Proceedings of the International Workshop “Advanced Techniques for Energy Sources Investigation and Testing” 4 – 9 Sept. 2004, Sofia, Bulgaria

THE TECHNIQUE OF THE DIFFERENTIAL IMPEDANCE ANALYSIS Part I: BASICS OF THE IMPEDANCE SPECTROSCOPY

Daria Vladikova

Institute of Electrochemistry and Energy Systems– Bulgarian Academy of Sciences

Acad. G. Bonchev Str., bl.10, 1113 Sofia, BULGARIA, e-mail: d.vladikova@astratec.net

Abstract

This review paper gives basic knowledge about the Electrochemical Impedance Spectroscopy. The main working hypotheses for its correct performance are discussed. A detailed presentation of the structural modelling is made. A brief introduction of the basic elements and electrochemical models is done. The different steps of the impedance data analysis: data pre-processing, parametric identification and selection of the best model are all described in more details. Keywords: Electrochemical Impedance Spectroscopy, working hypotheses, data analysis, structural modelling, parametric identification, electrochemical models. 1. Introduction Although successfully developed for application in electrochemical systems [1-9], the Electrochemical Impedance Spectroscopy (EIS) may provide useful and reliable information about the object under study only when the impedance investigation is performed correctly. The major difficulties originate from the complexity of electrochemical systems. Thus, the successful performance of the impedance investigation needs adherence to some basic requirements.

The impedance method is based on the classical method of transfer function (TF). The system under investigation is perturbed with a sinusoidal wave input and the response is measured at the output*. If the system is linear, the response is also sinusoidal with the same frequency and a different phase and amplitude. The ratio between the response and the input signal determines the complex transfer coefficient for the corresponding frequency. The frequency dependence of this coefficient defines the transfer function of the system. When determined in a sufficiently wide frequency range, the TF describes entirely the dynamic properties of the linear system. Defined in this way, the TF represents the steady-state, as well as the non-steady state properties of the system. Its analysis needs Laplas Transforms. If the system is in a steady-state for the measured frequencies, the response is also steady-state. Then the Laplas Transforms can be replaced with the simpler Fourier Transforms. In this case the TF is formulated as the ratio of the response to the input signal obtained in the frequency domain. It describes totally the properties of a linear, steady-state system. The direct application of the classical TF method for electrochemical systems is impossible due to their non-linear behaviour. The problem solving approach is based on the theory of Friedhol-Voltera and lies in the local application of the theory of linear systems. It allows the approximation of the non-linear system with linear terms, if the equivalent linear equations are known at every point of its steady-state non-linear characteristic [10-11]. In

* From a theoretical point of view any type of perturbing signal – white noise, step, pulse, or sin wave may be applied for analysis of linear systems [2]. In practice, however, the sinusoidal wave signal is found to be the most suitable for the electrochemical impedance technique.

L8-1

electrochemical systems the local analysis is implemented by measuring the Transfer Function with small amplitude of the perturbation signal, taking into account only the linear part of the response. If the current is applied as an input and the voltage – as an output signal, the TF is identified as impedance Z (iω). In the opposite case its quantity is admittance Y(iω) = Z-1(iω). The interpretation of EIS as a TF determines it as a local linearised and full (in a frequency aspect) description of the investigated electrochemical system, which is assumed to be steady-state. Real electrochemical objects, however, behave as large statistical systems with distributed parameters in macro and micro scale. During experimentation, processes of mass and energy transfer could take place. They change the object’s structure and parameters. Thus, the system can show non-linear, non steady-state behaviour and memory properties. Obviously, the impedance analysis of a complicated electrochemical system needs many simplifications, which can be generalised in a number of working hypotheses [2,9,12]. 2. Working Hypotheses

The working hypotheses may be subdivided into two categories. The first one concerns the system analysis. The second one refers to the electrochemical behaviour of the system. 2.1. Working Hypotheses concerning the System Analysis Linearity. As it has already been mentioned, the requirement for linearity could be fulfilled, if the amplitude of the input sine wave perturbation signal is small enough to keep the selected state of the system unchanged. The measuring instrument, however, should analyse only the linear component of the output signal frequency spectrum. The requirement for small amplitude of the perturbed signal concerns the potential and the current, as well as the quantity of electricity running in one direction for one half of the period. The last requirement is extremely important in the infra-low frequency range. Steady state. The measure of the complex transfer coefficient should be independent from the moment of the measurement. This requirement has to be valid for the whole frequency range. Finiteness. The real and imaginary parts of the impedance should take finite values over the entire frequency range. Single input, single output. This requirement could be achieved if only the rest of all the parameters – temperature, concentration, pH, etc. are kept constant by passive or active conditioning. Lack of memory properties. The system should not “remember” the history of the experiment and thus the result would be independent from the sequence of measurements. Observability. The investigated phenomena should be observable in the measured frequency range.

2.2. Working Hypotheses from an Electrochemical Point of View Additiveness of the Faradaic current and the charging current of the double layer. Electroneutrality of the electrolytic solution. Practically this assumption implies that the total density of the charges in every point of the solution could be accepted as zero. Lack of convection and migration. This assumption leads to the concept that there are no other changes in the local concentration of the electrolyte than those caused by the diffusion or charge transfer. Lack of lateral mass and charge fluxes at the electrode surface. The well designed experiment must obey all working hypotheses. The applied methods of measurement and instrumentation technique are responsible for the system analysis hypotheses.

L8-2

The simplifications defined by the second type of hypotheses help for the construction and solution of the theoretical electrochemical impedance model. It is obvious that the experimental conditions determine the precision of the EI experiment.

