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Calculation of impedance spectra by Laplace transformation of voltage transients generated by current-step excitation
33
Journal of Electroanalytical Chemistry, 372 (1994) 33-37
Calculation of impedance spectra by Laplace transformation transients generated by current-step excitation Michael Neumann-Spallart
of voltage
a and Mohamed Etman b
a Laboratoire de Physique des Solides, CNRS, 1 place Aristide Briand, F-92195 Meudon (France) b Laboratoire d’Electrochimie des Interfaces, CNRS, I place Arktide Briand, F-92195 Meudon (France) (Received 16 April 1993; in revised form 15 October 1993)
Abstract Impedance data over a broad frequency range were obtained by converting the transient voltage response of a passive network to a current-step excitation into the frequency domain using Laplace transformation. The merits and limitations of the method and the type and quality of the required data are discussed. Examples of impedance plots obtained using this method are given.
1. Introduction
Impedance spectra are of great importance for the analysis of electrochemical interfaces and cells. Transient perturbations (single potential or current steps) probe the system during a very short period of time. This is particularly important for rapidly evolving systems, such as electrodes in a corroding electrolyte, systems where electrode and electrolyte compositions change (accumulators during charge-recharge cycles) and oscillating chemical systems. The response contains all the information needed for the calculation of the impedance Z of the system over the whole frequency range. Z can be extracted from transient data by using the Laplace transformation [l-4]. In this article we describe the calculation of impedance spectra from voltage transients generated by current-step excitations. The advantage of the current-step method for electrochemical cells lies in the accessibility of the high frequency range, since fast potential control is not necessary. On the contrary, this must be suppressed by deliberately slowing down the response of a potentiostat connected to the circuit [31. However, in the low frequency range (< 1 Hz for a reasonable damping of the potentiostatic response) potential control sets in and this precludes the use of data in the corresponding time domain. Therefore this method is complementary to the more frequently used voltage step method. 0022-0728/94/$7.00 SSDI 0022-0728(93)03265-Q
Single-event excitation waveforms and the response to such perturbations can be processed by Laplace transformation: U_,,(w) =
/,,f( t)
e-“’
dt;
Ii,
= irnf( t) e-j”’ dt
(1) where I,, and Ii, are the real and imaginary axis Laplace transforms respectively, o/r-ad s-l is the angular frequency, j = m and f(t) denotes either the excitation function i(t) or the response U(t). For the imaginary axis transformation, f(t) must be split into two parts before integration to guarantee convergence of the calculation. The constant value at t + 03 is subtracted from f(t) and transformed separately. The calculations are carried out for a series of predetermined frequencies in the range of interest for the particular circuit. In this way, time domain data are transformed into the frequency domain. The impedance is the ratio of the Laplace transforms of the potential and the current transients. The real axis Laplace transformation leads to the so-called operational impedance Z, and the imaginary axis Laplace transformation yields 2. Bode plots (log IZ Ivs.log(o) or log(Z,,) vs. log(o)) can be used to derive the values of series and parallel resistances and capacitances of simple circuits consisting of a resistance in series with sections of parallel resistance and capacitance (R,+ R,,(I C, +
0 1994 -
Elsevier Science S.A. All rights reserved
M. Neumann-Spallart, M. Etman / Calculation of Impedance Spectra
34
limiting value of log(Z,,) or log I 2 I at sufficiently high or low frequencies represents the series resistance and the sum of all resistances respectively. In log 1Z I plots, there are inflection points with tangents of slope - 1; the extrapolation of these to log(w) = 0 yields - log C. In log(Z,,) plots, however, the tangents depend on the circuit parameters and
the information on the impedance, including the phase angle, it can be extracted using imaginary axis Laplace transformation. We have applied this procedure to synthetic transients, different test circuits, and electrochemical interfaces.
