### Nyquist plot

```Electrochemistry MAE-295
Dr. Marc Madou, UCI, Winter 2012
Class X Electrochemical Impedance Analysis (EIS)
Table of Content
 Introduction
 Impedance Concept and Representation of Complex Impedance







2
Values
Nyquist plot or Cole-Cole plot
Bode plot
Summary Nyquist and Bode Plots
Review of Circuits Elements
Equivalent Circuit of a Cell
Back to Electrochemistry
Summary
Introduction
 In EIS an electrochemical system is perturbed with an alternating
current or voltage signal of small magnitude and one observes the way in
which the system follows the perturbation at steady state.
 EIS measures the impedance of a circuit to an applied voltage:
Z(t)=E(t)/I(t)
 When E (or I) is applied as a sinusoidal function in a linear system, the I
(or V) response can be represented by a sum of sinusoidal functions with
phase shifts.
 If an equivalent circuit for the system being probed can be constructed,
then the resistance or capacitance values for each circuit element can be
backed out from Z.
3
Introduction
•Electrical circuit theory distinguishes between linear and non-linear systems
(circuits). Impedance analysis of linear circuits is much easier than analysis of non-linear
ones.
•A linear system ... is one that possesses the important property of superposition: If the
input consists of the weighted sum of several signals, then the output is simply the
superposition, that is, the weighted sum, of the responses of the system to each of the
signals.
•Mathematically, let y1(t) be the response of a continuous time system to x1(t) ant let
y2(t) be the output corresponding to the input x2(t).
Then the system is linear if:
1)The response to x1(t) + x2(t) is y1(t) + y2(t)
4
2) The response to ax1(t) is ay1(t) ...
Introduction
•For a potentiostated electrochemical cell, the input is the
potential and the output is the current (for a galvanostated
cell it is the other way around).
•Most electrochemical cells are not linear! Doubling the
voltage will not necessarily double the current.
•However, electrochemical systems can be pseudo-linear. When you look at a small
enough portion of a cell's current versus voltage curve, it seems to be linear.
•In normal EIS practice, a small (1 to 10 mV) AC signal is applied to the cell. The signal is
small enough to confine you to a pseudo-linear segment of the cell's current versus voltage
curve.You do not measure the cell's nonlinear response to the DC potential because in EIS
you only measure the cell current at the excitation frequency.
5
Introduction
 Measure Z(,Vbias) = V() / I()
6
Introduction
•Let us assume we have an electrical element to which we apply an electric field E(t) and get
the response I(t), then we can disturb this system at a certain field 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.
•If we plot the applied sinusoidal signal on the X-axis
of a graph and the sinusoidal response signal I(t) on
the Y-axis, an oval is plotted. This oval is known as a
"Lissajous figure". Analysis of Lissajous figures on
oscilloscope screens was the accepted method of
impedance measurement prior to the availability of
lock-in amplifiers (LIAs) and frequency response
analyzers (FRAs).
7
Introduction
Multiplier:
Vx(t)sin(t) &
Vx(t)cos(t)
Integrator: integrates multiplied
signals
Display result:
a + jb = Vsign/Vref
Impedance:
Zsample = Rm (a + jb)
But be aware of the input impedance of
the FRA!
8
Introduction
Vpwr.amp = A k Vk
A= amplification
Vwork – Vref =
Vpol. + V3 + V4
Current-voltage converter
provides virtual ground for
Work-electrode.
Source of inductive effects
General
schematic
9
Introduction
10
Introduction
FRA: Frequency Response Analysis
I
I0
I0 +  I sin (t + 
E0
11
Potentiostatic or galvanostatic
measurements
E
E0 +  E sin t
• Electrochemical Impedance Spectroscopy (EIS) is also called AC Impedance or just
Impedance Spectroscopy. The usefulness of impedance spectroscopy lies in the ability
to distinguish the dielectric and electric properties of individual contributions of
components under investigation.
