kit.edu/wet
IntroductionBattery Modeling Today
IAM-WET
EIS Measurement
electrochemical impedance measurements
non parametric representation of linear dynamics
variation of operating points (SOC,T)
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 5, 10.07.2018
www.iam.kit.edu/wet
Model selection and fit
a priori knowledge necessary
stabilization of fit procedure critical
subtraction of differential capacity
R0 RQ1 RQ2 RQ3 ZGFLWC0
differential capacity
IntroductionBattery Modeling Today
IAM-WET
EIS Measurement
R0 RQ1 RQ2 RQ3 ZGFLWC0 fractional impedance elements
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 6, 10.07.2018
www.iam.kit.edu/wet
IntroductionBattery Modeling Today
IAM-WET
EIS Measurement
Model selection and fit
Approximation of fractional impedance elements
Fractional impedance elements cannot be simulated directly in time domain
Approximation with sum of RC-Elements:
empirical relation [1]
analytical expansion to sum
discretization of the distribution of relaxation times
RQ
RC1 RC2 RC3
Analytical expression for the measured impedance ha s to be known!
[1] Stephan Buller, Phd-Thesis, Aachener Beitrage des ISEA, Shaker Verlag GmbH, 2002
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 7, 10.07.2018
www.iam.kit.edu/wet
Approximation of fractional impedance elements
IntroductionBattery Modeling Today
IAM-WET
EIS Measurement
Model selection and fit
Time Domain Model and Simulation
OCV
SOC
UO
CV …+
PLECSt
U
t
I
Loss ofaccuracy
time effort
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 8, 10.07.2018
www.iam.kit.edu/wet
g(τ): Dirac pulses
g
t
1
p o l
RR
2
p o l
RR
τ1 τ2
ideal process: RC-Elements
RC1 RC2
The Distribution of Relaxation Times
IAM-WET
pol0
( )( )
1polg
Z j R djτω τωτ
∞=
+∫ g(τ): distribution function of relaxation times (DRT)
H. Schichlein, A.C. Müller, M. Voigts, A. Krügel and E. Ivers-Tiffée, Journal of Applied Electrochemistry, Vol. 32, No. 8, pp. 875-882, 2002
The Continuous Distribution of Relaxation Times
g
tτ1 τ2general distribution function
RQ1 RQ2
real process: RQ-Elements
Calculation of the Discrete Distribution of Relaxat ion Times
N is the Number of RC-Elementsdiscrete distribution: � is logarithmical equally spacedg(�n) = 0 at highest and lowest �n
g
log(τ)
ττω δωτ=
=+∑
1
( )( )
1
Nn
polnn
gZ j
j
…
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 9, 10.07.2018
www.iam.kit.edu/wet
Approximation of fractional impedance elements
IntroductionBattery Modeling Today
IAM-WET
EIS Measurement
Model selection and fit
Time Domain Model and Simulation
OCV
SOC
UO
CV …+
PLECSt
U
t
I
Loss ofaccuracy
time effort
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 10, 10.07.2018
www.iam.kit.edu/wet
IntroductionBattery Modeling Today
IAM-WET
EIS Measurement
Method of the Distribution of Relaxation Times
Time Domain Model and Simulation
OCV
SOC
UO
CV +
PLECSt
U
t
I
…Re(Zth)
Im(Z
th)
deconvolution of EIS
…
τωωτ=
=+∑
1
( )( )
1
Nn
polpolnn
Z jg
Rj
distribution ofrelaxation times
…
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 11, 10.07.2018
www.iam.kit.edu/wet
Verification of the DRT ModelExperimental
IAM-WET
2 Ah Li-ion cell from KOKAM
Pouch-cell
Anode: Graphite
Cathode: NCA/LCO-Blend
REM: Anode
2µm 2µm
Anode: Graphite Cathode: NCA / LiCoO2 Blend
LiCoO2
NCA
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 12, 10.07.2018
www.iam.kit.edu/wet
pulse height 1C
pulse length = 10s
f = [100kHz … 5mHz]
Uampl = 10mV / 5mV
Verification of the DRT ModelExperimental
IAM-WET, D. Klotz, M. Schönleber, J. P. Schmidt, and E. Ivers-Tiffée, Electrochim. Acta, 56, 8763 (2011).
