Optimization of impedance measurements using ‘chirp’ type

1. Optimization of impedance measurements using ‘chirp’ type
perturbation signal
P. Slepski *, K. Darowicki
Department of Electrochemistry Corrosion and Materials Engineering, Gdansk University of Technology, 11/12 Narutowicza Street, 80-233 Gdansk, Poland
a r t i c l e i n f o
Article history:
Received 20 March 2008
Received in revised form 9 February 2009
Accepted 12 June 2009
Available online 17 June 2009
Keywords:
Chirp signal
Impedance
Variable sampling frequency
a b s t r a c t
The paper presents a novel method of impedance measurements using the non-stationary
‘chirp’ signal. Variable sampling frequency has been employed in a process of acquisition of
the voltage perturbation and current response signals. Such solution allows significant
decrease in a number of data necessary to obtain an impedance spectrum as well as sim-
plification and acceleration of the calculation process. It has been found that an accuracy of
the results obtained with this method was directly proportional to the frequency resolu-
tion, with which impedance values were obtained.
Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction
For many years electrochemical impedance spectros-
copy technique has been used in analysis of electrochemi-
cal phenomena. Universal character and thus popularity of
impedance spectroscopy stems from a big number of infor-
mation provided in a single measurement cycle. However,
just as other investigation techniques also electrochemical
impedance spectroscopy (EIS) possesses certain limita-
tions. They result mainly from some discrepancies in
behaviour of real electrochemical systems and the electri-
cal equivalent circuits used for modelling and analysis of
experimental results. A correct impedance measurement
requires three fundamental conditions to be fulfilled: line-
arization, causality and stationarity [1,2]. The first two
requirements can now be easily met but the stationarity
condition is still a serious limitation of impedance spec-
troscopy due to a non-stationary nature of electrochemical
corrosion processes.
For many years impedance measurements have been
dominated by the frequency response analysis (FRA) tech-
nique, which consists in sequential perturbation of the
investigated system with sinusoidal signals having defined
frequencies. In such case determination of an impedance
spectrum requires application of consecutive perturbations
of different frequencies. Despite high accuracy of the re-
sults obtained this method is very time-consuming. Pertur-
bation in the form of a composition of several elementary
sinusoidal signals in measurement of electrode impedance
can provide considerable reduction of measurement time
[3–11]. In such case analysis of results is mostly based on
Fourier transformation, while use of short time Fourier
transformation (STFT) enables additionally to obtain time
dimension – dynamic electrochemical impedance spec-
troscopy (DEIS) technique [12,13]. Minimum time neces-
sary to obtain a single impedance spectrum is equal to a
period of the perturbation signal having the lowest fre-
quency. A constraint of this method is so-called frequency
limit. An increase in the number of elementary frequencies
in the perturbation signal might yield an increase in the
resultant amplitude of the perturbation. In order to obey
linearity regime, optimization methods of multisine per-
turbation signal are applied, which base mostly on ade-
quate selection of phase shifts of components [7,10]. In
many cases this fact may make it difficult to describe the
investigated system properly.
0263-2241/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.measurement.2009.06.005
* Corresponding author. Tel./fax: +48 58 347 10 92.
E-mail addresses: pawelkor@chem.pg.gda.pl (P. Slepski), zak@chem.
pg.gda.pl (K. Darowicki).
Measurement 42 (2009) 1220–1225
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2. A compromise between accuracy of the FRA and
dynamics of the multisine methods can be the method
utilizing a non-stationary, voltage signal of ‘chirp’ type
[14–16]. In practice, it is the only method, which makes
it possible to obtain continuous impedance spectrum with
respect to frequency, for relatively short duration time of
the perturbation signal. Therefore it is possible to achieve
impedance values for every frequency from initial to final
perturbation frequency. Receiving similar results with
FRA technique would require excitation with infinite num-
ber of sequentially changed frequencies. The white noise
would be required in the case of multisine signal excitation
as bend pass. That may lead to unsatisfying as for its qual-
ity achieved impedance spectra.