•For example, if the behavior of a coating on a metal when in salt water is required, by
the appropriate use of impedance spectroscopy, a value of resistance and capacitance
for the coating can be determined through modeling of the electrochemical data. The
modeling procedure uses electrical circuits built from components such as resistors
and capacitors to represent the electrochemical behavior of the coating and the metal
substrate. Changes in the values for the individual components indicate their behavior
and performance.
•Impedance spectroscopy is a non-destructive technique and so can provide time
dependent information about the properties but also about ongoing processes such as
corrosion or the discharge of batteries and e.g. the electrochemical reactions in fuel
cells, batteries or any other electrochemical process.
12
1. Useful on high resistance materials such as paints and coatings.
2. Time dependent data is available
3. Non- destructive.
4. Quantitative data available.
5. Use service environments.
6. System in thermodynamic equilibrium
7. Measurement is small perturbation (approximately linear)
8. Different processes have different time constants
9. Large frequency range, Hz to GHz (and up)
10. Generally analytical models available
11.Pre-analysis (subtraction procedure) leads to plausible models and starting values
1. Rather expensive equipment,
2. Low frequencies difficult to measure
3. Complex data analysis for quantification.
13
AC Circuit Theory and Representation of
Complex Impedance Values
•Concept of complex impedance: from R to Z
•Ohm's law defines resistance in terms of the ratio between voltage E and current I :
R
E (t )
I (t )
•The 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 of frequency.
• AC current and voltage signals though a resistor are in phase with each other
14
AC Circuit Theory and Representation
of Complex Impedance Values
• In practice circuit elements exhibit much more complex behavior. This forces one to
abandon the simple concept of a resistance only. In its place we use impedance, which is a
more general circuit parameter (Z instead of R).
• Like resistance R, impedance Z is a measure of the ability of a circuit to resist the flow of
electrical current.
• Electrochemical impedance is usually measured by applying an AC potential (current) to an
electrochemical cell and measuring the current (voltage) through the cell.
• Suppose that we apply a sinusoidal potential excitation. The response to this potential is an
AC current signal, containing the excitation frequency and it's harmonics. This current signal
can be analyzed as a sum of sinusoidal functions (for a linear system-see earlier).
• The current response to a sinusoidal potential will be a sinusoid at the same frequency but
shifted in phase.
15
AC Circuit Theory and Representation of
Complex Impedance Values
•The excitation signal, expressed as a function of time, has the form of:
E(t )  E0 cos(t )
•E(t) is the potential at time t, Eo is the amplitude of the signal, and  is the radial frequency.
The relationship between radial frequency  (expressed in radians/second) and frequency f
(expressed in Hertz (1/sec).
16
 =2 f
AC Circuit Theory and Representation
of Complex Impedance Values
•In a linear system, the response signal, the current I(t), is shifted in phase () and has a
different amplitude, I0:
I (t )  I0 cos(t   )
•An expression analogous to Ohm's Law allows us to calculate the impedance (=the AC
resistance) of the system :
E0 cos(t )
E (t )
cos(t )
Z (t ) 

 Z0
I (t ) I 0 cos(t   )
cos(t   )
•The impedance is therefore expressed in terms of a magnitude, Z0, and a phase shift,.
•This impedance may also be written as a complex function (see next slide) :
17
AC Circuit Theory and Representation of Complex
Impedance Values
Z (t ) 
E0 cos(t )
E (t )
cos(t )

 Z0
I (t ) I 0 cos(t   )
cos(t   )
•Using Eulers relationship:
exp(i )  cos   i sin 
it is possible to express the impedance as a complex function. The
potential is described as:
E(t) = E0 exp(it)
and the current response as:
I (t )  I0 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
18
AC Circuit Theory and Representation
of Complex Impedance Values
E(t) = E0cos(t), =2f
I(t) = I0 cos(t-)
Or, if one writes in complex notation:
E(t) = E0 exp(it)
I(t) = I0 exp(it - i)
Z(t) = Z0 exp(i) = Z0 (cos  + isin )
19
E/I response for a resistor (=0)
E/I response for a capacitor (=-90)
E/I response for an inductor (=90)
Data Presentation:
Nyquist Plot with Impedance Vector
E
 Z 0 exp(i )  Z 0 (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” (also Cole-Cole
plot). Notice that in this plot the y-axis is negative and that each point on the Nyquist plot is the
impedance Z at one frequency.