C = (l/R;
The experimental arrangement has been described in detail elsewhere [3]. A Wenking 68FA0.5 potentiostat and a signal generator were coupled in parallel via large resistances to the cell which was connected in series to a calibrated measuring resistance. The potentiostat was used to control the working electrode potential, but its response was slowed down. A capacitor in parallel with the reference electrode ensured that the voltage transients to be measured were not suppressed, at least for a reasonable period of time (a few tenths of a second). The potential responses to a current step of both the series combination of the cell
R,, II C, + R,, II C, + . . . ). The
+ 1/R,R,)1’2/2wina
(2)
where minn denotes the frequency at an inflection point
[51. For more complicated circuits, sections corresponding to different R, 1)C subsections are not well resolved in log(Z,,) vs. log(o) diagrams. Thus additional information about the phase angle or a representation of the real and imaginary parts of the complex impedance is required for circuit analysis and modelling (plots of log(Z,,) and log(Zi,) as functions of log(w), or -.Zi, vs. Z,). As the transients contain all
Fig. 1. Example of the time dependence S = (0.5/o)C[U(ti)
2. Experimental
of the sum [61
+ U(ti+,)][Sin(oti+,)
- SiIl(wti)]
used in the calculation of the real part of the Laplace transform of a synthetic voltage transient: R, = 50 R, R, = 1000 0, C = 9.8 PF and I Sfep= 10 FA; dynamic spacing over seven orders of magnitude, each containing 90 equally spaced points, from At = 1 X lo-’ s to At = 1 X 10-l o=WOOrads-l;---~=2000rads-1.Upperinset, s. U(t) is the voltage response after subtraction of the plateau value U(t + 03).transient calculated using eqn. 3; lower inset, impedance spectrum. The points were obtained by Laplace transformation and connected with a solid line in order to show their sequence with increasing o.
35
M. Neumann-Spaliart, M. Etrnan / Calculation of Impedance Spectra
under test and, the measuring resistor and the cell alone were amplified and recorded simultaneously on the two channels of a Tektronix A2221 storage oscilloscope (2 x 2000 points, 8 bit resolution) and subsequently transferred to a microcomputer. The excitation current waveform was obtained from the two signals by subtraction. 3. Results and discussion It can be shown that the transient
response
R, + (RPI( C) circuit to a current step perturbation
of a Z is
given by U(t)=Z(
R,+R,)-ZR, exp( -t/R,C)
(3)
Transients with different circuit parameters were calculated using eqn. (3) and transformed by imaginary axis Laplace transformation using an algorithm described by Ye and Doblhofer [6] (The integrations were considerably simplified and precision at high frequency was enhanced by substituting U, = w-isin and ui = --w-i co4 ot) in eqn. (1)). The transients had either equally spaced data points or a fixed number of data points per decade of time in order to check the reaction of the integration algorithm to these features and the maximum frequency that could be analysed. After transformation and calculation of the impedance, transients with equally spaced data points with sampling
times up to 2 Z.LSgave smooth Nyquist plots up to 600 krad s-l coinciding perfectly with the theoretical curves. Also, transients with a fixed number of data points per decade (typically 10 points per decade), which had the advantage of requiring a much smaller number of points for the whole transient, gave good results. However, in the latter type of transient, aberrant points in the impedance spectrum (lower inset in Fig. 1) became quite abundant above a certain frequency. We analysed the reason for the production of these aberrant points in detail. In the calculation, the integral in eqn. (1) was solved by numerical integration for each frequency. The intermediate sum of the integration values was a convergent oscillating function, (e-jmt can be written as the sum of sine and cosine terms). Its frequency was given by the o for which the transformation was carried out. At high w, upon transition to the next domain of (larger) data spacing the density of data points was no longer sufficient to match the oscillation in o. The oscillation then appeared with a lower frequency (aliasing). Depending on the actual value of the response function, this may have caused a sudden jump in the value of the integral, leading to an aberrant value in the result (see an example in Fig. 1 for o = 5000 rad s-‘; the aberrant point occurring with the spacing transition is marked by an arrow>. The risk of this occurrence becomes increasingly important with increasing frequency w used for the
10
rad/s
100 ,,,,,,,,,,,,,,,,,,,,,.,,,,,,.,~....~.,,,,,,,~,,,!.~,~~,,,l krad/s
0”
400
800
1200 Re(Z/ n)
Fig. 2. Example of a Nyquist plot calculated by Laplace transformation
1600
2000
2400
of a transient obtained by current-step perturbation of a test circuit with
R, = 50 0, R,, = R,, = 1000 a, C, = 1 PF and C, = 10 pF. The points were calculated from experimental data and the solid line was calculated
from the values of the elements in the circuit.