Electrochemical Measurements
quasi stationary OCV
discharge/charge at C/40
Variation of Imax = C/4 … 2C � 0.5A � 4A
average discharge with Imax/2
transformationto frequencydomain1
merger of high andlow frequency range
impedance spectraf = [100kHz …10µHz]
completely autom
ated process
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 13, 10.07.2018
www.iam.kit.edu/wet
impedance spectroscopy time domain measurements
Impedance Measurements in the Frequency and Time Domain
IAM-WET
0.1 0.2 0.3 0.40
0.05
0.1
0.15
Z´ [Ω]
-Z´´
[Ω]
1 2 3
5
10
15
20
25
30
35
40
45
50
55
Z´ [Ω]
-Z´´
[Ω]
1 2 3
5
10
15
20
25
30
35
40
45
50
55
Z´ [Ω]
-Z´´
[Ω]
0.1 0.2 0.3 0.40
0.05
0.1
0.15
Z´ [Ω]
-Z´´
[Ω]
cell
00
00
SOC = 100 %
Û = 10 mV
fmin = 5,88 µHz
fmax = 10 kHzΔSOC = 1,95 %
tMess = 19 days
nIntegration = 3...8
LiCoO2 / C
C = 120 mAh
cellLiCoO2 / C
C = 120 mAh
SOC = 100 %
ΔU = 50 mV
fmin = 6,84 µHz
fmax,ZB = 8,63 Hzfmax = 10 kHz
ΔSOC = 5,01 %
tMess = 4 days
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 14, 10.07.2018
www.iam.kit.edu/wet
Measurement ResultsEIS
IAM-WET
f �0
f � ∞∞∞∞
SOC [%]
f �0
10mHz
extended frequency range via pulse measurements
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 15, 10.07.2018
www.iam.kit.edu/wet
Measurement ResultsDRT
IAM-WET
SOC ↓
SOC ?
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 16, 10.07.2018
www.iam.kit.edu/wet
Measurement ResultsDRT
IAM-WET
complex SOC-dependence
beneficial use of the DRT for identification and se paration of Processes 1,2
interpretation not necessary for parameterization of time domain model
[1] J.P. Schmidt, T. Chrobak, M. Ender, J. Illig, D. Klotz, and E. Ivers-Tiffée, J. Power Sources, 196 (2010) 5342-5348.
[2] J. Illig, T. Chrobak, D. Klotz, and E. Ivers-Tiffée, ECS Trans., 33 (2011) 3-15.
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 17, 10.07.2018
www.iam.kit.edu/wet
Measurement ResultsComparison: EIS / DRT-Model
IAM-WET
SOC [%]
DRT Model
Measurement
Very good accordance of measurement and DRT-model
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 18, 10.07.2018
www.iam.kit.edu/wet
Verification of the DRT-ModelScalability
IAM-WET
EIS Measurement / DRT-Model Variation of N
τωωτ=
=+∑
1
( )( )
1
Nn
polnn
gZ j
j
Results
N = 100 � 11.6 RC / decade
20 RC-elements show no significant difference to 100 RC-elements
N = 20 � 2.3 RC / decade
10 RC-elements show slight deviations
N = 10 � 1.1 RC / decade
4 RC-elements result in a bad fit of the impedance data
N = 4 � 0.5 RC / decade
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 19, 10.07.2018
www.iam.kit.edu/wet
Δ
Verification of the DRT-ModelScalability – Time Domain Simulation
IAM-WET
Comparison of dynamic behavior
seconds-timescale shows significant differences for 4 RC-model
slight increase of accuracy between 10 and 20 RC-model
no visible difference for 20 and 100 RC-model
10 RC-Model shows good accordance and modest calcul ation effort
Imax
-Imax Currrent
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 20, 10.07.2018
www.iam.kit.edu/wet
Comparison of Simulation Time
DRT-ModelReal-Time Capability
IAM-WET
Model Order
100
20
10
4
RCs / decade
11.6
2.3
1.1
0.5
Tsim / Tmeas [%]
9%
0.8%
0.5%
0.3%
Simulation Environment
PLECS
Matlab R2009awww.mathworks.com
PLECS 2.1www.plexim.com
Hardware:Intel® Core™ i7 CPU4GB RAM
����
����
����
����
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 21, 10.07.2018
www.iam.kit.edu/wet
Physically based models Behavioral models
Electrochemical models Equivalent circuit models Time -domain models
+ charge- and mass transport+ understand electrochemical
reactions+ high accuracy
+ dependence of battery characteristics on operating parameters
+ optimal battery design according to performance and capacity
+ estimation of the current battery condition
+ fast computation+ battery management systems
- time consuming computation - only applicable in the frequency domain
- no physical interpretation possible
Battery Models
IAM-WET
[1] Ma, Y. et al., RSC Advances, 6 (2016), 25435-25443
anode separator cathode
curr
entc
olle
ctor
curr
entc
olle
ctor
[1]
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 22, 10.07.2018
www.iam.kit.edu/wet
Physically based models Behavioral models