When the object obeys all the hypotheses and is in addition causal, i.e. all changes in the investigated system are caused only by the perturbing signal, then the Hilbert Transform can also be applied. In accordance with this theory only one of the impedance (TF) components (real or imaginary) is sufficient for the full description of the object. This theory, known in practice as Kramers-Kronig (K-K) relationship, is frequently used for verification of the impedance data, or more precisely – for verification of the assumed working hypotheses. The value of the second component is calculated from the first one with the help of Hilbert Transform and compared with the measured one. More detailed information may be found in [12-18]. It is worth mentioning that (K-K) transforms are valid only for a class of electrochemical systems, which obeys the requirements for causality, stability, linearity, steady-state and finiteness. 3. Trends in the Development of EIS The richness, complexity and importance of the electrochemical systems, as well as the sophistication of the experimental technique, increase permanently the demands for improving the analytical power of EIS. Although introduced for application of the impedance technique in electrochemical systems, the necessary working hypotheses could also be regarded as restrictions towards the experiment. A trend in the development of EIS is the elimination of the restrictions imposed by a given working hypothesis. Thus, the rejection of the requirement for linearity of the investigated system, i.e. the examination of the system as actually non-linear, brought to the development of the non-linear impedance [9]. The release from the restriction for steady–state conditions boosted the elaboration of the non-stationary impedance analysis [29-37]. The consideration of the system with a single input and multiple outputs permitted the development of the multi-transfer function analysis. This approach combines the classical impedance method with hydrodynamic impedance, electro-coulometry, electro-gravimetry, electro-optometry and other techniques [28-37], which increases the information capabilities of the electrochemical investigations. 4. Theoretical Modelling and Identification The information yield of EIS is accumulated in the impedance or admittance functions. From one side they contain all the information, which could be collected from a linear system, subjected to a time-dependent electrical perturbation, in case that the working hypotheses, necessary for this analysis, are obeyed. Thus the electrochemical impedance has the unique possibility to separate the kinetics of the different steps involved in the total process under investigation, because as a transfer function it gives a local, linear and full description of the system under study. From another side this information has to be extracted from the data. A number of processes, caused by the perturbation signal are taking place. The impedance, however, does not measure them, i.e. it is not a physical reality, but information property of the object. The problem-solving tool for the best interpretation of the experimental data is the construction of a working model. As the aim of impedance modelling is knowledge enhancement, the applied models should be of a physical type, i.e. they should describe the properties of the processes taking place in the object under study. The model could be regarded as a rational presentation of the existing phenomena and of the preliminary knowledge. In

L8-3

accordance to the system theory the construction of the model should be presented as an identification procedure.

In principle, the impedance data analysis may follow two different pathways: • Confirmation of a preliminary derived hypothetical model. This approach, known as

theoretical, dominates in EIS. Since the model structure is chosen a priori, the identification procedure is only parametric.

• Derivation of the working model from the experimental data, known as experimental approach. In this case the identification is both structural and parametric. It will be discussed in part II.

The essential part of the theoretical approach is the construction of a model, which should describe a preliminary stated theoretical hypothesis about the processes taking place in the object under investigation. Besides, two different procedures may be applied: classical (method of the full electrochemical and mathematical solution) and structural (construction of a mathematical model consisting of elements and digital solution of its impedance behaviour). 4.1. Classical Modelling The Classical Modelling derives a theoretical expression for the impedance of the investigated object, based on some considerations about the processes occurring at the electrode interface. It starts from an initial set of integral-differential equations, which describe the space distribution of the state variables potential E(N,t) and the concentration of species i Ci (N,t) at point N and time t. Their solution needs further mathematical simplification, which is obtained by applying the initial electrochemical hypotheses. Although the solution of the final impedance equation may follow various approaches, differing in the assumptions defining the model [2, 38-46], most of them obey the same sequence of procedures:

• A choice of the initial and boundary conditions. • Linearisation. The procedure is performed applying a small perturbation around the

working point. The corresponding response is obtained by Tailor’s series expansion. The effectiveness of the procedure is determined by the small magnitude of the perturbing signal. It should ensure minimisation of all terms in comparison to the first one, which is linear.

• Conversion of the system from the time domain to the frequency domain by the Laplas transform.

• Mathematical or numerical solution of the new set of equations. The final expression should give the frequency dependence of the real and imaginary components of the impedance.

The total procedure of the classical modelling and its application for the construction of some basic kinetic models is described in details in [2] (it can be downloaded free of charge from the web site of the European Internet Centre for Impedance Spectroscopy http://accessimpedance.iusi.bas.bg). 4.2. Structural Modelling The structural Modelling assumes that the mathematical impedance model can be represented directly in the frequency domain as a construction, consisting of elements. They are connected under different laws in accordance with the real behaviour of the system under investigation. Every element should describe a single physical process, taking place in the impedance object. Thus, the full model could be constructed as an electrical circuit corresponding to a given mechanism (Fig. 1a).

L8-4

An important advantage of the structural models is the reduced number of parameters, which makes them totally identifiable. They are also efficient and easy for manipulation. The structural modelling can be sophisticated by introducing a set of new and modified electrochemical elements as well as elements, representing selected mathematical functions [47]. The main electrochemical elements and models are given in Section 5.

0 100 200 300 4000

100

200b)

1mHz10mHz

0.1Hz

10Hz

100Hz 1Hz

- Im

/ Ω

Re / Ω

R2 R1 a)

C1 C2

c)

lg|Z

|

lgω

φ 0

Fig. 1. Equivalent circuit of two time constants model with Voigt’s structure [R/C R/C] (a)

and its graphical impedance presentation: complex plane impedance diagram (b); Bode plots (c).

4.3. Impedance Data Analysis The aim of the impedance analysis is identification of the appropriate working model, which corresponds to the experimental data. This final and extremely important stage of the impedance investigation includes few consecutive steps. They start from data pre-processing and finish with the choice of the appropriate model. Data pre-processing

In order to achieve a better identification, the experimental results should be evaluated in advance. The exactness could be increased by a preliminary evaluation of the experimental data quality. During this procedure, known as data pre-processing, the experimental errors are estimated and corrected. It is worth mentioning that the problems of data pre-processing are frequently underestimated. Data pre-processing includes several procedures: Data monitoring. The problem of impedance monitoring concerns the 3-dimensional nature of the data, which should be plotted in a 2-dimensional plane. The experimental set Z (Rei, Imi,

L8-5

ωi), ( i = 1,2, . . . N) of N points measured at different frequencies ωι, where Rei and Imi are the real and the imaginary part of the impedance for ωι, are presented typically in Cartesian co-ordinates

