Solid State lonics 18 & 19 (1986) 136-140
North-Holland, Amsterdam
136
A PACKAGE FOR IMPEDANCE/ADMITTANCEDATA ANALYSIS
Bernard A. BOUKAMP
Twente University of Technology, Department of Chemical Technology, Laboratory f o r Inorganic
Chemistry and Materials Science, P.O.Box 217, 7500 AE Enschede, The Netherlands.
An o u t l i n e is given of a Basic computer program which f a c i l i t a t e s the analysis of frequency
dispersion data. With t h i s program an equivalent c i r c u i t , and s t a r t i n g values f o r the corresponding c i r c u i t parameters, can be extracted from the dispersion data. A c i r c u i t description
together with crude parameter values form an essential requirement f o r a subsequent NLLSF
procedure. A b r i e f description is given of a frequency dispersion simulation program, also
w r i t t e n in Basic, which can be used to compare measured data with a calculated response.
Both programs employ the C i r c u i t Description Code (CDC), thus allowing the use of a v a r i e t y of
equivalent c i r c u i t s . The use of both programs is demonstrated with the analysis of a dispersion
measurement performed on a sample of Sn-doped AgCrS2, which is a pure ionic conductor.
adjusted simultaneously, in order to f i t
1. INTRODUCTION
Impedance spectroscopy is frequently used in
the
equivalent c i r c u i t response to the measured dis-
the study of electrochemical systems I-6. One of
persion. This can be accomplished through the
the advantages of this technique is that from
use of a special Non-Linear Least Squares F i t
the immittance plots of the dispersion data good
(NLLSF) procedure I0-12
visual information is obtained about the charac-
NLLSF procedure e f f e c t i v e l y one must know the
t e r i s t i c s of the electrochemical system. The
shape of the equivalent c i r c u i t and have a set
dispersion is generally analyzed with the use of
of adequate s t a r t i n g values f o r the adjustable
an equivalent c i r c u i t as model, in which the
c i r c u i t parameters.
various c i r c u i t elements are related to the
In order to use such a
This information can be obtained by a step-
respective processes in the system, e.g. i o n i c
wise analysis of the dispersion data in an
c o n d u c t i v i t y , double layer capacitance, Warburg
immittance diagram.
diffusion, etc..
are located f i r s t
In many cases the c i r c u i t
The most prominent parts
and the corresponding elements
parameters may be extracted using simple graph-
are removed from the c i r c u i t , substracting t h e i r
ical means. However, i f the time constants of
dispersion from the overal frequency dispersion.
the respective subcircuits are r e l a t i v e l y close
The use of personal computers, which have moder-
together the frequency dispersion cannot be
ate graphics c a p a b i l i t i e s , can be of great
advantage in this analysis.
devided in d i s t i n c t separate regions. This is
also true i f elements with a f r a c t i o n a l power
In t h i s paper an o u t l i n e is given of such an
dependence on frequency are present, e.g.
analysis program.
Warburg or Constant Phase Elements (CPE7-9),
w i l l y i e l d a s u i t a b l e equivalent c i r c u i t ,
In many cases t h i s program
because the modulus of such an element varies
together with an adequate set of s t a r t i n g values
sub-linear with frequency, extending i t s
For a subsequent NLLSF procedure.
influence over a large frequency range in the
diate and f i n a l f i t
dispersion.
the measured data using a dispersion simulation
In these cases a l l c i r c u i t parameters must be
0 167-2738/86/$ 03.50 © Elsevier Science Publishers B.V.
(North-Holland Physics Publishing Division)
The interme-
results may be compared with
program, which w i l l be discussed b r i e f l y .
Both
B.A. Boukamp / A package for impedance~admittance data analysis
137
programs use the C i r c u i t Description Code12'13
the t o t a l dispersion, thus exposing the disper-
(CDC), which allows the use of a v a r i e t y of
sion of the adjacent s u b c i r c u i t more c l e a r l y .
equivalent c i r c u i t s with these programs without
a need f o r r e w r i t i n g the source code.
Generally the next s u b c i r c u i t can be regarded
as a p a r a l l e l combination of a CPE and a resistance (e.g. grain boundary immittance) in series
2. OUTLINE OF CIRCUIT ANALYSIS PROGRAM
with another resistance.
Crude values f o r the adjustable parameters
This results in a
depressed semi-circle in the immittance diagrams.
are obtained through simple "graphical" means.
By choosing a set of three data points the
The i n t e r f a c e response of an i o n i c conductor can
program can f i t
often be modeled as a t o t a l i o n i c resistance
resistances, as well as the CPE parameters.