•In the Nyquist plot the impedance is represented as a vector
of length |Z|. The angle between this vector and the x-axis is
the phase angle .
Z
1
Y ()   jC
R
•Nyquist plots have one major shortcoming. When you
look at any data point on the plot, you cannot tell what
frequency was used to record that point.
•Low frequency data are on the right side of the plot and
higher frequencies are on the left. Y =1/Z.
•A semicircle is characteristic of a single "time constant". Electrochemical Impedance plots often contain
several time constants. Often only a portion of one or more of their semicircles is seen.
20
AC Circuit Theory and Representation
of Complex Impedance Values
 The magnitude of Z and phase angle are given by the following,
respectively (with R = real and Xc = imaginary, also a and b later
in the class):
Z  ( R 2  X C 2 )1/ 2
XC
1
tan  

R  RC
 The impedance Z is a kind of a generalized resistance R. The phase
angle expresses the balance between capacitive and resistance
components in the series circuit. For a pure resistance, φ=0; for a
pure capacitance, φ=π/2; and for mixtures, intermediate phase
angles are observed.
21
Data Presentation:
Nyquist Plot with Impedance Vector
 For a pure resistance R, E=IR, and the phase is zero.
 For a pure capacitance C:
E

i
sin(t  )
XC
2
 Where Xc is the capacitive reactance, 1/ωC
 A comparison of R and Xc shows that Xc must carry the
dimensions of a resistance, but the magnitude of Xc falls
with increasing frequency.
22
Data Presentation:
Nyquist Plot with Impedance Vector
•Take a look at the properties of a capacitor: C  A 0 
d
•Charge stored (Coulombs):
•Change of voltage results
in current, I:
•Alternating voltage (ac):
•Impedance:
23
Q  C V
dQ
dV
I
C
dt
dt
dV0  e jt
I (t )  C
 j C V0  e jt
dt
V ()
1
Z C   

I () jC
YC    Z () 1  jC
Data Presentation:
Nyquist Plot with Impedance Vector
Impedance  ‘resistance’
24
Representation of impedance value, Z = a +jb, in the complex plane (see also
http://math.tutorvista.com/number-system/absolute-value-of-a-complexnumber.html
Data Presentation:
Nyquist Plot with Impedance Vector
What is the impedance of an -R-C- circuit?
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
25
Semicircle
‘time constant’:
 = RC
Data Presentation:
Nyquist Plot with Impedance Vector
•The parallel combination of a resistance and a capacitance, start in the admittance
representation:
1
Y () 
R
 jC
R
•Transform to impedance representation:
1
1
1/ R  jC
Z () 



Y () 1/ R  jC 1/ R  jC
R  j R 2 C
1  j
R
2 2 2
1  R C
1  2 2
•A semicircle in the impedance plane!
26
C
Data Presentation:
Nyquist Plot with Impedance Vector
8.0E+04
fmax = 1/(6.3x310-9x105)=530 Hz
6.0E+04
-Zimag, [ohm]
518 Hz
R = 100 k
C = 3 nF
4.0E+04
2.0E+04
1 MHz
0.0E+00
0.0E+00
27
1 Hz
2.0E+04
4.0E+04
6.0E+04
Zreal, [ohm]
8.0E+04
1.0E+05
1.2E+05
Data Presentation:
Nyquist Plot with Impedance Vector
•What happens for  <<  and for  >>  ?