Z (iωi) = Rei+iImi , (1) or in polar co-ordinates

Z (iωi) =Z e ,iiφ (2)

where Z= (Rei

2 + Imi2)1/2 is the modulus and ϕi = Arc tg Imi/Rei is the phase, corresponding to

a given frequency [2,9,48]. The most common graphical representation of the experimental impedance data is the complex-plane impedance (Nyquist) diagram. It describes the dependence fi(Rei, Imi) in Cartesian co-ordinates (Fig.1b). The plot is more illustrative when the decade points and especially those, corresponding to 10-3,100, 103, 106 Hz, are marked in an explicit way. The Bode diagrams are an alternative plot [2,9,48]. They use polar co-ordinates presentation. Two plots are applied, which describe the dependencies lgZ/lgωi and ϕi/lgωi (Fig.1c). In some cases additional information may be extracted, if the data are presented in complex-plane plots of the admittance Y , of the capacitance (complex permittance) Yω-1 and elastance (complex modulus of the permittance) ωΖ [48-50]. Calibration of the measuring cell and instruments. The measurement cell contains the object under investigation and ensures the connections with the electrochemical interface. In order to meet the requirement of the basic working hypothesis for absence of lateral fluxes, the measurement cell should be of a planar or of a coaxial cylindrical configuration. Even with a perfect geometry, the measurement cell contains parasitic elements, which participate in the measurement circuitry and can deteriorate the accuracy of the measurements, especially at high frequencies. Their influence should be taken into account for the evaluation of the errors, and when possible, for errors correction. The most simplified consideration of the high frequency errors is the cell representation with a simple parasitic element. For low impedance objects the errors are dominated by the cell parasitic inductance Lp. It comes mainly from the electrical leads self-inductance. The presence of Lp produces typical shapes with positive imaginary values and can deform significantly the object’s impedance shape (Fig. 2b). The influence of Lp can be easily calculated and corrected.

The evaluation of the parasitic inductance is performed by calibration measurements: • Measurement at short circuit – evaluation of the cables impedance/inductance; • Measurement of a dummy object with high conductance – evaluation of the object’s

self-inductance. The calibration should be carried out with a fixed configuration of the measurement circuitry and cables, which should be used in the following object’s measurements. Fig. 2 shows the calibration procedure.

L8-6

0 30 60

-30

0

30

- Im

/ Ω

Re / Ω

a)

0 30 60

-30

0

30

- Im

/ Ω

Re / Ω

b)

0 30 60

-30

0

30

- Im

/ Ω

Re / Ω

c)

Fig. 2. Correction of the parasitic inductance of YSZ sample: calibration measurement at short circuitry (a);

measurement of the sample (b); impedance diagram of the sample after the correction (c). Filtration. The aim of this procedure is to detect and eliminate data points, which are strongly deteriorated by errors [9, 51-53]. Their use for identification of the model would only decrease the precision of the analysis. It is known that measurements at frequencies near to the power supply frequency, as well as to its harmonics – 50, 100, 150, 250 Hz, produce frequently erroneous data. These frequencies should be avoided in the frequency-scanning program or the corresponding points should be rejected from the experimental results when possible. Another type of erroneous data are the so-called “wild” points. This term denotes a point, which does not obey the general behaviour of a given ensemble of points (Fig. 3). The classification of an experimental point as a wild point should follow certain criteria. An effective filtration of the wild points can be performed by analysis of the local Tailor Spectra of the real and imaginary sets of data [9].

0 3 60

3

10 Hz

0.01 Hz

0.1 Hz

1 Hz

- Im

/ kΩ

Re / kΩ Fig. 3. Synthetic impedance diagram with one “wild” point (10 % deviation of Re).

L8-7

Data quality evaluation. This procedure serves for quick and direct monitoring of the experimental data quality. It could be applied also for effective filtration, as well as for weighting of the data during the model identification. One of the possible approaches for data quality evaluation is based on the property of electrochemical impedance objects to be quiet, resonance free, frequency continuous systems. Therefore, the dependencies of the impedance components (Re and Im) in respect of the logarithm of the frequency should exhibit smooth behaviour and the appearance of sharp deviations can be regarded as an indication for the presence of erroneous data (Fig. 4).

Fig. 4. Impedance diagram of polycrystalline solid electrolyte, deteriorated by two wild points “wp” and the correspondent data quality diagram.

The evaluation is also based on the respective Tailor’s spectra. More information about the data quality evaluation may be found in [9,47,49,51-54]. Validation of the System Analysis Hypotheses In this stage of the data analysis a validation of the initial hypotheses for linearity, steady- state and absence of memory is performed. When the investigated system obeys the requirement for linearity, the impedance should be independent from the amplitude of the input signal. This property is applied for experimental examination of the conditions for linearity. Consecutive impedance measurements with increased amplitude of the perturbation signal are performed in the whole frequency range [9]. The mean of the weighted differences between two series of measurements is used as a direct measure for the validity of the hypothesis for linearity. Although the criterion for a small magnitude of the perturbing signal is accepted to be 5 ÷ 3 mV, this range of values should be accepted as a simplification. In some experiments [9] the requirement for linearity is fulfilled only at amplitudes of 50 100 µV. ÷ The hypothesis for steady-state can be experimentally tested by performing two consecutive experiments at the same conditions [9]. In this case, the weighted differences, which are frequency dependent, can be used again as measures for the steady state. Obviously, the most pronounced deviation from the steady state will be observed at the lowest frequencies and thus the acceptable rate of errors will limit the low frequency range. Electrochemical systems are a priori very sensitive to memory effects. They originate from the typical for the electrochemistry connection between the charge transport and the mass transport. The charge carriers are ions. Their accumulation can lead to a formation of a new phase and to its growth. This phenomenon can change drastically the morphology and the behaviour of

L8-8

the electrode. In practice, the presence of a memory effect may be estimated by implementation of two consecutive experiments, which differ only in the direction of the frequency scanning [9]. The weighted differences are applied again as a measure for the validity of the working hypothesis. The appearance of a pronounced hysteresis is an experimental evidence for the presence of memory effect. Parametric Identification The identification is the most important step of the theoretical approach for data analysis. In the common case, the model may be presented by the matrix M, which includes the structure matrix SM and the parameters vector PM , with the unknown values of the parameters p1,…, pj,…, pm [7,9,4]:

M [ SM , PM ] (3) The structure is described with the elements EM and their connections CM:

SM [EM, CM] (4)

When the model is known by its structure and parameters, the corresponding impedance is produced by simulation (Simul.) [7]:

Z(iωi) = Simul. [M | ωi] , (5)

where the frequency range ωι, i = 1,2,…., N is predetermined. The opposite goal is the identification. In this case, the impedance is known and presented by the experimental data set:

D[ωi, Rei, Imi]. (6)

The theoretical approach accepts the model structure known in advance as a working hypothesis. The identification in this case is only parametric [9,13,47]. The procedure may be described as an estimation of the parameter’s vector P :

MP = Par.Ident.[ωi, Rei, Imi| SM ], (7) i = 1, 2, …, N,

where SM is the assumed or supposed model structure. The final procedure of the parametric identification is the model validation [9]. It includes the following steps:

• Model simulation. The impedance of the model can be calculated by simulation, using the estimates of the parameters [ , …, ]: MP 1p mp

Z (iω) = Simul. [ωi, SM, ]. (8) MP

The application of the structural simulation [7, 9] allows for the synthesis of the model’s impedance behaviour, taking into account only the structure and parameters and avoiding the solution of the impedance equations.

