Basics of Impedance
Spectroscopy
(<1% of the entire topic!)
B. Markovsky, markovb@mail.biu.ac.il
Summer Courses, Bar-Ilan University, September 2014
The main goal of this
presentation is a brief
“Introduction” to EIS
Electrochemical Techniques
CVs of LiMO2 electrode.
Dynamic technique.
50 V/s
t
40
30
20
3.76 V
3.69 V
10 V/s
10
T=300C
Time domain. Current = f(time)
Relaxation:
Current decays with time
An Electrochemical Impedance Spectrum
1 µHz
Frequency domain
from high to low frequencies
100 kHz
The number of papers on EIS has
doubled every 4 – 5 years !
AC vs. DC methods
Current
Potential
time
time
Once we apply DC methods, the cell is totally changed.
 Surface and volume changes
 Phase transitions
 Electrolyte oxidation/reduction
Current
Potential
time
time
For AC methods, very small perturbation is applied.
Nearly non-destructive! Cell is unchanged!
Alternating current (ac) methods: the
merits
• To grasp the entire features of the system:
Sometimes we need
X-ray
to see the inside of our
body.
Sometimes we need ac
methods
to see the inside of an
electrochemical cell.
ELECTROCHEMICAL IMPEDANCE
SPECTROSCOPY
 Electrochemical Impedance Spectroscopy (EIS) is actually a
special case among electrochemical techniques.
 It is based on the perturbation of an equilibrium state, while the
standard techniques are dynamic (e.g. CV) or are based on the
change from an initial equilibrium state to a different, final state
(e.g. potential step, chronocoulometry).
 Hence, EIS is a small-signal technique where, in the
analysis of the impedance spectra, a linear current-voltage
relation is assumed.
ELECTROCHEMICAL IMPEDANCE
SPECTROSCOPY
Impedance spectroscopy is a non-destructive technique and so
can provide time dependent information about the properties of
a system but also about ongoing processes such as:
- corrosion
of metals,
- discharge and charge of batteries,
- electrochemical reactions in fuel cells,
capacitors or any other electrochemical
process.
Resistance. Ideal Resistor
Everyone knows about the concept of electrical resistance.
What is resistance?
It is the ability of a circuit element to resist the flow of electrical
current.
Ohm's law defines resistance R in terms of the ratio between
voltage, E, and current, I.
E (t )
R
I (t )
Ideal Resistor
E (t )
R
I (t )
This relationship is limited to only one circuit element ---the ideal resistor !
An ideal resistor has several simplifying properties:
• It follows Ohm's law at all current and voltage levels.
• It's resistance value is independent on frequency.
• AC current and voltage signals through a resistor are in
phase with each other.
In a Real World:
 Circuit elements exhibit much more complex behavior. In place of
resistance, we use impedance, which is a more general circuit
parameter.
 Like resistance, impedance is a measure of the ability of a
circuit to resist the flow of electrical current.
 Electrochemical Impedance is normally measured using a
small excitation signal (3 – 10 mV). This is done so that the cell's
response is pseudo-linear.
 In a pseudo-linear system, the current response to a
sinusoidal potential will be a sinusoid at the same frequency
but shifted in phase.
Sinusoidal Current Response in a Linear System
Phasor (Vector)
diagram
for an ac-Voltage
E
time
I
time
 - phase angle
Phaseshift
Phase-shift
A purely sinusoidal voltage: Et=E0 sin t
E is the amplitude of the signal, and  is the radial (angular)
frequency.  (in radians/second) and frequency f (in Hertz (1/sec)
are related as:  =2 f
Phasor (Rotating Vector) diagram
Response dI of dE from the Current / Potential
relation:
We can disturb an electrical element at a
certain potential E with a small
perturbation dE and we will get at the
current I a small response perturbation
dI.
In the first approximation, as the
perturbation dE is small, the response dI
will be a linear response as well.
An oval is plotted.
This oval is known as a "Lissajous figure".
Complex Numbers
In Electrical Engineering to add together resistances, currents or DC
voltages “real numbers” are used.
But real numbers are not the only kind of numbers we need to use
especially when dealing with frequency dependent sinusoidal
sources and vectors.
Complex Numbers were introduced to allow complex
equations to be solved with numbers that are the
square roots of negative numbers, √-1.
i=-1
Complex Number =
Real number + Imaginary number
In electrical engineering, √-1 is called an “imaginary number”
and to distinguish an imaginary number from a real number the
letter “ j ” known commonly in electrical engineering as the
j-operator, is used.
The letter j is placed in front of a real number to signify its imaginary
number operation. Examples of imaginary numbers are:
j3, j12, j100 etc.
A complex number consists of two distinct but very much
related parts, a “Real Number ” plus an “Imaginary Number”.
Complex Numbers. Complex Plane
Complex Numbers represent points in a two-dimensional
complex plane that are referenced to two distinct axes.
The horizontal axis is called the “Real Axis” while the
vertical axis is called the “Imaginary Axis”.
The real and imaginary parts of a complex number,
Z are abbreviated as Re(z) and Im(z).
Two Dimensional Complex Plane (Four Quadrant
Argand Diagram)
Z= -8 – j5
Z = 5 + j0
Z = 0 + j4
Negative Imaginary Axis
Complex Numbers. Complex Plane
Imaginary axis
i=-1
Real axis
The plane of complex numbers spanned by the vectors 1
and i, where i is the imaginary number. Every complex
number corresponds to a unique point in the complex plane
(Argand or Gauss plane).
Complex writting
Using Euler’s relationship exp(i )  cos   i sin 
it is possible to express the impedance as a complex function.
The potential is described as,
Z (t ) 
E0 cos(t )
E (t )
cos(t )