 <<  :
 >>  :
1  j
2
Z ()  R

R

j

R


R

j

R
C
2 2
1  
Z ()  R
1  j
R
R
1
1


j


j
1  2 2 2 2
 2 RC 2
C
This is best observed in a so-called Bode plot
log(Zre), log(Zim) vs. log(f ), or
log|Z| and phase vs. log(f )
28
1.E+05
Bode plot (Zre, Zim)
Zreal
Zimag
Zreal, -Zimag, [ohm]
1.E+04
1.E+03
1.E+02
-1
1.E+01
-2
1.E+00
1.E-01
29
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, abs(Z), phase
1.E+05
90
1.E+04
75
60
45
1.E+03
30
15
1.E+02
1.E+00
30
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 (°)
The Bode Plot
1 1
1
 
Z R iC
•Another popular presentation method is the "Bode plot". The impedance is plotted with
log frequency on the x-axis and both the absolute value of the impedance (|Z| =Z0 ) and
phase-shift on the y-axis.
•The Bode plot for the RC circuit is shown below. Unlike the Nyquist plot, the Bode plot
explicitly shows frequency information.
31
Data Presentation:
Nyquist Plot with Impedance Vector
 An electrical layer of a device can often be described by a
resistor and capacitor in parallel: Voigt network.
32
Data Presentation:
Nyquist Plot with Impedance Vector
•When we plot the real and imaginary components of impedance in the complex plane
(Argand diagram), we obtain a semicircle or partial semicircle for each parallel RC Voigt
network: Nyquist plot or also Cole-Cole plot.
•The diameter corresponds to the resistance R.
•The frequency at the 90° position corresponds to 1/ = 1/RC
33
Data Presentation:
Nyquist Plot with Impedance Vector
•The Randles cell is one of the simplest and most common cell models. It includes a solution resistance, a
double layer capacitor and a charge transfer or polarization resistance. In addition to being a useful model
in its own right, the Randles cell model is often the starting point for other more complex models.
•The equivalent circuit for the Randles cell is shown in the Figure. The double layer capacity is in parallel
with the impedance due to the charge transfer reaction
•The Nyquist plot for a Randles cell is always a semicircle. The solution resistance can found by reading the
real axis value at the high frequency intercept. This is the intercept near the origin of the plot.This plot was
generated assuming that Rs = 20  and Rp= 250  .
The real axis value at the other (low frequency) intercept is the sum of the polarization resistance and the
solution resistance. The diameter of the semicircle is therefore equal to the polarization resistance (in this
case 250 ).
34
Data Presentation:
Nyquist Plot with Impedance Vector
35
Data Presentation:
Nyquist Plot with Impedance Vector
36
Data Presentation:
Nyquist Plot with Impedance Vector
37
Summary Nyquist and Bode Plots
38
Review of Circuit Elements
Very few electrochemical cells can be modeled using a single equivalent
circuit element. Instead, EIS models usually consist of a number of elements
in a network. Both serial and parallel combinations of elements occur.
Impedances in Series:
Impedances in Parallel
39
Zeq  Z1  Z2  Z3
1 1 1 1
  
Zeq Z1 Z 2 Z3
Review of Circuit Elements
•EIS data is commonly analyzed by fitting it to an equivalent electrical circuit model. Most of the circuit elements in the model are
common electrical elements such as resistors, capacitors, and inductors. To be useful, the elements in the model should have a basis
in the physical electrochemistry of the system. As an example, most models contain a resistor that models the cell's solution
resistance.
•Some knowledge of the impedance of the standard circuit components is therefore very important. The Table below lists the
common circuit elements, the equation for their current versus voltage relationship, and their impedance:
Component
Current Vs.Voltage
Impedance
resistor
E= IR
Z=R
inductor
E = L di/dt
Z = iL
capacitor
I = C dE/dt
Z = 1/iC
•Notice that the impedance of a resistor is independent of frequency and has only a real component. Because there is no imaginary
impedance, the current through a resistor is always in phase with the voltage.