L8-9

• Selection of a criterion for proximity. Usually the distance ∆ i between the initial data (6) and the estimated model data (8)

∆ i(iω) = Φi ( - ΖiZ i) (9)

is accepted as a measure of the model proximity to the object. The function Φi can be either a direct difference, or a weight of a more sophisticated function [4,6, 8,9, 49-51,55,56].

• Analysis of the residuals. It provides information about the adequacy of the selected a priori model. The distances ∆ i(iω) are complex and frequently are called “residuals”. The analysis of the residuals as a function of “log f” is the last necessary and important step. This analysis must show that all the residuals have properties of a noise (Gausian noise in the ideal case). When this requirement is obeyed, the model corresponds to the object. In the opposite case, when significant deviations are observed in a given frequency range(s), the model does not correspond to the object, i.e. the assumed model structure is not adequate.

Selection of the best model After testing of a number of possible models, the best one should be selected. It corresponds to the “best fit”, which gives a minimum value of the functional distances. Usually the mean square of the weighted distances is used as a general criterion:

2∑∆i ⇒ min. (10)

In the general case, the parametric identification (7) requires solution of complex nonlinear equations. The most frequently applied algorithm is the complex nonlinear least square (CNLS), based on the Newton-Marqueward method [57]. For practical use, professional software tools have been developed [8, 58-63]. Discussing generally the “classical” parametric identification, two main features must be emphasised:

• The identification procedure (7) is performed simultaneously over the entire frequency range;

• The measure of the model proximity (10) is mean and integral.

The main problem of the theoretical approach is the need of a preliminary set-up of working hypotheses. Sometimes the accepted assumptions are supported by new physical considerations, but frequently the hypotheses are generated in a more or less subjective manner or are induced by a supposed similarity. In some cases there are no theoretical models. A typical situation is the investigation of a new object, about which little is known. That makes the development of a theoretical model impossible. The situation is similar when the investigation concerns an object of practical interest, like batteries, when knowledge-based models are an exception, rather than the rule. The discrepancy between the power of the impedance measurement technique, which ensures a large volume of precise data, and the interpretation of the results, often based on subjective and oversimplified hypotheses, is the main driving force for the development of the principally new structural identification approach for impedance data analysis (part II).

L8-10

5. Basic Electrochemical Models

In the impedance studies of electrochemical systems three main structures of electrical models are known and used:

Maxwell structure. In this model the currents flowing through the branches of the model are additive – their sum forms the total current. As a consequence

Y (iω) = ∑ Yk (iω), (11)

]

where Y(iω) is the total admittance and Yk(iω) denotes the admittances of the individual branches. Maxwell structure is used for modelling of parallel processes. Fig. 5 presents a model

with Maxwell structure. Its impedance is expressed as:

(12) ( )[11

2

11

11R )i(

−−

=

−−

+++= ∑n

kkk CiRCiZ ωωω

R3 C3

R1

R2 C2

C1

Fig. 5. Maxwell structure. Voigt structure. This model consists of meshes with impedances Zk(iω), connected in series. The total impedance is

Z (iω) = ∑ Zk (iω). (13)

The flowing current is equal for all meshes and the phenomena modelled by each mesh start instantaneously. Their rates depend only on their own time-constants.

Voigt model is used basically for description of the impedance of solid state bulk samples. Fig. 6 presents a model with Voigt’s structure. Its impedance is described as:

(14) ( 1

1

1−

=

−∑ +=n

kkk CiR)i(Z ωω )

R1

C1

R2

C2 C3

R3

Fig. 6. Voigt structure.

L8-11

L8-12

Ladder structure. This structure consists of a number of kernels correspondent to the modelled phenomena, which occur consequently. The model has a typical “ladder” structure. Its total impedance is

( )1111

431

21 ...)i()i()i()i()i(−−−−−

++++= ωωωωω ZZZZZ (15)

where Zi (iω) denotes the impedances of the ladder elements. The impedance of a ladder model with a structure presented in Fig. 7 is:

...i1

1i

1)i(

22

1

1

++

++=

CR

CRZ

ω

ωω . (16)

Fig. 7. Ladder structure.

As it was seen in Section 4, there are few approaches for presentation of the impedance models. The structural (electrochemical circuit) modelling approach is very convenient for impedance studies of electrical properties. In this case the electrical circuit has a response identical to that obtained from the measurement of the investigated system.

The electrochemical circuit can be regarded as a construction of different electrical and electrochemical elements (structural elements) connected under given laws. If the model is not formal, the values of its elements could give a significant contribution to the physical understanding of the investigated system.

5.1. Modelling Elements

The Impedance elements are described with one or more parameters, in accordance with their dimension (Table 1). The impedance elements can be divided in 2 basic groups:

• Lumped elements: resistance R; capacitance C; inductance L. They are directly adopted form the electrical engineering, i.e. they are electrical elements and can describe only homogeneous systems.

• Frequency dependent elements – they describe frequency unhomogeneity. They are developed for description of some electrochemical processes, i.e. they are electrochemical elements.

Resistance. Its impedance (Fig. 8) is given as

ZR (iω) = R , (17) i.e. Re = R and Im = 0.

C1 C2

R2 R1

C3

R3

0 10 20

-10

0

-Im

/ Ω

Re / Ω

10 mHz

1mHz

0 9 18

0

9

10 mHz

-Im

/ Ω

Re / Ω

1mHz

Fig. 8. Impedance diagram of Resistance simulated in the frequency range 103 ÷10-3 Hz (R = 90 Ohm , 200 Ohm, 300 Ohm, 400 Ohm).

0 100 200 300 4000

100

200

-Im

/ Ω

Re / Ω

Capacitance. Its impedance (Fig. 9) is described as:

ZC (iω) = (iω C)-1 = - i(ω C)-1 , (18) i.e. the capacitance has only an imaginary part (Re = 0). Its values are negative at positive C, which corresponds to a 90o phase retardation.

Fig. 9. Impedance diagram of Capacitance simulated in the frequency range 103 ÷10-3 Hz (C = 1E-3 F). Inductance. Its impedance (Fig. 10) is:

ZL (iω) = iωL . (19)

It has only an imaginary part (Re = 0) with positive values at positive inductance. Fig. 10. Impedance diagram of Inductance simulated in the frequency range 103 ÷10-3 Hz (L = 1E-3 H).