Electrochemical models Equivalent circuit models Time -domain models
Battery Models
IAM-WET
� New approach combines advantages
� High-resolution electrochemical characterisation needed for physical interpretability
physical interpretability fast computation in the time domain
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 23, 10.07.2018
www.iam.kit.edu/wet
Methodology
IAM-WET
Voltage behavior of a battery when applying a curre ntLoss processes that determine the overvoltage
< �� Open circuit voltage ���� ���
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 24, 10.07.2018
www.iam.kit.edu/wet
Methodology
IAM-WET
< �� Open circuit voltage
�� �� Instantaneous voltage drop
���� ���
���� �
Voltage behavior of a battery when applying a curre ntLoss processes that determine the overvoltage
Lithium transport in the electrolyte, electron transportin the electrodes and current collectors
1
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 25, 10.07.2018
www.iam.kit.edu/wet
Methodology
IAM-WET
< �� Open circuit voltage
�� �� Instantaneous voltage drop
�� �� Fast kinetic processes
���� ���
���� � ���� ���,�/�����,�
Voltage behavior of a battery when applying a curre ntLoss processes that determine the overvoltage
Lithium transport in the electrolyte, electron transportin the electrodes and current collectors
Interface losses at anode and cathode2
1
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 26, 10.07.2018
www.iam.kit.edu/wet
Methodology
IAM-WET
< �� Open circuit voltage
�� �� Instantaneous voltage drop
�� �� Fast kinetic processes
�� �� Slow diffusion processes
���� ���
���� � ���� ���,�/�����,�
���� �����,�/�
Voltage behavior of a battery when applying a curre ntLoss processes that determine the overvoltage
Lithium transport in the electrolyte, electron transportin the electrodes and current collectors
Interface losses at anode and cathode
Lithium solid state diffusion in anode and cathode3
2
1
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 27, 10.07.2018
www.iam.kit.edu/wet
Methodology
IAM-WET
< �� Open circuit voltage
�� �� Instantaneous voltage drop
�� �� Fast kinetic processes
�� �� Slow diffusion processes
���� ���
���� � ���� ���,�/�����,�
���� �����,�/�
Voltage behavior of a battery when applying a curre ntLoss processes that determine the overvoltage
Lithium transport in the electrolyte, electron transportin the electrodes and current collectors
Interface losses at anode and cathode
Lithium solid state diffusion in anode and cathode
Contact losses included in 4 1
3
2
1
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 28, 10.07.2018
www.iam.kit.edu/wet
Methodology
IAM-WET
Voltage behavior of a battery when applying a curre ntLoss processes that determine the overvoltage
< �� Open circuit voltage
�� �� Instantaneous voltage drop
�� �� Fast kinetic processes
�� �� Slow diffusion processes
���� ���
���� � ���� ���,�/�����,�
���� �����,�/�
Lithium transport in the electrolyte, electron transportin the electrodes and current collectors
Interface losses at anode and cathode
Lithium solid state diffusion in anode and cathode
Contact losses included in 4 1
3
2
1
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 29, 10.07.2018
www.iam.kit.edu/wet
Methodology
IAM-WET
How to determine the operating voltage
„fast“ lossprocesses (< 1 Hz)
„slow “ lossprocesses (> 1 Hz)
UOCV
Uop
ηloss
ηDiff
η0
ηCT/SEI
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 30, 10.07.2018
www.iam.kit.edu/wet
Methodology
IAM-WET
How to determine the operating voltage
Z´´
[]
g(f
) [
s]
Electrochemical Impedance Spectroscopy
DRT-Analysis:
Description of the loss processes over the frequency
Equivalent Circuit Modelling
Fast loss processes
P1
P2P3
� �� � �� !" #$�%1 ' ��� (�
)
*
„fast“ lossprocesses (< 1 Hz)
„slow “ lossprocesses (> 1 Hz)
UOCV
Uop
ηloss
ηDiff
η0
ηCT/SEI
���� Measurement of the frequency dependent resistance
�process identification
���� process quantification
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 31, 10.07.2018
www.iam.kit.edu/wet
Methodology
IAM-WET
How to determine the operating voltage
Current interruption method in the time domain
Time domain model fit
Slow loss processes
η � , ∙ �. ∙ 1 exp 2�.