(grain boundary + i o n i c resistance) in series
The f i t t i n g
a c i r c l e , and give values f o r the
of a set of subcircuits is per-
with a CPE element, sometimes combined with a
formed in the impedance representation to a
capacitance 14. In t h i s p a r t i a l dispersion, in
selected part of the dispersion curve.
An ana-
the impedance representation, a s u i t a b l e point
litical
is selected through which a tangential l i n e is
described elsewhere12, is used. The e r r o r func-
drawn.
tion f i t s
The i n t e r s e c t i o n with the real axis
gives the t o t a l resistance, while the f r a c t i o n a l
exponent, n, of the CPE is found from the slope.
The dispersion r e l a t i o n f o r the CPE is given by9
Y*(w) = Yo . (j~)n
(1)
search procedure, which has been
the real and imaginary parts simul-
taneously, using one weight f ac t o r :
(Z~-Z'(w~)2+ (Z" Z"(~ )
i
• i- - iwhere Z*-Z'+'Z"
is
the
measured
data set and
i- i J i
Z*(m) represents the calculated f i t
response.
The f a c t o r Yo is found from the imaginary value
The weight factors are inversely proportional
of the selected data p o i n t , Z~':
to the square of the modulus of the measurements:
Yo : -sinn-~-~ / Z" . con
1
(2)
1
An estimate f o r the capacitance can be found
by substracting at the low frequency l i m i t , ~ I '
the calculated CPE response from the imaginary
part of the measurement, ZIi I..
(3)
A s i m i l a r procedure is used f o r the analysis
of the bulk response at the high frequency l i m i t
of the dispersion, but now in the admittance
Here a resistance in p a r a l l e l
with a CPE and/or capacitance is obtained.
When there is l i t t l e
interference with the
dispersion of the adjacent s u b c i r c u i t , these
values can be used f o r a subsequent simple
NLLS-fit by the program, see below.
Otherwise,
the parameters f o r the adjacent s u b c i r c u i t must
(5)
which insures that a l l data sets contribute
equally to the er r or function 15.
The c a l c u l a t i o n of the c i r c u i t response in
the f i t
C = I / wI (Z~' - ml n. s i n - ~ / Y o )
representation.
wi = I / IZ*
-i i 2 ,
procedure is based on a C i r c u i t Des-
c r i p t i o n Code (CDC)12'13, which uniquely represents the equivalent c i r c u i t . The CDC, which is
in t e r p r e t e d by the program, acts as a set of
pointers to a corresponding set of subroutines
which calculate the i n d i v i d u a l responses of the
d i f f e r e n t types of c i r c u i t elements.
The par-
t i a l d e r i v a t i v e s of the e r r o r function with respect to the adjustable parameters, are calculated in the same subroutines and at the same
time.
This provides f o r a very compact source
code f o r the f i t
procedure.
be obtained in order to perform a NLLS-fit f o r
the combined subcircuits.
To t h i s purpose, the
CPE (and capacitance) dispersion is removed from
3. DISPERSION SIMULATION PROGRAM
Through the use of the CDC t h i s program can
B.A. Boukamp / A package for impedance~admittance data analysis
138
calculate the frequency dispersion of d i f f e r e n t
complex equivalent c i r c u i t s , which may include
various d i f f u s i o n related dispersions 13'16. The
5
T<
calculated dispersion can be compared to actual
measurements in both immittance diagrams and in
a Bode p l o t .
O
O3
O
The q u a l i t y of a NLLS-fit r e s u l t is best seen
in a " F i t Quality" plot (FQ-plot).
,~2
In t h i s p l o t
the r e l a t i v e deviations of the r e a l , Are, and
imaginary, Aim, parts are plotted against log(~J,
with:
1
-
Arei -
i Z*i I
For a good f i t
and Aim.=1
2
3
4
5
6
Z' , (*103ohm)
-
I Z;i
(6
these deviations should be d i s -
t r i b u t e d randomly around the frequency axis.
4. EXAMPLEOF DATA ANALYSIS
FIGURE 2
Impedance diagram showing the tangential l i n e
f o r the estimation of the interface parameters.
first
The use of the data analysis program is demon-
observation the equivalent c i r c u i t seems
quite simple, a bulk CPE in p a r a l l e l with the
strated with a frequency dispersion measurement
i o n i c resistance, and an interface element
of a sample of AgCrS2 doped with 10% SnS~. This
(Warburg or CPE).
layer compound is a pure i o n i c conductor ~7.