L8-13

The frequency dependent elements are developed especially for the description of some electrochemical processes, i.e. they have electrochemical origin.

Warburg element (W) [2,9,48,65] is the first electrochemical element, which has been introduced in the impedance description of linear semi-infinite diffusion, obeying the second Fick’s law. Its impedance (Fig 11) is presented as:

Zw(iω) = σ (iω )-1/2 , (20)

where the coefficient σ is known as the Warburg parameter (Table 1). The values of the real and imaginary components of the impedance are equal and thus the phase shift is 45o and it is frequency independent.

.

0 6 120

6

121mHz

10mHz

- Im

/ Ω

Re / Ω

0,1Hz

Fig. 11. Impedance diagram of Warburg element simulated in the frequency range 103 ÷10-3 Hz (σ = 400 Ω /s1/2).

Bounded Warburg element (BW) is introduced in impedance description of linear diffusion in a homogeneous layer with finite thickness [9,48]. Its impedance description (Fig. 12) as a structural element is:

ZBW (iω) = (iω )-1/2 th(iωR02/σ)1/2 , (21)

where σ and R0 are two independent structural parameters, which can be determined from the impedance analysis (Table 1).

0 1000

100

100Hz

10Hz

1Hz

- Im

/ Ω

Re / Ω

0,1Hz

Fig. 12. Impedance diagram of Bounded Warburg element simulated in the frequency range 103 ÷10-3 Hz (σ = 0,01

Ω /s1/2, R0 = 100 Ω).

L8-14

Constant Phase Element (CPE). It represents an empirical relationship, which is introduced for the description of the frequency dependence, caused by surface roughness or by non-uniformly distributed properties of the irregular electrode surface [8,9,48,66-68]. Its impedance (Fig. 13) is described as:

ZCPE (iω) = A-1 (iω )-n, (22)

where A is a factor of proportionality and n is the CPE exponent (Table 1), which characterises the phase shift.

.

1Hz0,1Hz

0 60

6

n = 0.8- I

m /

Ω

Re / Ω

n = 0.2

n = 0.5

R0=100 Ω

Fig. 13. Impedance diagram for CPE simulated in the frequency range 103 ÷10-3 Hz at different values of n (A = 100).

CPE is a generalised element, since it can model R, C, L, W, as well as a distortion of these elements, which is following a definite type of frequency dependence.

Bounded Constant Phase Element (BCP). It represents the impedance of a bounded homogeneous layer with CPE behaviour of the conductivity in the elementary volume and a finite conductivity at d.c. (ω 0) [9,47,69]. Its impedance (Fig. 14) is described as: →

ZBCP (iω) =A-1(iω )-n th[R0A(iω )n] , (23)

where A, n and R0 are the structural parameters of the element (Table 1).

.

0 100 200 300 4000

4000.1 Hz 1 Hz

R0=400 ΩR0=300 ΩR0=200 Ω

- Im

/ Ω

Re / Ω

n = 0.45

Fig. 14. Impedance diagram of BCP simulated in the frequency range 103 ÷10-3 Hz

at different values of R0 (A = 0,01).

L8-15

BCP is the most generalised explicit model for homogeneous layers, as it can represent BW as well as CPE with its transformations in R, C, L and W. At high enough frequencies BCP tends to the classical CPE. Table 1. Description of Impedance Elements Element No of Parameters Parameters Units (Dimensions) R 1 R Ohm C 1 C Farad L 1 L Henry W 1 σ Ohm/s1/2 CPE 2 A, n *, - BW 2 σ , R0 Ohm/s1/2, ohm BCP 3 A, n, R0 *, - , Ohm

* depends on the value of n 5.2. Physical Meaning of the Modelling Elements

The modelling elements have exact mathematical descriptions. Their physical meaning, however, has a much broader sense and depends on the modelling approach and on the physical nature of the phenomena which they represent. The simplified list of some possible interpretations of the most commonly applied modelling elements is given below:

• Resistance – energy losses, dissipation of energy, potential barrier, proportionality between the state parameters, electronic conductance or conductance due to very fast carriers, unobservable in the accessible frequency range; • Inductance – accumulation of magnetic energy, self-inductance of current flow or of charge carriers movement; • Capacitance – accumulation of electrostatic energy, accumulation of charge carriers, integral relation between the parameters; • Warburg – pure linear, semi-infinite diffusion; • Bounded Warburg – pure linear diffusion in a layer with a finite thickness and an ideally reversible Faradaic reaction at the end of the layer; • CPE – exponential distribution of time-constants.

5.3. Model Description For easier presentation of the models a symbolic description is introduced. The applied convention includes the following components:

• Structures – La (Ladder); Vo (Voigt); Ma (Maxwell); • Elements: R, C, L, W, BW, CPE, BCP; • Connections: “ “ – in series; “/” – in parallel; • Parameters – dimensions in SI (Table 1), delimiters – “;”, separator of multiple parameters element “\” (the values are ordered in accordance with Table 1).

Example: M = La: R CPE/R. par: 120; 1E-2\0.8; 1000 5.4. Basic Electrochemical Models

The modelling elements correspond to the simplest mathematical relations of proportionality, integration, differentiation, partial differentiation, etc., expressed in the frequency domain. Thus, the equations of the structural impedance models can be presented by correspondent combinations of elements. The resulting equivalent circuits illustrate the mathematical relations and the graphical plots emphasise the impedance behaviour of the different models.

L8-16

The basic impedance models correspond to the simplest electrochemical phenomena; in the same time the models have a large scale of applications and could serve for the construction of modified and more complicated models.

The classical theoretical impedance models contain in an implicit way the initial assumption for the homogeneity of the modelled processes. As a result, the modelling elements are lumped – the resistances and capacitances are ideal, the diffusion coefficients are constant, the surface is homogeneous, the surface coverage is described with a simple proportional coefficient, etc.

In reality, all those assumptions are never exactly obeyed. The measurement cell’s geometry is not ideal, the double layer structure is more complicated and has a space distributed nature, the electrode surface is inhomogeneous and could contain active centres or less active zones, the energy state of the adsorbed species clusters is strongly frequency-time distributed. These and other possible reasons lead to deviations of the measured impedances from the idealised theoretical models. The modelling of the parameters distribution can follow different space or/and frequency distribution approaches. The simplest solution of the problem is the application of CPE. This element, however, has a frequency distribution from zero to infinity, and has a direct physical meaning only in very rare cases. Ideally Polarizable Electrode. The assumption accepted for this simplest model is the absence of any process at the surface of the electrode. The double layer is presented by a simple capacitance Cdl and the impedance contains only the resistance of the electrolyte Rs and Cdl:

Z IPE (iω) = Rs - i (ωCdl) -1. (24) The impedance diagram of the model is presented in Fig. 15.