', ∙ �3 ∙ 1 exp 2�3
„fast“ lossprocesses (< 1 Hz)
„slow “ lossprocesses (> 1 Hz)
UOCV
Uop
ηloss
ηDiff
η0
ηCT/SEI
���� Measurement of the low-frequent loss contributions
���� process quantification
U [
V]
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 32, 10.07.2018
www.iam.kit.edu/wet
Investigated Cell
IAM-WET
anode:
graphite
2 µm
LiNixCoyAl1-x-yO2LiCoO2carbon black
10 µm
SEM imagesHigh power pouch cell
Manufacturer KOKAM
Nominal capacity 350 mAh
Nominal voltage 3,7 V
Max. charge current 700 mA (≙ 2C)
Max. discharge current 7 A (≙ 20C)
Voltage limits (Umin- Umax) 2,7 - 4,2 V
cathode:
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 33, 10.07.2018
www.iam.kit.edu/wet
„fast“ lossprocesses (< 1 Hz)
„slow “ lossprocesses (> 1 Hz)
UOCV
Uop
ηloss
ηDiff
η0
ηCT/SEI
Experimental
IAM-WET
Fast loss processes
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 34, 10.07.2018
www.iam.kit.edu/wet
ExperimentalAnalysis of high and middle frequency Electrochemical Processes
IAM-WET
Electrochemical impedance spectroscopy: measurement of the frequency-dependent resistance
Excitation and response signal
0
0
( )( ) Re( ) Im( ) ' ''
( )juu t
Z e Z j Z Z j Zi t i
ϕω ⋅= = ⋅ = + ⋅ = + ⋅
T = 25 °CSOC = 60 %
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 35, 10.07.2018
www.iam.kit.edu/wet
ExperimentalAnalysis of high and middle frequency Electrochemical Processes
IAM-WET
Electrochemical impedance spectroscopy: measurement of the frequency-dependent resistance
Excitation and response signal Impedance in a Nyquist diagram
0
0
( )( ) Re( ) Im( ) ' ''
( )juu t
Z e Z j Z Z j Zi t i
ϕω ⋅= = ⋅ = + ⋅ = + ⋅
T = 25 °CSOC = 60 %
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 36, 10.07.2018
www.iam.kit.edu/wet
Requirements for validity
ExperimentalAnalysis of high and middle frequency Electrochemical Processes
IAM-WET
Kramers Kronig validity test to ensure the validity o f the measurement
http://www.iam.kit.edu/wet/Lin-KK.php
If all these requirements are fulfilled, the Kramers Kronigrelations can be derived:
Linearity of the measured system
Time invariance of the measured system
Causality of the measured system
2 20
2 ' Im( ( '))Re( ( )) '
'
ZZ d
ω ωω ωπ ω ω
∞ ⋅=−∫
2 20
2 ' Re( ( '))Im( ( )) '
'
ZZ d
ω ωω ωπ ω ω
∞ ⋅= −−∫
If Re(Z) is known, Im(Z) can be computed.
� Comparison of computed part and measured partreveals the Kramers Kronig residuals .
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 37, 10.07.2018
www.iam.kit.edu/wet
Requirements for validity
ExperimentalAnalysis of high and middle frequency Electrochemical Processes
IAM-WET
Kramers Kronig validity test to ensure the validity o f the measurement
Example: change of SOC
http://www.iam.kit.edu/wet/Lin-KK.php
If all these requirements are fulfilled, the Kramers Kronigrelations can be derived:
Linearity of the measured system
Time invariance of the measured system
Causality of the measured system
2 20
2 ' Im( ( '))Re( ( )) '
'
ZZ d
ω ωω ωπ ω ω
∞ ⋅=−∫
2 20
2 ' Re( ( '))Im( ( )) '
'
ZZ d
ω ωω ωπ ω ω
∞ ⋅= −−∫
If Re(Z) is known, Im(Z) can be computed.
� Comparison of computed part and measured partreveals the Kramers Kronig residuals .
1. Approach the desired SOC
2. Wait for x hours until the cell is in stationary mode
� What is x ?