Applying the perturbation of exponential time–fre-
quency characteristic allows full characterization of an
electrochemical system. Despite these advantages the
method, described previously by the authors, possesses
also certain drawbacks connected with registration of
numerous data connected with the perturbation and re-
sponse signals. Moreover, ambiguity in frequency localiza-
tion due to applied data analysis imposes a serious
problem. In the paper the authors present an alternative
way of obtaining impedance spectra using the ‘chirp’ signal
of exponential characteristics, aimed at elimination of the
drawbacks mentioned earlier.
2. Analysis of signal
Perturbation of an investigated system with the ‘chirp’
signal of exponential time–frequency characteristic results
in the current response signal, which similarly to the per-
turbation is a non-stationary signal.
DiðsÞ ¼ DuðsÞYðxðsÞÞ ð1Þ
where Di(s) is the current response signal of exponential
time frequency characteristic, Du(s) – the potential pertur-
bation signal of exponential time frequency characteristic,
and Y(x(s)) is the admittance of investigated system.
Determination of impedance spectra based on the
recorded courses requires the knowledge of changes of the
amplitudes and phase shifts versus frequency. This informa-
tion can be obtained via classical Fourier transformation
[4,5,17] as well as via time–frequency transformations,
including the STFT [18,19]. In the case of non-stationary sig-
nals analysis a disadvantage of these transformations is the
spectrum dispersion effect, which results in appearance of
additional error [20] and limited number of impedance val-
ues,originatingfromthepropertiesofa discretetransforma-
tion [16]. To eliminate this effect a modification of the STFT
transformation has been proposed. It consists in introduc-
tion of an additional element in the form of a window with
variable length dependent on a value of analyzed frequency
[16]. The window function prevents from the phenomenon
of spectrum leakage and its variable length allows the anal-
ysis of any frequency from the investigated range.
The modifications introduced enabled obtaining, con-
tinuous impedance spectrum, however, they did not elim-
inate two basic disadvantages of this method. The first one
is a big number of data connected with registration of the
entire perturbation and response courses. Constant sam-
pling frequency during the acquisition process, dependent
on the highest frequency of the perturbation signal, may
lead to oversized files in the case of wide range of analyzed
frequencies. They may not be able to be handled by the
measurement software or by the PC operation system.
The second disadvantage is ambiguity in frequency locali-
zation. Application of a typical symmetric analyzing
window for an analysis of the signal with exponential
time–frequency characteristic results in the fact that a
determined frequency is not equal to the mean value of
frequency in a given range, what is schematically pre-
sented in Fig. 1.
An alternative approach, able to overcame the draw-
back described above, can be the method utilizing variable
sampling frequency at a digitization stage, the value of
which is a multiple of the perturbation signal (Fig. 2):
fSðtÞ
fiðtÞ
¼ const: ¼ l ð2Þ
where fS(t) is the variable sampling frequency, fi(t) – the
perturbation frequency, and l is the variable sampling fre-
quency to perturbation frequency ratio.
Such approach significantly decreases size of the files
with current and voltage registered during impedance
measurement, thus accelerating the following analysis
stage.
Application of the variable sampling frequency in acqui-
sition of the perturbation and response signals provides that
successive values are recorded in the time intervals depen-
Fig. 1. A scheme representing ambiguity in instantaneous frequency
localization; (------) change of frequency of analyzed signal, (——)
window function, Df difference between the frequency localized with
window function and the mean frequency within the analyzing window.
Fig. 2. Exemplary changes of frequency versus time of the perturbation
signal (——) and the data acquisition timing signal (Á Á ÁÁ Á Á), fF – the final
frequency of perturbation signal, f0 – the initial frequency of perturbation
signal.
P. Slepski, K. Darowicki / Measurement 42 (2009) 1220–1225 1221

3. dent on an instantaneous sampling frequency Dt = f(fS(t)).