0 10 20 30 40 50 600

10

20

30

0.1 Hz

-Im

/ Ω

Re / Ω

0.01 Hz

Fig. 15. Impedance diagram of Ideally Polarizable Electrode simulated in the frequency range 103 ÷10-3 Hz at different values of Rs (Cdl = 0,01 F).

Following the notations, discussed in Section 5.3, the structure of this model can be presented as a Ladder one:

M IPE = La: Rs Cdl . (25)

The elements of this structural model have a direct physical meaning and correspond to the electrochemical parameters.

The electrolyte impedance is presented as a simple resistance, which for water electrolytes is reasonable for a large frequency range up to 1 ÷ 5 MHz.

L8-17

The modelling of the double layer by a simple capacitance is a simplification, accepted to be reasonable for concentrated electrolytes. In the case when the electrode surface is inhomogeneous or rough, the impedance diagram is deformed. In the cases when the double layer has a more complicated structure, which has to be taken into account, the model is modified by presentation of Cdl as a CPE of capacitive nature (Fig. 16):

M mIPE = La: Rs CPEdl . (26)

0 100 2000

100

-Im

/ Ω

Re / Ω

n = 1n = 0.8

0.01 Hz

Fig. 16. Impedance diagram of Modified Ideally Polarisable Electrode simulated in the frequency range 103 ÷10-3 Hz

at different values of n (Rs = 100 Ohm , A = 0,01 ).

Polarisable Electrode This simple model accounts for the presence of an electrochemical reaction at the electrode surface. Following the basic working hypothesis, the current correspondent to this reaction is treated as additive to the current of the double layer charging. As a result, the model includes additional impedance, which is connected in parallel to Cdl. It is called Faradaic impedance ZF [2]. The structure of this generalised model is

M F = La: Rs Cdl / Z F (27) The model of a polarisable electrode accepts the assumption for the absence of diffusion limitation and for a presence of a simple single step electrochemical reaction. As a result, the Faradaic impedance is simplified to a resistance, called charge transfer resistance Rct (Fig. 17) and the model notation becomes

M PE = La: Rs Cdl / Rct . (28)

0 100 200 300 400 500 6000

100

200

300

10 Hz

-Im

/ Ω

Re / Ω

100 Hz

Fig. 17. Impedance diagram of Polarizable Electrode simulated in the frequency range 103 ÷10-3 Hz at different

values of Rct (Rs = 100 Ohm, Cdl = 1E-4 F).

L8-18

The impedance diagram of the model is a semi-circle with a diameter Rct (Fig.17). For ω →∞ this semi-circle intercepts the real axis in a point with a value Rs and for ω →0 – in a point with value Rs + Rct. The imaginary component reaches a maximum at frequency ωo

ωo = (CdlRct)–1 = T –1, (29) called characteristic frequency. The product Cdl Rct is the time-constant T of the impedance. It controls the distribution of the total current between the charging of Cdl and the electrochemical reaction.

The solution of model (28) in the frequency domain gives the impedance equation

Z PE (iω) = Rs + Rct (1+ ω2T2)-1 - iωRct T(1+ ω2T2)-1 . (30)

For equilibrium of a partially reversible reaction

Rct = (RT/nF).(1/Io), (31)

where Io is the exchange current, n is the number of electrons implied in the transfer, F is the Faraday constant and RT is the product of the molar gas constant and the absolute temperature.

For an exponential dependence of the reaction rate constants kfO and kf

R on the potential, the exchange current is:

Io = nF A kfO cO* exp[ - αnFEeq/(RT) ] = nF A kf

R cR* exp[ (1- α) nFEeq/(RT) ], (32) where A is the electrode surface, α is the transfer coefficient, cO* and cR* are the concentrations of the species at the equilibrium potential Eeq.

The value of Rct depends on the rate of the reaction, which is potential dependent. Thus Rct varies with the potential, which leads to changes of the semicircle diameter, of the time-constant T and of ωo

Ideally Non-Polarisable Electrode. For an ideally reversible reaction Rct = 0 and the impedance turns into a simple resistance Randles Model. This model is based on the assumptions of the polarisable electrode with account of the diffusion limitations. The diffusion is accepted as linear, semi-infinite [70,71]. According to the solution, given by Graham [72], the model is described as

M RND = La: Rs Cdl / Rct W. (33)

The impedance of this model is:

ZRND (iω) = Rs + [ i ωC dl + (Rct + σ ω–1/2- i σ ω–1/2 )-1 ] –1. (34)

The structural parameter σ is related to the electrochemical parameters

σ = Rct [ kfO (DO) –1/2 + kf

R (DR) –1/2 ], (35)

where DO and DR are the diffusion coefficients of the species.

L8-19

As it could be seen, the structural model contains four parameters (Rs, Cdl, Rct and σ), while the electrochemical impedance model contains seven parameters (Rs, Cdl, Io, kf

O, DO, kfR,

DR ). The independent determination of all those seven parameters is impossible. Fig. 18 presents the impedance diagrams of the model with variation of the parameter Cdl.

0 2 4 6 80

2

4

0,01 Hz

-Im

/ Ω

Re / Ω

10 Hz

Fig. 18. Impedance diagram of Randles model at different values simulated in the frequency range 103 ÷10-3 Hz of

Cdl: 3E-4 F, 1E-3 F, 3E-3 F, 1E-2 F (Rs = 100 Ohm, Rct = 5000 Ohm, σ = 100). Bounded Randles Model. This model accounts also for a linear, but finite diffusion, taking place in a layer of finite thickness [9]. The structure of the impedance is

M BRND = La: Rs Cdl / Rct BW (36) The correspondence between the structural and the electrochemical parameters is given in [2].