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 38, 10.07.2018
www.iam.kit.edu/wet
ExperimentalAnalysis of high and middle frequency Electrochemical Processes
IAM-WET
Distribution of relaxation times for the clear sepa ration of loss processes
T = 25 °CSOC = 60 %
Z´´
[]
15 Hz100 Hz
MHz
µHz
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 39, 10.07.2018
www.iam.kit.edu/wet
ExperimentalAnalysis of high and middle frequency Electrochemical Processes
IAM-WET
Distribution of relaxation times for the clear sepa ration of loss processes
T = 25 °CSOC = 60 %
Z´´
[]
15 Hz100 Hz
MHz
µHz
relaxation time of process n
2,0
2
1,0
1
22
2
11
1
1111)(
ωτωτωωω
jR
jR
CRjR
CRjR
jZ+
++
=+
++
=
:nnn0, CR=τ
distribution function of relaxation timespol
0
( )( )
1g
Z j R djτω τωτ
∞
=+∫
# � :[2]
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 40, 10.07.2018
www.iam.kit.edu/wet
ExperimentalAnalysis of high and middle frequency Electrochemical Processes
IAM-WET
Distribution of relaxation times for the clear sepa ration of loss processes
T = 25 °CSOC = 60 %
Z´´
[]
15 Hz100 Hz
MHz
µHz
relaxation time of process n
2,0
2
1,0
1
22
2
11
1
1111)(
ωτωτωωω
jR
jR
CRjR
CRjR
jZ+
++
=+
++
=
:nnn0, CR=τ
distribution function of relaxation timespol
0
( )( )
1g
Z j R djτω τωτ
∞
=+∫
# � :[2]
process 1 process 2
ggeneraldistributionfunction
�*,3�*,. �
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 41, 10.07.2018
www.iam.kit.edu/wet
ExperimentalAnalysis of high and middle frequency Electrochemical Processes
IAM-WET
Distribution of relaxation times for the clear sepa ration of loss processes
T = 25 °CSOC = 60 %
Z´´
[]
15 Hz100 Hz
MHz
µHz
relaxation time of process n
2,0
2
1,0
1
22
2
11
1
1111)(
ωτωτωωω
jR
jR
CRjR
CRjR
jZ+
++
=+
++
=
:nnn0, CR=τ
distribution function of relaxation timespol
0
( )( )
1g
Z j R djτω τωτ
∞
=+∫
# � :[2]
P1
P2 P3
15 Hz
100 Hz
process 1 process 2
ggeneraldistributionfunction
�*,3�*,. �
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 42, 10.07.2018
www.iam.kit.edu/wet
process 1 process 2
ggeneraldistributionfunction
�*,3�*,. �
relaxation time of process n
2,0
2
1,0
1
22
2
11
1
1111)(
ωτωτωωω
jR
jR
CRjR
CRjR
jZ+
++
=+
++
=
:nnn0, CR=τ
distribution function of relaxation timespol
0
( )( )
1g
Z j R djτω τωτ
∞
=+∫
# � :[2]
ExperimentalAnalysis of high and middle frequency Electrochemical Processes
IAM-WET
Distribution of relaxation times for the clear sepa ration of loss processes
T = 25 °CSOC = 60 %
Z´´
[]
15 Hz100 Hz
MHz
µHz
P1
P2 P3
15 Hz
100 Hz
Physical meaning of P1, P2 and P3 ?Physical meaning of P1, P2 and P3 ?
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 43, 10.07.2018
www.iam.kit.edu/wet
ExperimentalAnalysis of high and middle frequency Electrochemical Processes
IAM-WET
Experimental cells for the assignment of loss proce sses
3-electrode setup
IAM-WET-housing
6$2%78!!
6$2%98:3 ;$2%78!!6$2%98:.
Working electrode: Anode or Cathode
Reference electrode: Al-mesh with LTO coating
Counter electrode: Lithium
T = 25 °CSOC = 40 %
• inhouse development• 18 mm electrode diameter• 3-electrode setup[3,4]
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 44, 10.07.2018
www.iam.kit.edu/wet
ExperimentalAnalysis of high and middle frequency Electrochemical Processes
IAM-WET
Experimental cells for the assignment of loss proce sses
3-electrode setup
IAM-WET-housing
6$2%78!!
6$2%98:3 ;$2%78!!6$2%98:.