Analysis of such records by means of classic transformations
requires very complicated mathematical approach. How-
ever, there is another, very simple method of signal decom-
position, which makes possible to obtain information about
the amplitudes and phase shifts versus frequency. Sampling
of the signal during measurement with the frequency fS(t)
being a constant multiple of the perturbation signal’s fre-
quency causes that values of amplitude of registered signal
represents next frequencies as well. Frequency difference
between two adjacent registered samples is constant and
equal to Df *
(Fig. 3):
DfÃ
¼
fF À f0j j
N
ð3Þ
where Df *
is the frequency interval between of recorded
samples, and N is the number of recorded samples.
Thus, the instantaneous frequency fi can be determined
from the relationship below:
fi ¼ f0 þ DfÃ
Á ið Þ for f0 < fF ð4Þ
fi ¼ f0 À DfÃ
Á ið Þ for f0 > fF
where i is the sample index in a record, and fi is the instan-
taneous frequency.
Decomposing a fragment of the signal having length L
and midpoint at fi, one obtains full characteristic for the
frequency of interest:
DFT ¼
XL
i¼0
sðiÞ exp
Àj2pni
L
ð5Þ
where: s(i) – the recorded perturbation or response signal,
L – the number of samples of portion decomposing signal.
Application of above discrete Fourier transformation in
the presented method requires the following rule to be
obeyed: a length of the record subjected to decomposition
for the frequency fi should be equal to:
L ¼ n Á l where n ¼ 1; 2; . . . ð6Þ
Otherwise, an additional error connected with spec-
trum dispersion occurs. It should be emphasized that the
operations in Eq. (5) are executed for a single value of n.
Analysis of the successive n multiples corresponding to
harmonic components can be an effective tool verifying
whether the linearity regime was maintained during the
measurement.
In practice, a single measurement cycle enables obtain-
ing impedance values for almost all frequencies from the
scanned range. In this way a continuous series of instanta-
neous impedance spectra is created. An exceptions are the
terminal values, which are not completely engulfed by the
analyzing window (Fig. 4).
The missing values of the signal could be filled with
zero values and then such modified fragment of the signal
can be subjected to decomposition [20]. However, the re-
sults obtained in this way would be burdened with a sig-
nificant error. A solution can be application of the
perturbation consisting of individual signals of ‘chirp’ type
having alternate increasing and decreasing frequency
(Fig. 5).
Fig. 4. An example showing the problem with analysis of the terminal
frequencies.
Fig. 3. (A) Recording with variable sampling frequency – a signal in time
domain. (B) The registered signal, the successive values of which are
presented at constant intervals.
Fig. 5. An example of the continuous signal being a composition of successive individual signals of ‘chirp’ type.
1222 P. Slepski, K. Darowicki / Measurement 42 (2009) 1220–1225

4. It enables analysis of the full frequency range.
Moreover, analysis of two neighboring ‘chirp’ signals is
an excellent verification of stationarity of the investigated
object. Application of this continuous signal and mainte-
nance of stationarity within the time period when a single
‘chirp’ signal is generated create a possibility of tracing the
impedance changes versus time.
3. Accuracy
Accuracy of the experimental results is a fundamental
point of every measurement method. In case of the FRA
method an integration time is the main factor affecting
accuracy, while for the DEIS these are the character and
size of the analyzing window, which decide about accu-
racy. In the presented method, during the measurement
and analysis of records, there is a possibility of adjustment
of several parameters such as: time of perturbation dura-
tion T, number of points registered within one period l,
and number of periods n subjected to elementary decom-
position inside the analyzing window. These parameters
have a fundamental influence on a duration time of mea-
surement and analysis as well as on a number of recorded
data. They are also expected to have an impact on accuracy
of obtained results. In order to evaluate this impact a spe-
cial, dedicated set-up was prepared and a series of mea-
surements and analyses was carried out for various
values of the parameters T, l and n.
The measurement set-up utilized for impedance mea-
surements using the frequency scanning method is sche-
matically presented in Fig. 6.