Fig. 19 presents the impedance diagram with variation of R0:

0 100 200 3000

100

0,01 Hz

-Im

/ Ω

Re / Ω

100 Hz

Fig. 19. Impedance diagram of Bounded Randles model simulated in the frequency range 103 ÷10-3 Hz at different values of R0: 50 Ohm, 100 Ohm, 200 Ohm, 400 Ohm (Rs = 20 Ohm, Cdl= 1Е-4 F, Rct = 50 Ohm, A = 0.1, n = 0.45). Heterogeneous Polarisable Electrode (Faradaic Reaction Involving One Adsorbed Species). This model accounts for a heterogeneous reaction occurring in two steps in the absence of diffusion limitations. The intermediate species are adsorbed at the electrode’s surface. The model has a Ladder structure:

M HPE = La: R1 C1/R2 C2/R3 , (37)

L8-20

where the parameters R1 = Rs and C1 = Cdl have a direct physical meaning, and

R2 = Rct = [ (k1cB)–1 + k2–1] . [ FA (b1+ b2) ] -1; (38a)

R3 = Rct [(b1- b2) (k1cB - k2

)] . [ 2 (b2k1cB + b1k2)] –1 (38b)

C2 = [ β (b1+ b2)] . [ Rct (b1- b2) (k1cB - k2 )] –1 (38c)

are products of the electrochemical parameters [2]. As it could be seen, R3 and C2 could have positive or negative values, depending on the sign of the parameters combination (b1- b2).(k1cB - k2

). Fig. 20 presents impedance diagrams of this model for different combinations of the

parameters.

0 100 200 300 4000

100

200a)

1mHz10mHz

0.1Hz

10Hz

100Hz 1Hz

- Im

/ Ω

Re / Ω

0 100 200 300 4000

100

200b)

0.1Hz

10Hz

1Hz

- Im

/ Ω

Re / Ω0 500 1000

0

500c)

10Hz

1Hz- Im

/ Ω

Re / Ω

100 Hz

Fig. 20. Impedance diagrams of a Faradaic Reaction with one adsorbed species simulated in the frequency range 103 ÷10-3 Hz [(a): R1 = 50 Ohm, C1 = 1E-4 F, R2 = 100 Ohm, C2 = 1E-2 F, R3 = 200 Ohm;

(b): R1 = 50 Ohm, C1 = 1E-3 F, R2 = 100 Ohm,C2 = 1E-2 F, R3 = 200 Ohm and (c):R1 = 10 Ohm,C1 = 1E-3 F, R2 = 800 Ohm,C2 = −1E-2 F, R3 = −300 Ohm].

Voigt’s Model in solid state. A big group of materials and components applied in energy sources (electrodes and electrolytes for fuel cells, electrodes for batteries, intercalation processes) are investigated in solid state, where models with Voigt’s structure are applied. The idealised Nyquist plot of solid oxide electroceramic consists of three semi-circles (Fig. 21). Its corresponding equivalent circuit includes three parallel R-C elements, connected in series (Fig. 21), i.e. in Voigt’s modelling structure. A detailed analysis has been performed for the elucidation of the physical meaning of each semi-circle by Schouler [73] and Bauerle [74]. It was found, that the first semicircle corresponds to the bulk properties, the second one – to the grain boundaries, while the last one is due to the electrode response. Polycrystalline materials with uniform composition of the grains and grain boundaries show only one bulk and one grain boundary semicircle regardless of the fact whether the grains and grain boundaries are of uniform

L8-21

L8-22

thickness or not. That conclusion comes from the independence of the characteristic time-constant on the thickness.

The experimentally obtained spectra often differ strongly from the idealised ones presented in Fig. 21. The semi-circles may overlap and their centres may lie below the horizontal axis.

- Im

ReR1 R3

R3 R2R1

C1 C2 C3

R2

Fig. 21. An equivalent circuit and schematic impedance diagram of solid oxide electroceramics.

The degree of overlapping depends on the ratio of the time-constants. The closer they are,

the bigger is the mixing. Fig. 22 presents a two time-constants Voigt’s model with very high degree of overlapping, i.e. a model of two parallel R-C elements connected in series. Every R-C mesh in the model is characterised with one time-constant. The depression of every semi-circle could be attributed also to the heterogeneity of the properties.

0 300 6000

300

1000 Hz 1 Hz

10 Hz

- Im

/ Ω

Re / Ω

100 Hz

R2 R1

C1 C2

Fig. 22. Impedance diagram of a two time-constants model with high degree of mixing of the time-constants simulated in the frequency range 103 ÷10-3 Hz (R1 = 200 Ω, C1 = 1 E-4F; R2 = 200 Ω, C2 = 1E-4F).

Thus the advantage of the Impedance Spectroscopy for electroceramic studies is its

unique ability to separate the bulk, grain boundary and electrode behaviour. The impedance investigations of the conductivity are operating with the first two semicircles, since they are corresponding to the bulk and grain boundary properties and thus they ensure a quantitative evaluation of their resistances Rb and Rgb, respectively the conductivities and the capacitances Cb, Cgb.

For a better modelling of real data with depressed semicircles, the C components may be replaced with CPE [9,75]. The performance of measurements in a wide temperature interval

ensures the calculation of the bulk and grain boundary activation energies Ερ using the Arrhenius relation:

ρ = ρ 0(expΕρ /kT) , (39)

where ρ is the corresponding conductivity, ρ 0 is the pre-exponential factor, k is Boltzman’s constant and T is the temperature in K.

Impedance studies of the electrode reaction ensure a wide scope of information concerning the efficiency of the electroceramic materials as building blocks of the device, in which they are applied, as well as characterisation of the device functionality.

For interpretation of the electrode reaction the charge transfer concept from liquid electrochemistry, occurring across an interface is adopted. The idealised model and its corresponding impedance diagram are shown in Fig. 23.

For real systems the electrode response differs strongly from a semi-circle (Fig. 23). The electrode reaction in solid state electrochemical systems requires participation of species from the electrode, from the electrolyte and from the gas phase. For oxygen-ion conducting devices those species are electrons, oxygen vacancies and oxygen molecules. For the direct involvement of the gas-phase species the so called “three-phase boundary” concept (solid/liquid/gas) [76] has been introduced and additionally modified. The electrode reaction is divided into individual steps, involving charge transfer across the interface, as well as non-charge transfer processes such as adsorption, solid state diffusion and gas phase diffusion. The impedance data analysis tries to find the best model and to derive some parameters [3, 18, 38, 52]. Unfortunately, the interpretation is not always straightforward. For one and the same system different models could be proposed.

- Im

ReRct

Rct Rel

Cdl

Rel

Fig. 22. Idealized impedance diagram of the electrode reaction and its equivalent circuit.