 Z0
I (t ) I 0 cos(t   )
cos(t   )
E (t )  E0 exp( jt )
and the current response as,
I (t )  I 0 exp(it  i )
The impedance is then represented as a
complex number:
E
Z
 Z 0 exp(i )  Z 0 (cos   i sin  )
I
Data Presentation: Nyquist Plot with Impedance Vector
Z
E
 Z0 exp(i )  Z0 (cos   i sin  )
I
The expression for Z() is composed of a real and an
imaginary part. If the real part is plotted on the X axis and the
imaginary part on the Y axis of a chart, we get a "Nyquist
plot“. (Harry Nyquist, 1889-1976).
C
Imaginary part
1 1
1
 
Z R iC
Real part
R
The Nyquist plot results from
the RC circuit. The semicircle
is characteristic of a single
"time constant".
Semicircle in Nyquist plot
Et=E0 sin t
(1)
It=I0 sin (t + )
(2)
The impedance of an ohmic resistance R and a capacitance C
in parallel can be written as follows:
R
C
General formulae of
a circle:
X 2 + Y 2=r2
(r is the radius)
The Bode Plot
Another popular presentation method is the "Bode plot". The
impedance is plotted with log frequency (log ) on the X-axis
and both the absolute value of the impedance (|Z| =Z0 ) and
phase-shift on the Y-axis.
Unlike the Nyquist plot, the Bode plot explicitly shows frequency
information.
A parallel R-C combination
The parallel combination of a resistance and a
capacitance, start in the admittance representation:
1
Y ()   jC
R
Transform to impedance representation:
R
C
1
1
1/ R  jC
Z () 



Y () 1/ R  jC 1/ R  jC
General formulae of
R  j R 2 C
1  j
R
a circle:
2 2 2
1  R C
1  2 2
X 2 + Y 2=r2
(r is the radius)
A semicircle in the impedance plane!
Semicircles in Nyquist plots
 The semicircle is characteristic of a single
“RC-constant”.
 Electrochemical impedance plots often contain
several semicircles.
 Often only a portion of a semicircle is seen.
Bode plot (Zre, Zim)
1.E+05
Zreal
Zimag
Zreal, -Zimag, [ohm]
1.E+04
1.E+03
1.E+02
1.E+01
1.E+00
1.E-01
1.E-02
1.E+00
1.E+01
1.E+02
1.E+03
frequency, [Hz]
1.E+04
1.E+05
1.E+06
Bode plot: absolute (Z), phase vs. frequency
1.E+05
90
1.E+04
75
60
45
1.E+03
30
15
1.E+02
1.E+00
1.E+01
1.E+02
1.E+03
Frequency, [Hz]
1.E+04
1.E+05
0
1.E+06
Phase (degr)
abs(Z), [ohm]
abs(Z)
Phase (°)
Different Bode representations
Zimag
Zreal
Other representations
Bode graph. “Double log” plot
1
Z R  R ; YR 
R
1
Z ()  R 
 R  j / C
j C
1
Y () 