36
M. Neumann-SpaNart, M. Etman / Calculation of Impedance Spectra
calculation. At sufficiently low w, spacing transitions only lead to a more irregular course of the function (Fig. 1, broken line, w = 2000 rad s-l>. This analysis allowed us to determine the spacing necessary to calculate the Laplace transformation up to a given frequency o, or to determine the maximum explorable frequency for a circuit with given parameters and the available sampling rate and memory of a transient digitizer or digital storage oscilloscope. According to eqn. (31, the relaxation time RC for the circuit in Fig. 1 was 0.01 s. The recorded length of a transient should be not much less than five times the time constant in order to reach a value close to the plateau; in our example this is 0.05 s. For instance, if 1000 samples are collected, this would correspond to a time spacing of 5 X 10m5 s (= 20 kHz). Five points per period is the absolute minimum to describe a sine wave but 10 points were needed to describe it adequately. Therefore the highest explorable frequency was less than or equal to ten times the sample frequency, resulting in 0 max= 2000 Hz (12566 rad s-l) for this example. Actual transients produced by sending a current step through a test circuit and recording the response on the transient recorder also proved to be sensitive to the quality of the data (noise). Three different methods of measuring these transients were adopted: (i> use of a single time transient on the transient recorder; (ii> use of three different time transients (i.e. repeating the experiment three times) and combination of the three transients by rejecting in transients 2 and 3 points that had already been recorded in transients 1 and 2 respectively; (iii) averaging several (up to 16) transients using either method (i) or method (ii). Noise and the use of several time domains to record the transient (see above) usually limited the highest frequency for which reasonable results could be obtained here to ca. 500 krad s-l. An example for a circuit with two subsections is given in Fig. 2. In summary, the Nyquist and Bode plots obtained for most test circuits investigated so far were of good quality and agreed well with the theoretical curves. In general, a larger number of samples (> 1024) was desirable. Clearly, data of excellent quality were needed for the imaginary axis Laplace transformation. They can be obtained by averaging if the system does not change too fast. The real axis Laplace transformation was much less sensitive to the quality of the data, and moreover subtraction of a constant plateau value at t -+ 03 was not needed for convergence so that meaningful results could be obtained for transients which did reach a plateau. This is shown in Fig. 3 where spectra of ZoP are calculated with (Fig. 3(a)) and without (Fig. 3(b)) subtraction of the final plateau value. The results were calculated using the same syn-
000
Fig. 3. The operational impedance Z,r obtained by Laplace transformation of synthetic transients: (a) calculated with subtraction of the plateau values of the transients; (b) calculated without subtraction. The transients were identical with that shown in Fig. 1 except for the length which was varied, giving rise to a series of spectra: A 1 s; B 0.1 s; C 0.01 s; D 0.001 s; E 0.0001 s; F 0.00001 s. In (a) the first two impedance spectra are superposed.