•The impedance of an inductor increases as frequency increases. Inductors have only an imaginary impedance component. As a
result, an inductor's current is phase shifted 90 degrees with respect to the voltage.
•The impedance versus frequency behavior of a capacitor is opposite to that of an inductor. A capacitor's impedance decreases as the
frequency is raised. Capacitors also have only an imaginary impedance component. The current through a capacitor is phase shifted 90 degrees with respect to the voltage.
40
Review of Circuit Elements
•Suppose we have a 1 and a 4  resistor is series. The impedance of a resistor is the
same as its resistance . We thus calculate the total impedance Zeq:
R1
R2
Zeq  Z1  Z2  R1  R 2  1 4  5
•Resistance and impedance both go up when resistors are combined in series.
•Now suppose that we connect two 2 µF capacitors in series. The total capacitance of the
combined capacitors is 1 µF
C1
C2
1
1
1
 Z1  Z 2 

Zeq
iC1 iC2

1
1
1


 1 µF
6
6
e 6
i (2e ) i (2e ) i (1 )
•Impedance goes up, but capacitance goes down when capacitors are connected in series.
41 This is a consequence of the inverse relationship between capacitance and impedance.
Review of Circuits Elements
Resistance:
Capacitance:
42
Z R
Z 
1
 C
I
E
E
I
 0
o
  90
o
Equivalent Circuit of a Cell
 In a general sense, we ought to be able to represent the
performance of a cell by an equivalent circuit of resistors
and capacitors under a given excitation.
 The elements of equivalent circuit of a cell: double-layer
capacitance Cd, faradaic impedance Zf, solution resistance
Rs, charge transfer resistance Rct, Warburg impedance Zw.
43
Equivalent Circuit of a Cell: Rs and Cd
•Electrolyte resistance R is often a significant factor in the impedance of an electrochemical
cell. A modern 3 electrode potentiostat compensates for the solution resistance between the
counter and reference electrodes. However, any solution resistance between the reference
electrode and the working electrode must be considered when you model your cell.
•The resistance of an ionic solution depends on the ionic concentration, type of ions,
temperature and the geometry of the area in which current is carried. In a bounded area
with area A and length l carrying a uniform current the resistance is defined as:
l
A
where r is the solution resistivity.
R
The conductivity of the solution,  , is more commonly used in solution resistance calculations.
Its relationship with solution resistance is:
1 l
l
R=

 A
RA
44
Equivalent Circuit of a Cell: Rs and Cd
•Standard chemical handbooks list  values for specific solutions. For other solutions and
solid materials, you can calculate  from specific ion conductances. The units for  are
Siemens per meter (S/m). The Siemens is the reciprocal of the ohm, so 1 S = 1/ohm
•Unfortunately, most electrochemical cells do not have uniform current distribution
through a definite electrolyte area. The major problem in calculating solution resistance
therefore concerns determination of the current flow path and the geometry of the
electrolyte that carries the current. A comprehensive discussion of the approaches used to
calculate practical resistances from ionic conductances is beyond the scope of this class.
•Fortunately, you don't usually calculate solution resistance from ionic conductances.
Instead, it is found when you fit a model to experimental EIS data.
45
Electrochemical Impedance Spectroscopy
Equivalent Circuit of a Cell: Rs and Cd
A Resistance and capacitance
in series
f is low:
f is high:
Z
1
 C
ZR
  90
0
In electrochemical cell:
R=Rs: solution resistance
C=Cd: double layer capacitance


Bode
Nyquist
9
10
|Z|, 
Zj, 
-600
-400
-200
0
500.00
46
Zr, 
10
10
10
8
-90
6
-80
Phase
-800x10
4
-60
2
10
-70
1
10
2
10
3
10
4
10
Frequency, Hz
5
10
6
10
1
10
2
10
3
10
4
10
Frequency, Hz
5
10
6
Equivalent Circuit of a Cell: Rs and Cd
•A electrical double layer exists at the interface between an electrode and its surrounding
electrolyte.