R  j / C
2C 2 R
C
j
2 2 2
1  C R
1  2C 2 R 2
‘time constant’:
Semicircle
 = RC
Warburg Impedance
The rate of an electrochemical reaction: charge-transfer,
diffusion…
Whenever diffusion effects completely dominate the
electrochemical reaction mechanism, the impedance is called the
Warburg Impedance.
For diffusion-controlled electrochemical reaction, the current is
45 degrees out of phase with the imposed potential.
In this case, (450) , the real and imaginary components of the
impedance vector are equal at all frequencies.
In terms of simple equivalent circuits, the behavior of Warburg
impedance (a 450 phase shift) is midway between that of a
resistor (a 00 phase shift) and a capacitor (900 phase shift).
Warburg impedance
Diffusion: Warburg element
Semi-infinite diffusion,
Flux (current) : J   D C
First Fick’s Law
x
x 0
RT
Potential
:EE 
ln C
nF
ac-perturbation: C (t )  C  c(t )
2

C

C
Second Fick’s Law
:
D 2
t
x
Boundary
condition
: C ( x, t )
C
x 
PITT and Impedance spectroscopy
PITT – Small potential steps E from Eeq, I vs. t is measured.
Kinetic limitations other than diffusion are ignored.
 = l2/D
It1/2 is the time invariant at t<< (short-time domain)
D(E)=[1/2l It1/2/QmX(E)]2= [(1/2l (It1/2/E)/ Cint(E)]2
EIS – the semi-infinite (Warburg) domain for finite-space diffusion
response: Z’’ vs. Z’ at the low frequency is analyzed.
D = 0.5 l2 [CintAw]-2
-1/2
PITT and EIS for the same electrode potential, should provide the constant:
Aw [(It1/2)/E]= (2)-1/2
This is a proof that the measurements are correct.
Real thin (1500 ) cathode LixV2O5
450, Warburg
element
M. Levi, Z. Lu, D. Aurbach, JPS, 2001, 97, 482
Diffusion time .
Li+ diffusion coefficient in LiV2O5
Li+ diffusion coefficient in thin film graphite electrodes
M. Levi, D. Aurbach, J. Phys. Chem., 1997, 101, 4641
Li+ intercalation cathode LixCoO2
Li+ diffusion in LixCoO2 at low frequencies
Thin Na-V2O5 electrodes, 3 , 1 – 2 mg/cm2
450, Warburg
element
Thin Na-V2O5 electrodes. Li+ diffusion coefficient
Equivalent Circuit Concept
RRsol