A module NI PXI-1031 DC with a measurement card
PXI-6120 was used for acquisition of the voltage perturba-
tion and current response signals. The ‘chirp’ signal of
amplitude 10 mV and frequency range 10 Hz–10 kHz as
well as suitable timing signal were generated by a com-
puter equipped with a measurement card PCI-6111. The
perturbation signal was applied to the system under
investigation using a potentiostat KGL. Controlling and
analyzing software was prepared in LabView 7.1 by the Na-
tional Instruments. The measurements were carried out on
a Dummy Cell 2 made by Eco Chemie BV, on a circuit sche-
matically presented in Fig. 7. Real values of investigated
elements were measured with Agilent 34410A. Difference
between nominal and real values did not exceed 0.2%.
The measurements were performed for T = 1,2,5,
10,20 s; n = 2,5,10,20; l = 5,7,10. Measurements were re-
peated 10 times for every setup. Analysis of the impedance
spectra, each composed of 50 impedance values (due to
limitations of the analyzing program), was done with the
ZSimpWin software.
Table 1 presents average values of relative errors deter-
mined basing on real values of the investigated elements of
electric equivalent circuit. for different parameters T, n and
l. The values within a single column, that is for fixed T, n
and for different l, are similar to each other. It means that
in the presented method a magnitude of error does not de-
pend on the number of samples used for recording of a sin-
gle period of sinusoid. Thus, one can use adequately low
value of this parameter in order to decrease a size of re-
corded data files. The lowest possible value of parameter
l is limited by the Nyquist rule, which defines that a single
Fig. 6. A scheme of the measurement set-up utilized for impedance
measurements using the frequency scanning method.
Fig. 7. A scheme of the electric al circuit subjected to measurements,.
Table 1
Relative errors obtained for different values of T and n.
Error/%
T = 1 s T = 2 s T = 5 s
n = 2 n = 10 n = 5
l = 5 R1 0.79 1.53 0.42
C2 0.71 1.49 0.40
R3 0.70 1.38 0.37
l = 7 R1 0.67 1.53 0.39
C2 0.63 1.47 0.36
R3 0.59 1.38 0.34
l = 10 R1 0.67 1.50 0.35
C2 0.64 1.44 0.33
R3 0.59 1.35 0.30
P. Slepski, K. Darowicki / Measurement 42 (2009) 1220–1225 1223

5. period of sinusoid cannot be described properly by fewer
than 2 values.
Unlike parameter l, the time of perturbation duration T
and the number of periods subjected to elementary
decomposition n have a fundamental impact on accuracy
of obtained impedance results. It is so due to the fact that
these parameters influence directly on the frequency reso-
lution df, with which impedance results are obtained. This
dependency is described by the following equation:
df ¼
n
T
3:454 ð7Þ
Figs. 8–10 depict the magnitude of relative error versus
frequency resolution.
In all cases the magnitude of error is a linear function
of frequency resolution. Minimum value of overall error
(about 0.3%) is due to the instrument error of the appa-
ratus as well as noise. Thus, minimization of the error
can be generally achieved in two ways. The first one con-
sists in application of the perturbation with longer dura-
tion time (increased T), while the second one involves
utilization of the analyzing window engulfing the small-
est possible number of periods of recorded signals (de-
creased parameter n).
4. Conclusion
In order to eliminate its inherent disadvantages, the
impedance measurements employing ‘chirp’ type signal
have been modified by application of variable sampling
frequency being a multiple of an instantaneous frequency
of the registered signal. Such approach substantially re-
duces the number of data necessary to be recorded, thus
allowing the impedance measurements to be carried out
in any frequency range. Moreover, the form of recorded
data enables simple analysis employing the classic STFT
transformation with the analyzing window of constant
length. In this way the ambiguity in frequency localization
occurring for the perturbation with non-linear time–fre-
quency distribution is eliminated. The results are burdened
with an error, which is proportional only to a time of the
perturbation duration and a length of the analyzing win-
dow applied. Accordingly, one can minimize the error by
application of long enough perturbation signal, which
would also be adjusted to the conditions of particular
experiment. The error can also be minimized at the later
stage that is during analysis of the measurement results.
The modifications presented in this paper open a possibil-
ity of further development of the method, for instance via
analysis of higher harmonics or by application of continu-
ous perturbation with the ‘chirp’ signal.
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Fig. 10. Relative error of parameter R3 versus frequency resolution.
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