thetic transient as in Fig. 1 except that the length was varied from 1 s to 10 ps (see caption to Fig. 3). In both cases the high frequency range showed good values down to frequencies determined by the length of the transient. The uppermost curves in the two graphs (obtained from the transient with a length of 1 s) coincided with the theoretical curve over the whole frequency range displayed. With shorter transients, data transformed after subtraction were accurate at somewhat lower frequencies for identical transient lengths. For both methods (transformation with and without plateau subtraction) with integration up to a time which does not include the plateau, the condition for Laplace transformation (integration to infinity) was not fulfilled. Therefore these calculated values deviated from those of the ideal Laplace transform. It was found that the extent of the deviation upon truncation was higher
M. Neumann-Spallart, M. Etman / Calculation of Impedance Spectra
for the voltage function where the plateau value was not subtracted since this function exhibited a maximum. The deviation of the integral of the current function upon truncation was insensitive to the calculation mode. The combined trends led to the behaviour shown in Fig. 3: truncation was more serious for the non-subtraction mode, but a reasonably extended high frequency region of acceptable results existed for both modes. It is clear from this analysis that very fast operation is possible if only the high frequency domain is of interest. Also, extrapolation of Z,, to infinite frequency is easy with such data. In contrast, results represented in the complex plane and phase-angle plots (obtained from the imaginary axis Laplace transformation) sometimes provided a clearer picture of the general behaviour of the system under investigation, particularly if several connected R, II C subsections
3-l
were involved. In this case, system modelling was facilitated. We propose that the two methods are used together since they are complementary. Experiments are now under way to evaluate the complex impedance of electrochemical circuits and to compare the results with those obtained by conventional methods of impedance measurement. References 1 E. Levart and E. Poirier d’Ange d’Orsay, J. Electroanal. Chem. 19 (1968) 335. 2 A.A. Pilla, J. Electrochem. Sot. 117 (1970) 467. 3 M. Etman, C. Koehler and R. Parsons, J. Electroanal. Chem. 130 (1981) 57. 4 C. Gabrielli, M. Keddam, and J.F. Lizee, J. Electroanal. Chem. 205 (1986) 59. 5 M. Neumann-Spallart, unpublished results. 6 J. Ye and K. Doblhofer, J. Electroanal. Chem., 261 (1989) 11.
Journal of Electroanalytical Chemistry, 372 (1994) 33-37
Calculation of impedance spectra by Laplace transformation transients generated by current-step excitation Michael Neumann-Spallart
of voltage
a and Mohamed Etman b
a Laboratoire de Physique des Solides, CNRS, 1 place Aristide Briand, F-92195 Meudon (France) b Laboratoire d’Electrochimie des Interfaces, CNRS, I place Arktide Briand, F-92195 Meudon (France) (Received 16 April 1993; in revised form 15 October 1993)
Abstract Impedance data over a broad frequency range were obtained by converting the transient voltage response of a passive network to a current-step excitation into the frequency domain using Laplace transformation. The merits and limitations of the method and the type and quality of the required data are discussed. Examples of impedance plots obtained using this method are given.