45°
Rsol
Analysis and Modeling. Data Validation
Before starting the analysis and modeling of the experimental
results one should be certain that the impedances are valid.
There is a general mathematical procedure
(Kramers-Kronig), which allows for the verification
of the impedance data.
The impedance measured is valid provided that the following
4 criteria are met: linearity, causality, stability, finiteness.
1. Linearity:
 A system is linear when its response to a sum of individual input
signals is equal to the sum of the individual responses.
 Electrochemical systems are usually highly non-linear
and the impedance is obtained by linearization of equations
for small amplitudes.
 For the linear systems the response is independent of the
amplitude.
 It is easy to verify the linearity of the system:
if the obtained impedance is the same when the
amplitude of the applied ac-signal is halved
then the system is linear.
Electrochemical systems are, in general,
not linear.
A very small portion
of the I vs. V curve
appears to be
linear (pseudo-linear)
In normal EIS practice, a small (1 to 10 mV) AC signal is applied to the cell.
With such a small potential signal, the system is pseudo-linear.
AC Current / A
AC Current / A
Lissajous plot
AC Potential / V
A typical Lissajous plot for
A Lissajous plot showing
a linear system
a non-linear response
2. Causality:
The response of the system must be entirely
determined by the applied perturbation.
The impedance measurements must also be
stationary.
The measured impedance must not be time
dependent !
3. Stability:
The stability of a system is determined by its response to
inputs. The system is stable if its response to the impulse
excitation approaches zero at long times.
The measured impedance must not be time dependent.
This condition may be easily checked by
repetitive recording of the impedance spectra;
then the obtained Bode plots should be identical.
Steady-State Systems
 Measuring an impedance spectrum takes time
(minutes - hours).
The system being measured must be at a steady-state
throughout the time required to measure the spectrum.
 A common cause of problems in EIS measurements and
analysis is drift in the system being measured.
 Standard EIS analysis tools may give wildly inaccurate
results on a system that is not at steady-state.
Possible tests for the validity of EIS data
Kronig-Kramers (KK) test
 The Kronig-Kramers (KK) relations are mathematical properties
which connect the real and imaginary parts of any complex
function.
 During the KK test, the experimental data points are fitted using
a special model circuit which always satisfies the KK relations.
Is the impedance data stable?
• Application of K-K test for system
stability
4. Finiteness:
The real Zreal and imaginary Zim components of the
impedance must be finite-valued over the entire
frequency range 0 < ω < ∞.
In particular, the impedance must tend to a constant
real value for ω → 0 and ω → ∞.
Instrumentation:
 3-electrode cell in a thermostat
 Potentiostat
 Frequency Response Analyzer
40 years ago…..
From Prof. B. Boukamp’s lecture, Intern. Symp. on EIS, 2008
Measuring impedance by means of
oscilloscopes, Sept. 1960
Impedance analysis in the old days
I
PC
FRA
BTU
RE
CE
WE
Pouch-cell
Frequency response analyzer (FRA)
R(t) cos(t)
osc.
t
cos(t)
R(t) sin(t)
sin(t)
∫
Z"
∫
Z'
Harmonic components
Vo sin(t)
R(t )  I 0 sin(t   )   Ak sin( kt  k )  noise(t )
Cell
This is necessary.
∫
1
Tint
1
Tint