The
As the interface dispersion is most pronoun-
a c - c o n d u c t i v i t y was measured on a sample with
ced i t is used f o r the f i r s t
i o n i c a l l y blocking gold electrodes, using a
lysis.
Solartron 1250 FRA. The data a c q u i s i t i o n and
l i n e option (eqs 2,3) is used, r e s u l t i n g in
step in the ana-
With a selected point the tangential
correction was performed with an Apple I I com-
crude values for the i o n i c resistance and the
puter. The frequency range was 655 mHz to 65.5
CPE parameters, f i g .
kHz.
optimized set is obtained.
The dispersion, measured at 298 K, is
given in the admittance plot of f i g .
T2
Ago.9CrogSno.lS2
i.
From
T = 298 K
S
2.
With the NLLS-fit an
Next the response
of the interface CPE is substracted from the
e n t i r e data set.
The r e s u l t i n g dispersion is
given in f i g . 3.
This admittance p l o t obvious-
l y represents two i n t e r f e r i n g s u b c i r c u i t s .
The parameters of the high frequency (bulk)
CPE are estimated using the tangential l i n e
option in the admittance representation ( f i g .
%
65535o Hz
o
oooOO o ° °
°
,~o-g~T55.35imHz i
Y'
° ° Oooo o°
i
,
3).
A subsequent NLLS-fit is not useful as the
influence of the medium frequency dispersion
o
might extend i n t o the high frequency d i s p e r s i o n
,
( x1 mho}
As an approximation the bulk CPE dispersion is
substracted from the modified data set, leaving
a depressed semi-circle.
A c i r c l e is f i t t e d by
the program through three selected points in
FIGURE 1
Admittance diagram of the measured dispersion.
t h i s dispersion, y i e l d i n g values for the r e s i s -
B.A. Boukamp / A package for impedance~admittance data analysis
139
1
T
-S
CDC
0.5
(IRP) ( R P ) P
x
FIGURE 5
Schematic representation of the equivalent
c i r c u i t used in the dispersion analysis.
ooOOOOoo°°
0
I
35
,
2
2.5
Y'
3
, ( xlO - 3 m h o )
>
this p l o t i t can be seen that the extracted
c i r c u i t and parameter set adequately describe
FIGURE 3
Admittance diagram of the modified data set
showing the tangential l i n e used in the e s t i mation of the parameters of the bulk CPE.
the measured dispersion.
In t h i s case the
general NLLSF procedure is only needed f o r
obtaining a set of error estimates f o r the
c i r c u i t parameters.
The r e s u l t of a complete
tances and the corresponding (grain boundary)
fit
performed with the program "EQIVCT" is
CPE.
given in the l a s t column of table i .
These values, together with the parameters of
the bulk CPE, are used in a f i n a l NLLS-fit of
5. DISCUSSION
the modified data set in order to optimize the
parameter set.
The c i r c u i t used in t h i s f i t
is
No concise and formal procedure can be given
f o r the analysis of frequency dispersion data.
based on the equivalent c i r c u i t of f i g . 5, CDC:
Each procedure strongly depends on the type of
(RP)(RP).
equivalent c i r c u i t assumed to represent the
The f i n a l r e s u l t is given in table i .
The frequency dispersion, based on the f i n a l
measured data best.
The i n t e r a c t i v e dispersion
parameter set, is compared with the t o t a l
analysis program, o u t l i n e d in t h i s paper, can be
measured dispersion in the FQ-plot of f i g . 6,
of help in many cases.
using the dispersion simulation program.
experience in frequency dispersion analysis.
T
From
t'
O.1
0
c-
0
Its use does require some
0
a
Are
•
Aim
0
o
o
E
_
o
-
0
'='",~o
.
o
o
x
.
o
g
o
0
E
!
.
-1
!.,
4/
iI
J
I
I
I
2.4
2.5
y' , { x 10-3 mho )
I
I
2.6
~-
FIGURE 4
Admittance diagram of the r e s u l t i n g (grain
boundary) dispersion.
-20
I
I
I
1
2
3
Iogca , (u in rod/s)
1
4
)
I
5
FIGURE 6
FQ-plot f o r the f i n a l parameter set obtained
with the dispersion analysis program.
B.A. Boukamp / A package for impedance~admittance data analysis
140
TABLE 1
Parameter r e s u l t s obtained with the data analysis
program and the general immittance f i t program
comparatively slow execution.
"EQIVCT''12. The f i n a l r e l a t i v e error estimates
are also given.
or Pascal.
