1. Faradaic impedance Introduction
V.S.Muralidharan
CSIR Emeritus scientist

2. Zero current crossing technique

3. For a resistance

4. For a capacitor

5. Electrochemical Impedance Spectroscopy
FRA: Frequency Response Analysis
Potentiostatic or galvanostatic measurements

6. AC & The Double Layer

7. DC & The Double Layer
• Surface atoms ionize and the electron flow towards the dc power supply. " Faradaic Current "
• The distance between the mobile layer and the electrode depends on dc voltage. DC voltage makes a net current flow through the double layer, from plate to plate, The double layer capacitor 'leaks'.

8. Behavior of The Double Layer
With AC, The double layer behaves like a capacitor (Cdl) With DC, The double layer behaves like a resistor, The Faradaic resistor, (Rf)

9. AC & DC Behavior of The Double Layer
With low frequency, Cdl is high. ⇒ current through Re (electrolyte resistor) and Rf, dc behavior rules.
With high frequency, Cdl is low. ⇒ current through Cdl, ac behavior rules

10. eR eR Resistive Circuit : angular frequency = 2
No Phase difference between. eR and iR ER

11. ec ec ec Ec Capacitive Circuit 90° Phase difference between. ec and ic
Define
: capacitive reactance ec Ec
90° Phase difference between. ec and ic

12. i = (E/R) sin  t (Ohm’s law)
Resistive Circuit E • I • e
I = E/R
E = I R •
e = E sin  t
i = (E/R) sin  t (Ohm’s law)
phasor notation I •
Capacitive Circuit Re e • • Im
E = -j XcI
e = E sin  t
i =  CE cos  t = (E/Xc) sin ( t +/2 ) • •
E = -j XcI

13. AC Voltage & Current Relation of the R-C circuit R C
iR = I
iC = I e
E = ER + EC
E = I ( R – jXC)
E = I Z •
Z = R – jXC
Z() = ZRe – jZIm • • • I
ER = I R R   • • • •
-j Xc
EC = -j XcI
E = I Z Z

14. Nyquist Plot & Bode Plot
The variation of the impedance with frequency can be displayed in different way.
1) Nyquist plot : displays ZIm vs. ZRe for different values of 
2) Bode plot : log |Z| and  are both plotted against log 

15. Series Connection of the R-C Circuit R C
iR = I
iC = I 
ZRe = R , ZIm = -1/(wC)
( Xc : capacitive reactance )
-ZIm
High frequency : Z = R
Low frequency : Z =  Xc
Z(w) 
tan = ZIm/ZRe = Xc/R = 1/RC R
ZRe

16. Actually this is in Fourth quadrant

17. Series Connection of the R-C Circuit
low freq.
 |Z| = 
100  F
High freq.
 |Z| = 0
ZIm w
low freq.
  = -90 o
High freq.
  = 0 o 
ZRe

18. Parallel Connection of the R-C Circuit
 w ,
ZIm
High frequency : Z = 0
Low frequency : Z = R
ZRe R

19. Parallel Connection of the R-C Circuit
100 
|Z| = R
10-4 F
|Z| = 0
 w
ZIm
 = -90 o 
 = 0 o
ZRe R

20. , Randle Circuit  w High frequency : Z = Rs
Rs : solution resistance
Rct : charge transfer resistance
Cdl : double layer capacitance
 w
ZIm ,
High frequency : Z = Rs
Low frequency : Z = Rs+Rct
Rs Rs+Rct
ZRe

21. 10-4 F 10  100  wmax = 100 = 2,  = 100/(2)  16 Hz
Randle Circuit
10-4 F
10 
Rs+Rct
100  Rs
wmax = 100 = 2,  = 100/(2)  16 Hz
wmax=1/(RctCdl)
max
-Rct/2 
high freq.
max 
Rs Rs+Rct/ Rs+Rct

22. Full Randle Circuit  w High freq. : W = 0 Low freq. : W = 
Warburg impedance, W
diffusion control reaction
Usually observed in low frequency region
High freq. : W = 0
Low freq. : W = 
 : Warburg impedance coefficient
R : ideal gas constant
T : absolute temperature
n : the number of electron transferred
F : Faraday’s constant
Cox : conc. of oxidation species
Cred : conc. of reduction species
Dox : diffusivity of oxidation species
Dred : diffusivity of reduction species
 w
slope=1
Rs Rs+Rct

23. Full Randle Circuit
 w
Nyquist plot
Bode plot

24. Nyquist & Bode Plot

25. Conditions for a good impedance
1.Linear system should be identified .linear Systems wouldnot exhibit hysteresis ( dE < 8/n ) 2.Casuality:Physically realizable systems– system does not generate Noise 3.Stablity :Input to output response reproducible 4. Finiteness: The impedance must tend to a constant real Value for w 0 and w a.

26. Measurement methods

27. setup schematic used in BHU 1972

32. Useful Frequency Range : > 10 Hz
WHEATSTONE
Z1 Z4 = Z2 Z4
Useful Frequency Range : > 10 Hz
VSMURALIDHARAN

33. Useful for limited frequency range of > 1 Hz
Simultaneous plotting of the current and voltage
Useful for limited frequency range of > 1 Hz
VSMURALIDHARAN

34. LISSAJOUS FIGURE
VSMURALIDHARAN

35. Phase Sensitive Method Not useful for frequencies < 1 Hz
Generator 
/2
Phase Detector
System Under Test
Output
Not useful for frequencies < 1 Hz
VSMURALIDHARAN

36. Digital Frequency Response Analyser
s(t) cost
Generator Im X 
/2 X  Re
x(t)
s(t) sint
s(t)
System under test
VSMURALIDHARAN