•This double layer is formed as ions from the solution "stick on" the electrode surface. Charges in
the electrode are separated from the charges of these ions. The separation is very small, on the
order of angstroms.
•Charges separated by an insulator form a capacitor. On a bare metal immersed in an electrolyte,
you can estimate that there will be approximately 30 µF of capacitance for every cm2 of electrode
area.
•The value of the double layer capacitance depends on many variables including electrode
potential, temperature, ionic concentrations, types of ions, oxide layers, electrode roughness,
Principle of the
Electric Double-Layer: Here C electrodes
47
Equivalent Circuit of a Cell: CPE
Constant Phase Element (for double layer capacity in real electrochemical cells)
•Capacitors in EIS experiments often do not behave ideally. Instead, they act like a constant
phase element (CPE) as defined below.
Z  A(i )
•When this equation describes a capacitor, the constant A = 1/C (the inverse of the
capacitance) and the exponent  = 1. For a constant phase element, the exponent a is less
than one.
•The "double layer capacitor" on real cells often behaves like a CPE instead of like a
capacitor. Several theories have been proposed to account for the non-ideal behavior of the
double layer but none has been universally accepted (fractal explanation!). In most cases,
you can safely treat  as an empirical constant and not worry about its physical basis.
48
Equivalent Circuit of a Cell: CPE
Constant Phase Element:
YCPE = Y0 n {cos(n /2) + j sin(n /2)}
n=1
 Capacitance: C = Y0
• n = ½  Warburg:  = Y0
• n = 0  Resistance:
R = 1/Y0
• n = -1  Inductance:
L = 1/Y0
All other values, ‘fractal?’
‘Non-ideal capacitance’, n < 1 (between 0.8 and 1?)
49
Equivalent Circuit of a Cell: CPE
General observations:
• Semicircle (RC )
• vertical spur (C )
 depressed
 inclined
• Warburg
 less than 45°
Deviation from ‘ideal’ dispersion:
Constant Phase Element (CPE),
(symbol: Q )
YCPE
50
n 
 n
 Y0 ( j)  Y0 cos  j sin 
2
2

n
n
n = 1, ½,
0, -1, ?
Equivalent Circuit of a Cell: CPE
How to explain this non-ideal behaviour?
1980’s: ‘Fractal behaviour’ (Le Mehaut)
= fractal dimensionality
i.e.: ‘What is the length of the coast line of England?’
 Depends on the size of the measuring stick!
 Self similarity 
51
Equivalent Circuit of a Cell: CPE
Fractal line
Self similarity!
‘Sierpinski carpet’
52
Equivalent Circuit of a Cell
Mixed kinetic and diffusion control
Cdl or CPE
Z  1 / (Q)
R
RP
with 0n1
ZW
Bode
Nyquist
-200
-40
100
-100
7
6
5
Phase
|Z|, 
Zj, 
-150
-30
-20
4
53
-50
3
0
2
0
50
100
Zr, 
150
200
-10
0
0
10
2
10
Frequency, Hz
4
10
0
10
2
10
Frequency, Hz
4
10
n
Equivalent Circuit of a Cell: Rp(or ct) and Cd
•A charge transfer resistance is formed by common kinetically controlled electrochemical reaction
•Consider a metal substrate in contact with an electrolyte. The metal molecules can electrolytically dissolve into
the electrolyte, according to:
or more generally:
In the forward reaction in the first equation, electrons enter the metal and metal ions diffuse into the electrolyte.
Charge is being transferred.
This charge transfer reaction has a certain speed. The speed depends on the kind of reaction, the temperature, the
concentration of the reaction products and the potential.