Param- data analysis
Element: eter:
program:
R-ionic
R
CPE-bulk IYn°
R-gb.
CPE-gb.
CPE-int.
R
~Yo
~n
~Yo
)n
391
9.00.55xI0 -7
38
7 . 1 x 10-s
O.84
6.66x I0 -s
0.502
NLLSF-prog.
"EQIVCT":
rel.
err:
389
8.2 x i0 -~
0.56
0.3%
0.7%5
%
62
12 %
1.5 x 10-" 18 %
O.7O
5 %
6.64x 10-s 0.1%
0.505
0.2%
The NLLS-fit option is an useful tool f o r
obtaining simultaneously the parameters of two
interfering subcircuits.
I t s use requires,
however, that the data has been corrected
previously for instrumental d i s t o r t i o n s .
Also
special care must be taken as the precission and
range of the f l o a t i n g point numbers is rather
l i m i t e d in the present generation of personal
computers.
To prevent overflow errors, the data
must be m u l t i p l i e d with a constant scaling
=
1 / / IZ~l
Z~
I hI
(7)
where IZ~I and IZ*h I are the vector length of the
impedances at the lowest and highest measured
frequencies, ml and wh. This procedure centers
the values of the dispersion measurements around
1.
This may be
improved by t r a n s l a t i n g the programs in FORTRAN
Also the new generation of 16-bit
personal computers w i l l g r e a t l y improve the
execution time of these programs.
The source code of the programs discussed in
t h i s paper may be obtained from the author.
REFERENCES
1. P.H. Bottelberghs, Low-frequency measurements on s o l i d e l e c t r o l y t e s and t h e i r
i n t e r p r e t a t i o n s , in: Solid E l e c t r o l y t e s , eds.
P. Hagenmuller and W. van Gool (Academic
Press, New York, 1978) pp. 145-172.
2. J.R. Macdonald and J.A, Garber, J. Electrochem. Soc. 124 (1977) 1022.
3. I.D. R a i s t r i c k and R.A. Huggins, in: Proc.
Symposium and Workshop on Advanced Battery
Research and Design, Argonne National
Laboratory 76-8 (1976) B277.
4. J.R. Macdonald, in: Electrode Processes in
Solid State l o n i c s , eds. M. K l e i t z and
J. Dupuy (D. Reidel, Dordrecht, 1976)
p. 149,
5. D.R. Franceschetti, J. Schoonman and J.R.
Macdonald, Solid State lonics 5 (1981) 617.
6. B.A. Boukamp and G.A. Wiegers, Solid State
lonics 9/10 (1983) 1193.
f a c t o r , fsc:
fsc
The disadvantage of Basic programs is the
The obtained parameters must be re-scaled
accordingly.
I t is often possible t h a t , w i t h i n the error
7. P.H. Bottelberghs and G.H.J. Broers, J.
Electroanal. Chem. 67 (1976) 155.
8, I.D. R a i s t r i c k , C. Ho and R.A. Huggins, J.
Electrochem. Soc. 123 (1976) 1469.
9. J.R. Macdonald, Solid State lonics 13 (1984)
147.
IO.J.R. Macdonald, A. Hooper and A.P. Lehnen,
Solid State lonics 6 (1982) 65.
I I . J . R . Macdonald, J. Schoonman and A.P.
Lehnen, J. Electroanal. Chem. 131 (1982) 77.
l i m i t s , the same data set may be interpreted
12 B.A. Boukamp, submitted f o r p u b l i c a t i o n .
with more than one equivalent c i r c u i t .
13 B.A. Boukamp, Internal Report CT85/177/128,
Twente U n i v e r s i t y of Technology, 1985.
The
example presented here can be analysed with the
bulk-CPE p a r a l l e l to Ri and the grain boundary
dispersion (CDC: '(P(R(RP)))P') but also with a
Randless type c i r c u i t f o r the i n t e r f a c e , without
grain boundary, (CDC: '(RP)(P(RP))'). Measurements at d i f f e r e n t temperatures are needed in
order to elucidate the correct equivalent c i r c u i t .
14.B.A. Boukamp and R.A. Huggins, Mat. Res. Bull
13 (1978) 23.
15.P. Zoltowski, d. Electroanal. Chem. 178
(1984) 11.
16.B.A. Boukamp, Internal Report CT85/178/128,
Twente U n i v e r s i t y of Technology, 1985.
17.T. Hibma, D. Br~esch and S. Str~ssler, Solid
State lonics 5 (1981) 481.