37. Schematic instrumentation for conducting EIS

38. IDENTIFICATION
ELECTROCHEMISTRY

39. ELECTROCHEMICAL REACTION Activation controlled
Cdl Rs
Rct
NYQUIST PLOT Z” Rs
Rs+Rct Z’
VSMURALIDHARAN

40. Cdl Rs
Rct
VSMURALIDHARAN

41. Eliminating  from Z’ & Z’’
Plot of Z’ vs Z’’ is a semi circle of radius
VSMURALIDHARAN

42. Rate of reaction io = Capacitance Cdl =
Bode Plot
Rs+Rct
log |Z| Rs
log f
Rate of reaction io =
Capacitance Cdl =

43. Diffusion Controlled Reaction
Cdl Rs
Rct W Z” Rs
Rs+Rct Z’
VSMURALIDHARAN

44. Randle’s Plot
Re|Z|
Rs+Rct
-1/2
VSMURALIDHARAN

45. Sluyters analysis

46. 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?)

47. Non-ideal behaviour vertical spur (C )  inclined
General observations:
Semicircle (RC )  depressed
vertical spur (C )  inclined
Warburg  less than 45°
Deviation from ‘ideal’ dispersion:
Constant Phase Element (CPE),
(symbol: Q )
n = 1, ½,
0, -1, ?

48. The Fractal Concept 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 INDIA?’
 Depends on the size of the measuring stick!

49. Fractals
Fractal line
Self similarity!
‘Sierpinski carpet’

50. Impedance of the network:
‘Fractal electrode’
‘Cantor bar’
arrangement
Impedance of the network:

51. Arriving at the ‘CPE’ Frequency scaling relation:
In the low frequency limit this reduces to:
Which is satisfied by the formula:
with n = 1 – ln2/lna
Fractal dimension of Cantor bar, d = ln2/lna
Hence: n = 1 –d

52. CDC W CDC = ‘instruction string’ for response calculation
Uses brackets:
[ … ] series combination, e.g.: [RC]
( … ) parallel combination, e.g. (RC)
Randles circuit: R(C[RW]) W

53. ( C [ ( Q [ R ( R Q ) ] ) ( C [ R Q ] ) ] )
Determining the CDC
( C [ ( Q [ R ( R Q ) ] ) ( C [ R Q ] ) ] )

54. CNLS ( Complex nonlinear least square data) analysis
Model function, Z(,ak), or equivalent circuit.
Adjust circuit parameters, ak, to match data,
Minimise error function:
with: (weight factor)
Non-linear, complex model function!
for k = 1 ..M
Effect of minimisation

55. Function Y (a1..aM) is not linear in its parameters, e.g.:
Non-linear systems
Function Y (a1..aM) is not linear in its parameters, e.g.:
(‘Gerischer’)

56. Linearisation: Taylor development around ‘guess values’, ajo:
Derivative of error sum with respect to aj :

57. NLLS-fit A set of M simultaneous equations, in matrix form:
  a =  , solution: a = -1  =    :
Derivatives are taken in point ao1..M.
Iteration process yields new, improved values: a’j = aoj + aj.

58. Marquardt-Levenberg
Analytical search: fast and accurate near true minimum
slow far from minimum
(and often erroneous)
Gradient search or steepest descent (diagonal terms only):
fast far from minimum
slow near minimum
Hence, combination!
Multiply diagonal terms with (1+).
 << 1, analytical search
 >> 1, gradient search
Successful iteration:
Snew < Sold
 decrease  (= /10).
Otherwise increase
 (= 10)
Bottom line: good starting parameter estimates are essential!

59. Error estimates
For proper statistical analysis the weight factors, wi, should be established from experiment.
Other (dangerous) method:
Step 1: set weight factors, wi = g  i-2
Step 2: assume variances can be replaced by parent
distribution, hence 2  (with  = N –M – 1)
Step 3: Hence proportionality factor, g = S/.

60. Error analysis NLLS-fit
Based on this assumption we can derive
the variances of the parameters:
Error matrix, , also contains the covariance of the parameters:
g j,k  0, no correlation between aj and ak.
g j,k  1, strong correlation between aj and ak.
Only acceptable for many data points AND
random distribution of the ‘residuals’

61. Weight factors and error estimates
Errors in parameters:
estimates from CNLS-fit procedure
assumption: error distribution equal to ‘parent distributio only valid for random errors,
no systematic errors allowed!
Residuals graph:
Large error estimates:
strongly correlated
parameters (+ noise).
Option: modification of weight factors.

62. Two different CNLS-fits
Example of correct error estimates:
R(RC)(RC)
CDC: R(RQ)(RQ)
2 2.410-5
R %
R %
Q3 1.03 %
-n %
R %
Q5 1.03 %
-n %
2 3.810-3
R %
R %
C  %
R %
C  %
And of incorrect error estimates:
CDC: R(RC)(RC)
Values seem O.K. but look at the residuals!

63. Good fit (not bad for a straight simulation!)
Residuals plot!
Systematic deviation,
‘Trace’, bad fit
Good fit (not bad for a straight simulation!)

64. Classification of capacitance source approximate value
‘Fingerprinting’
Classification of capacitance
source approximate value
geometric pF(cm-1)
grain boundaries nF (cm-1)
double layer / space charge F/cm2
surface charge /”adsorbed species” 0.2 mF/cm2
(closed) pores F/cm3
“pseudo capacitances”
“stoichiometry” changes large !!!!
vsmuralidharan

65. Conclusions on ‘fitting’
Many parameter, complex systems modelling:
Use Marquardt-Levenberg when quality starting values are available
Simplex (or Genetic Algorithm) for optimisation of ‘rough guess’ starting values, as input for M-L NLSF
Check residuals when calculating Error Estimates
Look for systematic error contributions, remove if feasible.
Provide error estimates in publications!