The general relation between the potential and the current holds:
io = exchange current density
Co = concentration of oxidant at the electrode surface
Co* = concentration of oxidant in the bulk
CR = concentration of reductant at the electrode surface
54
T = temperature
R = gas constant
a = reaction order
n = number of electrons involved
h = overpotential ( E - E0 )
Equivalent Circuit of a Cell: Rp(or ct) and Cd
The overpotential, h, measures the degree of polarization. It is the electrode potential minus the
equilibrium potential for the reaction.
When the concentration in the bulk is the same as at the electrode surface, Co=Co* and CR=CR*.
This simplifies the last equation into:
This equation is called the Butler-Volmer equation. It is applicable when the polarization
depends only on the charge transfer kinetics.
Stirring will minimize diffusion effects and keep the assumptions of Co=Co* and CR=CR* valid.
When the overpotential, h, is very small and the electrochemical system is at equilibrium, the
expression for the charge transfer resistance changes into:
From this equation the exchange current i0 density can be calculated when Rct is known (see
Rct in next figure).
55
Equivalent Circuit of a Cell: Rp(or ct) and
Cd
A resistance and capacitance in parallel (Randles circuit)
Z=Rs at high frequency
Z=Rct+Rs at low frequency
Nyquist
Bode
-300
2
-50
Zj, 
-200
-150
-40
|Z|, 
Phase
-250
-30
-20
0
10
-50
0
10
2
10
4
Frequency, Hz
0
56
0
100
200
Zr, 
300
8
6
4
-10
-100
100
10
6
2
0
10
2
10
4
10
Frequency, Hz
6
10
Equivalent Circuit of a Cell: Rp(or ct) and Cd
When there are two simple, kinetically controlled reactions occurring, the potential of the cell is again related to the
current by the following (known as the Butler-Volmer equation).
I is the measured cell current in amps,
Icorr is the corrosion current in amps,
Eoc is the open circuit potential in volts,
a is the anodic Beta coefficient in volts/decade,
c is the cathodic Beta coefficient in volts/decade
If we apply a small signal approximation (E-Eoc is small) to the buler Volmer equation, we get the following:
which introduces a new parameter, Rp, the polarization resistance.
If the Tafel constants i are known, you can calculate the Icorr from Rp. The Icorr in turn can be used to calculate a corrosion rate. The Rp
parameter comes again from the Nyquist plot.
57
Equivalent Circuit of a Cell: Rp(or ct) and Cd
58
Equivalent Circuit of a Cell: Warburg Impedance
Diffusion can create an impedance known as the Warburg impedance. This impedance
depends on the frequency of the potential perturbation. At high frequencies the Warburg
impedance is small since diffusing reactants don't have to move very far. At low frequencies
the reactants have to diffuse farther, thereby increasing the Warburg impedance.
The equation for the "infinite" Warburg impedance
On a Nyquist plot the infinite Warburg impedance appears as a diagonal line with a slope of 0.5.
On a Bode plot, the Warburg impedance exhibits a phase shift of 45°.
In the above equation,  is the Warburg coefficient defined as:
59
DO = diffusion coefficient of the oxidant
DR = diffusion coefficient of the reductant
A = surface area of the electrode
n = number of electrons transferred
C* = bulk concentration of the diffusing species (moles/cm3)
Equivalent Circuit of a Cell: Warburg
Impedance
Define impedance in Laplace space!