1. Introduction
Impedance spectra are of great importance for the analysis of electrochemical interfaces and cells. Transient perturbations (single potential or current steps) probe the system during a very short period of time. This is particularly important for rapidly evolving systems, such as electrodes in a corroding electrolyte, systems where electrode and electrolyte compositions change (accumulators during charge-recharge cycles) and oscillating chemical systems. The response contains all the information needed for the calculation of the impedance Z of the system over the whole frequency range. Z can be extracted from transient data by using the Laplace transformation [l-4]. In this article we describe the calculation of impedance spectra from voltage transients generated by current-step excitations. The advantage of the current-step method for electrochemical cells lies in the accessibility of the high frequency range, since fast potential control is not necessary. On the contrary, this must be suppressed by deliberately slowing down the response of a potentiostat connected to the circuit [31. However, in the low frequency range (< 1 Hz for a reasonable damping of the potentiostatic response) potential control sets in and this precludes the use of data in the corresponding time domain. Therefore this method is complementary to the more frequently used voltage step method. 0022-0728/94/$7.00 SSDI 0022-0728(93)03265-Q
Single-event excitation waveforms and the response to such perturbations can be processed by Laplace transformation: U_,,(w) =
/,,f( t)
e-“’
dt;
Ii,
= irnf( t) e-j”’ dt
(1) where I,, and Ii, are the real and imaginary axis Laplace transforms respectively, o/r-ad s-l is the angular frequency, j = m and f(t) denotes either the excitation function i(t) or the response U(t). For the imaginary axis transformation, f(t) must be split into two parts before integration to guarantee convergence of the calculation. The constant value at t + 03 is subtracted from f(t) and transformed separately. The calculations are carried out for a series of predetermined frequencies in the range of interest for the particular circuit. In this way, time domain data are transformed into the frequency domain. The impedance is the ratio of the Laplace transforms of the potential and the current transients. The real axis Laplace transformation leads to the so-called operational impedance Z, and the imaginary axis Laplace transformation yields 2. Bode plots (log IZ Ivs.log(o) or log(Z,,) vs. log(o)) can be used to derive the values of series and parallel resistances and capacitances of simple circuits consisting of a resistance in series with sections of parallel resistance and capacitance (R,+ R,,(I C, +
0 1994 -
Elsevier Science S.A. All rights reserved
M. Neumann-Spallart, M. Etman / Calculation of Impedance Spectra
34
limiting value of log(Z,,) or log I 2 I at sufficiently high or low frequencies represents the series resistance and the sum of all resistances respectively. In log 1Z I plots, there are inflection points with tangents of slope - 1; the extrapolation of these to log(w) = 0 yields - log C. In log(Z,,) plots, however, the tangents depend on the circuit parameters and
the information on the impedance, including the phase angle, it can be extracted using imaginary axis Laplace transformation. We have applied this procedure to synthetic transients, different test circuits, and electrochemical interfaces.
C = (l/R;
The experimental arrangement has been described in detail elsewhere [3]. A Wenking 68FA0.5 potentiostat and a signal generator were coupled in parallel via large resistances to the cell which was connected in series to a calibrated measuring resistance. The potentiostat was used to control the working electrode potential, but its response was slowed down. A capacitor in parallel with the reference electrode ensured that the voltage transients to be measured were not suppressed, at least for a reasonable period of time (a few tenths of a second). The potential responses to a current step of both the series combination of the cell
R,, II C, + R,, II C, + . . . ). The
+ 1/R,R,)1’2/2wina
(2)
where minn denotes the frequency at an inflection point
[51. For more complicated circuits, sections corresponding to different R, 1)C subsections are not well resolved in log(Z,,) vs. log(o) diagrams. Thus additional information about the phase angle or a representation of the real and imaginary parts of the complex impedance is required for circuit analysis and modelling (plots of log(Z,,) and log(Zi,) as functions of log(w), or -.Zi, vs. Z,). As the transients contain all
Fig. 1. Example of the time dependence S = (0.5/o)C[U(ti)
2. Experimental
of the sum [61
+ U(ti+,)][Sin(oti+,)
- SiIl(wti)]
used in the calculation of the real part of the Laplace transform of a synthetic voltage transient: R, = 50 R, R, = 1000 0, C = 9.8 PF and I Sfep= 10 FA; dynamic spacing over seven orders of magnitude, each containing 90 equally spaced points, from At = 1 X lo-’ s to At = 1 X 10-l o=WOOrads-l;---~=2000rads-1.Upperinset, s. U(t) is the voltage response after subtraction of the plateau value U(t + 03).transient calculated using eqn. 3; lower inset, impedance spectrum. The points were obtained by Laplace transformation and connected with a solid line in order to show their sequence with increasing o.