Tint
0

Tint
0
k
Vanishes by
orthogonality
2
2
I
I
R(t ) cos t dt  o Z o sin  ( )   o Z im ( )
2Vo
2Vo
2
2
I
I
R(t ) sin t dt  o Z o cos  ( )  o Z re ( )
2Vo
2Vo
64
.
..
S/N increase
by repeated measurements
More applications for Li-Batteries
Impedance spectra of Lithium electrodes
LiAsF6 0.25 M
-Z’’ / Ohm
LiAsF6 1 M
LiAsF6 1 M
+200 ppm H2O
Z’ / Ohm
Z’ / Ohm
3 hours aging
6 days aging
Z’ / Ohm
D. Aurbach, E. Zinigrad, A. Zaban, J. Phys. Chem., 100, 1996, 3091
Li-Intercalation Electrodes Li[Mn-Ni-Co]O2
-400
Initial state, 300C
After aging 4 weeks at 600C
5 mHz
-300
Z" / Ohm
5 mHz
-400
-200
4.7 V
32 mHz
-200
-100
2.5 Hz
12.6 Hz
50 kHz
2D Graph 5
2D Graph 2
0
0
100
200
0
400 -200 0
300
200
400
Z" / Ohm
-100
-100
32 mHz
20 Hz
50 kHz
2D Graph 1
2D Graph 4
0
-60
Z" / Ohm
800
5 mHz
-200
4.6 V
600
0
0
100
1-st
Semi
circle
-40
2-nd
Semi
circle
20 Hz
-20
50 kHz
200
158 mHz
300
400
0
20
40
60
Z' / Ohm
0
100
200
300
400
300
400
5 mHz
4.4 V
5 mHz
-100
W
20 Hz
R1C1 R2C2
0
-200
0
80
100
0
100
200
Z' / Ohm
AlF3-coated material
Li[Ni-Mn-Co]O2
Uncoated material
5 mHz
-3000
-1500
5 mHz
5 mHz
5 mHz
5 mHz
-2000
20 Hz
3000
50 Hz
5 mHz
-500
50 Hz
2000
1000
30
Cycle
20
10
numb
er
0
/O
hm
40
1000
0
40
Z'
0
0
500
30
Cycle
2D Graph 1
175
Uncoated
AlF3-coated
150
Rsf / Ohm.cm2
293-A15-2imp,Z' vs Col 17 vs 293-A15-2imp,Z''
293-A15-3imp,Z' vs Col 18 vs 293-A15-3imp, Z''
293-A15-4imp,Z' vs Col 19 vs 293-A15-4imp, Z''
293-A15-5imp, Z' vs Col 20 vs 293-A15-5imp,Z''
293-A15-6imp, Z' vs Col 21 vs 293-A15-6imp,Z''
125
2imp,Z' vs Col 17 vs 2imp,Z''
3imp,Z' vs Col 18 vs 3imp, Z''
4imp,Z' vs Col 19 vs 4imp, Z"
5imp,Z' vs Col 20 vs 5imp,Z''
6imp, Z' vs Col 21 vs 6imp, Z"
100
75
50
25
0
10
1500
20 Hz
20 Hz
20
30
40
50
Cycles
F. Amalraj, B. Markovsky et al., JES, 160, A2220, 2013
20
10
numb
er
0
0
/O
20 Hz
5 mHz
Z'
-1000
5 mHz
-1000
hm
Z" / Ohm
Z'' / Ohm
5 mHz
Thin-film Li[Ni-Mn-Co]O2 electrode (no PVdF, CB),
cycled, E=4.3 V. Impedance data fitting ZView
1-st
2-nd
Warburg element,
Li+ Solid-state diffusi
Slope=-1.076
Equivalent circuit
• Why we use it?
– Intuitive
– Practical
– Relatively easy to model
• Disadvantages
– Ambiguities
– Does not tell the mechanism of the reaction
– Explanation is difficult !
A Typical Equivalent Circuit
Rel
CPEsf
CPEct
Rsf
Rct
Zw
Intercalation electrodes can be described by
the following equivalent analog
Surface
films
Solution
resistance
Interfacial
charge
transfer
Solid state
diffusion
W

Intrcalation
capacitance
Building blocks of equivalent circuits
Fuel cell: Equivalent circuit analog
Conclusions-1
• Impedance is not a physical reality
– It is an alternating current technique.
– It is in the frequency domain.
– It is rigorously generalized concept that
contains whole features of an electrochemical
system.
75
Conclusions-2
• In many cases data fitting is not
needed.
– Exact plotting is essential.
– Graphical analysis is very useful.
• Experience is needed for data fitting.
• Anyway, enjoy EIS measurements!
– It is non-destructive technique.
– Good to understand your system
systematically.
The main question:
So, What is Electrochemical Impedance
Spectroscopy?
“Probing an electrochemical system with
small ac-perturbation over a range of
frequencies”.
Literature
1. Barsukov, E. and Macdonald, J. R. 2005. Impedance
Spectroscopy, 2nd ed. Wiley-Interscience, New York.
2. Conway, B. E. 1999. Electrochemical Supercapacitors, Kluwer
Academic/Plenum, New York.
3. A. Bard and L. Faulkner, Electrochemical Methods.
4. Orazem, M. and Tribollet, B. 2008. Electrochemical Impedance
Spectroscopy (The ECS Series of Texts and Monographs)
Wiley-Interscience, New York.
5. Solartron Analytical Frequency Response Analyzer (FRA).
Available at http://www.solartronanalytical.com/Pages/
1260AFRAPage.htm.
6. Lectures by Prof. Bernard Boukamp, University of Twente, Dept. of
Science &Technology, Enschede, The Netherlands.
……..Sunrise in Ein-Gedi, Dead Sea……