Working electrode: Anode or Cathode
Reference electrode: Al-mesh with LTO coating
Counter electrode: Lithium
Measurement results
P1 P2 P3
T = 25 °CSOC = 40 %
• inhouse development• 18 mm electrode diameter• 3-electrode setup[3,4]
Charge transfer of the
cathode PCT,C
SEI-process / charge transfer
of the anode PSEI/CT,A: side
peaks due to the porous
electrode structure
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 45, 10.07.2018
www.iam.kit.edu/wet
ExperimentalAnalysis of high and middle frequency Electrochemical Processes
IAM-WET
T = 25 °CSOC = 60 %
Quantification of loss processes by equivalent circ uit modeling
� Frequency dependent processes can be simplified simulated
with RQ elements
<=>,= <?@A/=>,B
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 46, 10.07.2018
www.iam.kit.edu/wet
ExperimentalAnalysis of high and middle frequency Electrochemical Processes
IAM-WET
T = 25 °CSOC = 60 %
Quantification of loss processes by equivalent circ uit modeling
� Frequency dependent processes can be simplified simulated
with RQ elements
� Good fit quality despite simplified model
� Fit delivers resistance and time constant of each process
Im(Z
) [
]
<=>,= <?@A/=>,B
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 47, 10.07.2018
www.iam.kit.edu/wet
Experimental
IAM-WET
Slow loss processes
„fast“ lossprocesses (< 1 Hz)
„slow “ lossprocesses (> 1 Hz)
UOCV
Uop
ηloss
ηDiff
η0
ηCT/SEI
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 48, 10.07.2018
www.iam.kit.edu/wet
ExperimentalAnalysis of low frequency Electrochemical Processes
IAM-WET
I = 350 mAT = 25 °C
Slow diffusion process PDiff,A/C takes place in the low frequency range
� EIS not applicable
Current interruption method in the time domain
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 49, 10.07.2018
www.iam.kit.edu/wet
ExperimentalAnalysis of low frequency Electrochemical Processes
IAM-WET
I = 350 mAT = 25 °C
Slow diffusion process PDiff,A/C takes place in the low frequency range
� EIS not applicable
� Time domain measurement
1. Start at OCV-level
2. Apply a current pulse
3. Analyze the voltage curve after cut-off of the current
� cell in stationary mode
Current interruption method in the time domain
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 50, 10.07.2018
www.iam.kit.edu/wet
ExperimentalAnalysis of low frequency Electrochemical Processes
IAM-WET
I = 350 mAT = 25 °C
Slow diffusion process PDiff,A/C takes place in the low frequency range
� EIS not applicable
� Time domain measurement
1. Start at OCV-level
2. Apply a current pulse
3. Analyze the voltage curve after cut-off of the current
� cell in stationary mode
Current interruption method in the time domain
4. Subtract η0, ηCT,C and ηCT/SEI,A from relaxing overvoltage
� diffusion overvoltage ηDiff,A/C remains
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 51, 10.07.2018
www.iam.kit.edu/wet
ExperimentalAnalysis of low frequency Electrochemical Processes
IAM-WET
I = 350 mAT = 25 °C
At least two diffusion processes in the cell: anode and cathode
� Fit delivers resistance and time constant :
CDE:: � , ∙ �DE::,. ∙ 1 exp GHIJKK,L ' , ∙ �DE::,3 ∙ 1 exp G
HIJKK,M
Time domain fit model
[5] Pop, V. et al., J. Electrochemical Society, 153 (2006), A2013-A2022
� Relative error below 0.1%
� good fit quality
[5]
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 52, 10.07.2018
www.iam.kit.edu/wet
Battery Model
IAM-WET
NOPQQ,B/=(SOC,T)ROPQQ,B/=(SOC,T)
OCV by linear interpolation ofcharge and discharge curve
EISDRT-AnalysisECM TDM
OCV(SOC,T) N (SOC,T)N=>,=(SOC,T)
R=>,=(SOC,T)
N?@A/=>,B(SOC,T)
R?@A/=>,B(SOC,T)
curr
ent
S � TUV, W, , �SXYZ TUV, W C* TUV, W, , C[\]/Y^,_ TUV, W, , CY^,Y$TUV, W, ,% CDE::,_/Y$TUV, W, ,%
Battery Model
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 53, 10.07.2018
www.iam.kit.edu/wet
Results
IAM-WET
�`a � ��� � ����/��,� ���,� �����,�/�Simulation of a cell discharge
I = 350 mAT = 25 °C
U [V
]
Quelle:
Institut für Angewandte Materialien Werkstoffe der Elektrotechnik
Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 54, 10.07.2018
www.iam.kit.edu/wet
Results
IAM-WET
Simulation of a cell discharge
� Absolute residuals below 30 mV in 90% of the SOC range
� Only at SOCs < 10% the residuals are higher than 30 mV
I = 350 mAT = 25 °C
U [V
]
resid
uals
[m
V]
�`a � ��� � ����/��,� ���,� �����,�/