E ( p)
RT
Z ( p) 

I ( p) (nF ) 2 C D  p
Take the Laplace variable, p, complex:
p = s + j . Steady state: s  0,
which yields the impedance:
RT
1/ 2
1/ 2
Z () 

Z
(


j

)
0
2
(nF ) C jD
In solution:
60

RT
1
1
Z 0  (  ) 2 2
 *
 *
n F A 2  CO DO CR DR




Equivalent Circuit of a Cell: Warburg
Impedance

45°
62
Equivalent Circuit of a Cell: Warburg
Impedance—Coating Capacitor
A capacitor is formed when two conducting plates are separated by a non-conducting media, called the
dielectric. The value of the capacitance depends on the size of the plates, the distance between the plates
and the properties of the dielectric. The relationship is:
C
 0 r A
d
With,
o = electrical permittivity
r = relative electrical permittivity
A = surface of one plate
d = distances between two plates
Whereas the electrical permittivity is a physical constant, the relative electrical permittivity depends on
the material. Some useful r values are given in the table:
Material r
vacuum
1
water
80.1 ( 20° C )
organic
coating
4-8
Notice the large difference between the electrical permittivity of water and that of an organic coating. The
capacitance of a coated substrate changes as it absorbs water. EIS can be used to measure that change
63
Equivalent Circuit of a Cell: Warburg
Impedance—Coating Capacitor
A metal covered with an undamaged coating generally has a very high impedance. The equivalent
circuit for such a situation is in the Figure:
R
C
The model includes a resistor (due primarily to the electrolyte) and the coating capacitance in series.
A Nyquist plot for this model is shown in the Figure. In making this plot, the following values were assigned:
R = 500  (a bit high but realistic for a poorly conductive solution)
C = 200 pF (realistic for a 1 cm2 sample, a 25 µm coating, and r = 6 )
fi = 0.1 Hz (lowest scan frequency -- a bit higher than typical)
ff = 100 kHz (highest scan frequency)
The value of the capacitance cannot be determined from the Nyquist plot. It can be determined by a
curve fit or from an examination of the data points. Notice that the intercept of the curve with the
real axis gives an estimate of the solution resistance.
10
64The highest impedance on this graph is close to 10  . This is close to the limit of measurement of
most EIS systems
Equivalent Circuit of a Cell: Warburg
Impedance—Coating Capacitor
The same data from the previous page shown in a Bode plot below. Notice that the capacitance can be
estimated from the graph but the solution resistance value does not appear on the chart. Even at 100
kHz, the impedance of the coating is higher than the solution resistance
65
Equivalent Circuit of a Cell: Warburg
Impedance—Coating Capacitor
Classification of types of capacitances
source
approximate value
geometric
2-20 pF (cm-1)
grain boundaries
1-10 nF (cm-1)
double layer / space charge
0.1-10 F/cm2
0.2 mF/cm2
(closed) pores
1-100 F/cm3
“pseudo capacitances”
“stoichiometry” changes
large !!!!
66
Equivalent Circuit of a Cell: Warburg
Impedance—Porous Coating
67
Summary
68
Summary
Below we show some common equivalent circuits models. Equations for both the admittance and
impedance are given for each element.
Equivalent
element
1/R
R
iC
C
1/iL
L
Y0(i)1/2
W (infinite
Warburg)
O (finite Warburg) Y0 i coth( B i )
Y0(i)
Q (CPE)
69
Impedance
R
1/1/iC
iL
1/Y0(i)1/2
tanh( B i ) / Y0 i 
1/Y0(i)
Summary
By using the various Cole-Cole plots we can calculate values of the elements of the equivalent circuit for
any applied bias voltage
By doing this over a range of bias voltages, we can obtain:
the field distribution in the layers of the device (potential divider) and the relative widths of the layers,
since C ~ 1/d
70
Note : Data validation
Kramers-Kronig relations
Real and imaginary parts are linked through
the K-K transforms:
Kramers-Kronig conditions:
• causality
• linearity
• stability
• (finiteness)
71
Response only due to
input signal
Response scales linearly
State
of system
may not
with
input signal
change during
measurement
Note: Putting ‘K-K’ in practice
Relations,
Real  imaginary:
Imaginary  real:

2 Z re ( x)  Z re ()
Z im () 
dx
2
2

 0
x 
not a singularity!

2 xZ im ( x)  Z im ()
Z re ()  R  
dx
2
2
0
x 
Problem:
Finite frequency range: extrapolation of dispersion  assumption
of a model.
72
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