35
M. Neumann-Spaliart, M. Etrnan / Calculation of Impedance Spectra
under test and, the measuring resistor and the cell alone were amplified and recorded simultaneously on the two channels of a Tektronix A2221 storage oscilloscope (2 x 2000 points, 8 bit resolution) and subsequently transferred to a microcomputer. The excitation current waveform was obtained from the two signals by subtraction. 3. Results and discussion It can be shown that the transient
response
R, + (RPI( C) circuit to a current step perturbation
of a Z is
given by U(t)=Z(
R,+R,)-ZR, exp( -t/R,C)
(3)
Transients with different circuit parameters were calculated using eqn. (3) and transformed by imaginary axis Laplace transformation using an algorithm described by Ye and Doblhofer [6] (The integrations were considerably simplified and precision at high frequency was enhanced by substituting U, = w-isin and ui = --w-i co4 ot) in eqn. (1)). The transients had either equally spaced data points or a fixed number of data points per decade of time in order to check the reaction of the integration algorithm to these features and the maximum frequency that could be analysed. After transformation and calculation of the impedance, transients with equally spaced data points with sampling
times up to 2 Z.LSgave smooth Nyquist plots up to 600 krad s-l coinciding perfectly with the theoretical curves. Also, transients with a fixed number of data points per decade (typically 10 points per decade), which had the advantage of requiring a much smaller number of points for the whole transient, gave good results. However, in the latter type of transient, aberrant points in the impedance spectrum (lower inset in Fig. 1) became quite abundant above a certain frequency. We analysed the reason for the production of these aberrant points in detail. In the calculation, the integral in eqn. (1) was solved by numerical integration for each frequency. The intermediate sum of the integration values was a convergent oscillating function, (e-jmt can be written as the sum of sine and cosine terms). Its frequency was given by the o for which the transformation was carried out. At high w, upon transition to the next domain of (larger) data spacing the density of data points was no longer sufficient to match the oscillation in o. The oscillation then appeared with a lower frequency (aliasing). Depending on the actual value of the response function, this may have caused a sudden jump in the value of the integral, leading to an aberrant value in the result (see an example in Fig. 1 for o = 5000 rad s-‘; the aberrant point occurring with the spacing transition is marked by an arrow>. The risk of this occurrence becomes increasingly important with increasing frequency w used for the
10
rad/s
100 ,,,,,,,,,,,,,,,,,,,,,.,,,,,,.,~....~.,,,,,,,~,,,!.~,~~,,,l krad/s
0”
400
800
1200 Re(Z/ n)
Fig. 2. Example of a Nyquist plot calculated by Laplace transformation
1600
2000
2400
of a transient obtained by current-step perturbation of a test circuit with
R, = 50 0, R,, = R,, = 1000 a, C, = 1 PF and C, = 10 pF. The points were calculated from experimental data and the solid line was calculated
from the values of the elements in the circuit.
36
M. Neumann-SpaNart, M. Etman / Calculation of Impedance Spectra
calculation. At sufficiently low w, spacing transitions only lead to a more irregular course of the function (Fig. 1, broken line, w = 2000 rad s-l>. This analysis allowed us to determine the spacing necessary to calculate the Laplace transformation up to a given frequency o, or to determine the maximum explorable frequency for a circuit with given parameters and the available sampling rate and memory of a transient digitizer or digital storage oscilloscope. According to eqn. (31, the relaxation time RC for the circuit in Fig. 1 was 0.01 s. The recorded length of a transient should be not much less than five times the time constant in order to reach a value close to the plateau; in our example this is 0.05 s. For instance, if 1000 samples are collected, this would correspond to a time spacing of 5 X 10m5 s (= 20 kHz). Five points per period is the absolute minimum to describe a sine wave but 10 points were needed to describe it adequately. Therefore the highest explorable frequency was less than or equal to ten times the sample frequency, resulting in 0 max= 2000 Hz (12566 rad s-l) for this example. Actual transients produced by sending a current step through a test circuit and recording the response on the transient recorder also proved to be sensitive to the quality of the data (noise). Three different methods of measuring these transients were adopted: (i> use of a single time transient on the transient recorder; (ii> use of three different time transients (i.e. repeating the experiment three times) and combination of the three transients by rejecting in transients 2 and 3 points that had already been recorded in transients 1 and 2 respectively; (iii) averaging several (up to 16) transients using either method (i) or method (ii). Noise and the use of several time domains to record the transient (see above) usually limited the highest frequency for which reasonable results could be obtained here to ca. 500 krad s-l. An example for a circuit with two subsections is given in Fig. 2. In summary, the Nyquist and Bode plots obtained for most test circuits investigated so far were of good quality and agreed well with the theoretical curves. In general, a larger number of samples (> 1024) was desirable. Clearly, data of excellent quality were needed for the imaginary axis Laplace transformation. They can be obtained by averaging if the system does not change too fast. The real axis Laplace transformation was much less sensitive to the quality of the data, and moreover subtraction of a constant plateau value at t -+ 03 was not needed for convergence so that meaningful results could be obtained for transients which did reach a plateau. This is shown in Fig. 3 where spectra of ZoP are calculated with (Fig. 3(a)) and without (Fig. 3(b)) subtraction of the final plateau value. The results were calculated using the same syn-
000
Fig. 3. The operational impedance Z,r obtained by Laplace transformation of synthetic transients: (a) calculated with subtraction of the plateau values of the transients; (b) calculated without subtraction. The transients were identical with that shown in Fig. 1 except for the length which was varied, giving rise to a series of spectra: A 1 s; B 0.1 s; C 0.01 s; D 0.001 s; E 0.0001 s; F 0.00001 s. In (a) the first two impedance spectra are superposed.
thetic transient as in Fig. 1 except that the length was varied from 1 s to 10 ps (see caption to Fig. 3). In both cases the high frequency range showed good values down to frequencies determined by the length of the transient. The uppermost curves in the two graphs (obtained from the transient with a length of 1 s) coincided with the theoretical curve over the whole frequency range displayed. With shorter transients, data transformed after subtraction were accurate at somewhat lower frequencies for identical transient lengths. For both methods (transformation with and without plateau subtraction) with integration up to a time which does not include the plateau, the condition for Laplace transformation (integration to infinity) was not fulfilled. Therefore these calculated values deviated from those of the ideal Laplace transform. It was found that the extent of the deviation upon truncation was higher
M. Neumann-Spallart, M. Etman / Calculation of Impedance Spectra
for the voltage function where the plateau value was not subtracted since this function exhibited a maximum. The deviation of the integral of the current function upon truncation was insensitive to the calculation mode. The combined trends led to the behaviour shown in Fig. 3: truncation was more serious for the non-subtraction mode, but a reasonably extended high frequency region of acceptable results existed for both modes. It is clear from this analysis that very fast operation is possible if only the high frequency domain is of interest. Also, extrapolation of Z,, to infinite frequency is easy with such data. In contrast, results represented in the complex plane and phase-angle plots (obtained from the imaginary axis Laplace transformation) sometimes provided a clearer picture of the general behaviour of the system under investigation, particularly if several connected R, II C subsections
3-l
were involved. In this case, system modelling was facilitated. We propose that the two methods are used together since they are complementary. Experiments are now under way to evaluate the complex impedance of electrochemical circuits and to compare the results with those obtained by conventional methods of impedance measurement. References 1 E. Levart and E. Poirier d’Ange d’Orsay, J. Electroanal. Chem. 19 (1968) 335. 2 A.A. Pilla, J. Electrochem. Sot. 117 (1970) 467. 3 M. Etman, C. Koehler and R. Parsons, J. Electroanal. Chem. 130 (1981) 57. 4 C. Gabrielli, M. Keddam, and J.F. Lizee, J. Electroanal. Chem. 205 (1986) 59. 5 M. Neumann-Spallart, unpublished results. 6 J. Ye and K. Doblhofer, J. Electroanal. Chem., 261 (1989) 11.