Recently a non-charge transfer approach, known as ALS model, has been developed for characterisation of oxygen reduction on porous mixed conducting oxygen electrodes [77-79]. The zero bias impedance Z of a symmetrical cell is described by the sum of the electrolyte resistance Rel, the electron- and ion-transfer, occurring respectively at the current collector/electrode and electrode/electrolyte interfaces Zinterfaces and the non-charge transfer processes Zchem as oxygen surface exchange, solid state diffusion and gas phase diffusion inside and outside the porous electrode (Fig. 24):

Z = Rel + Zinterfaces + Zchem. (40)

L8-23

Fig. 23. Schematic impedance diagram of mixed conducting oxygen electrode [78].

The experimental results show that Zinterfaces is very small and oxygen reduction is limited by a chemical reaction (surface exchange) (Fig. 25).

Fig. 24. Impedance diagram of a porous mixed conducting oxygen electrode (LSCO) [78].

At lower oxygen pressures the gas phase diffusion becomes the rate-determining step. According to the ALS model, the electrode reaction takes place at a distance from the electrolyte

L8-24

interface up to 20 µm, which shows that a big fraction of the total electrode area is involved in the reduction process. 5.5. Calculation of the Model Impedance

The calculation of the model impedance follows the rule, that when the elements are connected in series, their impedances are added to each other, while in the case of parallel connection it is their admittances, i.e. the reciprocals of the impedances, which are added.

The most frequently applied elements in the construction of models describing solid oxides are the resistance and capacitance. They can be connected in series or in parallel and thus the simplest models could be obtained.

The model of an ideally polarised electrode is a series connection of resistance and capacitance (Fig. 15). The calculations are performed in the following way: ZIP (iω) = ZR (iω) + ZC (iω) = R + (iωC)-1 = R – i(ωC)-1. (41) The parallel connection of R and C, (in liquid electrochemistry it corresponds to a Faradaic reaction), is the most frequently applied building model to solid oxide systems. Its impedance, which gives a semicirlce in the Nyquist plot (Fig. 17), is obtained in the following way:

=)i(

1

F ωZ

)i(1

R ωZ +

)i(1

C ωZ=

R1 +

)(i1

Cω− =

R1 -

iCω =

RCR

ii ω− (42a)

)i(1

F ωZ =

RωT

ii − (42b)

ZF (iω) = ωTR

−ii =

)i)(i()i(iωTωT

ωTR+−

+ = 222

2

iii

TRTR

ωω

−+ (43a)

ZF (iω) = 221 Tω

R+

- i221 T

RTω

ω+

(43b)

As it has been already discussed, the bulk and grain boundary properties in polycrystalline

electroceramic materials are usually described by a series connection of R and C elements in parallel. Taking into account the rules, one can calculate the impedance of the electroceramic ZEC as:

ZEC(iω) = Zb(iω) + Zgb(iω) (44a)

ZEC(iω) = 2b

2b

1 T

R

ω+ - i

2b

2bb

1 T

TR

ω

ω

+ +

2gb

2gb

1 T

R

ω+ - i

2gb

2gbgb

1 T

TR

ω

ω

+ (44b)

ZEC(iω) = 2b

2b

1 T

R

ω+ +

2gb

2gb

1 T

R

ω+ -

++

+ 2gb

2bgbg

2b

2bb

11 T

TR

TTR

ω

ω

ωω

i

L8-25

6. Conclusions The application of EIS for the investigation of electrochemical systems is strongly dependent on the correct performance of the experiment. The choice of proper conditions should satisfy the requirements of the working hypotheses. The applied technique should ensure measurements at a small perturbation signal, which meet the conditions for linearity. The wide frequency range should make the investigated phenomena observable. If there are deviations from the main working hypotheses, a part of the response will not correlate with the expected result. Thus, the observed impedance would contain an invalid part, i.e. a part with origin, which will not belong to the previewed study of the system. The sophistication of the model identification needs the development of new quantitative techniques for more objective determination of the appropriate model. One step forward is the model reduction method [64], which is based on a consecutive analysis of the residuals after simulation and subtraction of already identified elements. The general solution, however, should be searched in the principles of the structural identification approach. 7. Acknowledgements

The author is grateful to the EC (specific programme “Energy, Environment and Sustainable Development – Part B: Energy program”, contract No NNE5/2002/18) for the possibility to prepare and present this talk on the International Workshop “Advanced Techniques for Energy Sources Investigation and Testing”, as well as to Zdravko Stoynov for the fruitful discussions and help, to John Kilner, Stephen Skinner for the possibility to present examples on YSZ and to Gergana Raikova for her contribution in the preparation of the figures.

8. References 1. M. Sluyters-Rehbach, J. H. Sluyters in: A.J. Bard, (Ed.), Analytical Chemistry, vol. 4, Marcel

Dekker, New York, 1970. 2. C. Gabrielli, Identification of Electrochemical Process by Freguency Response Analysis,

Monograph Reference 004 /83, Solartron Instr.Group, Farnsborough, England, 1980. 3. D.D. Macdonald, M. C. H. McKubre, Electrochemical Impedance Technigues in Corrosion

Science: Electrochemical Corrosion Testing, STP 272, ASTM, Philadelphia, PA, 1981. 4. A. Lasia, in: B.E. Conway, J. Bockris, R. White, (Eds.), Electrochemical Impedance

Spectroscopy and Its Applications, Modern Aspects of Electrochemistry, Kluwer Academic/Plenum Publishers, New York, vol. 32, 1999, 143.

5. F. Mansfield, Corrosion 37 (1981) 301. 6. M. Sluyters-Rehbach, J.H. Sluyters in: E. Yeager, J. O’M. Bockris, B.E. Conway, S.

Sarangapani (Eds.), Comprehensive Treatise of Electrochemistry, Plenum Press, New York, 1984, 177.

7. Z.B. Stoynov, B. Savova-Stoynova, J. Electroanal. Chem. 209 (1986) 11. 8. J.R. Macdonald (Ed.) in: Impedance Spectroscopy - Emphasizing Solid Materials and

Systems, Wiley-Interscience, New York, 1987. 9. Z. Stoynov, B. Grafov, B. Savova-Stoynova, V. Elkin, Electrochemical Impedance,

Publishing House Science, Moscow, 1991. 10. J.W. White, Proc. IEEE 59, 1971, 98. 11. J.W. White, Proc. of the 6-th Annual Princeton Conf. on Information Sciences and Systems,

1972, 173. 12. M. Urguidi-Macdonald, S. Real, D.D. Macdonald, Electrochim. Acta 35 (1990) 1559. 13. J.R. Macdonald, Electrochim. Acta 38 (1993) 1883.

L8-26

14. H.A. Kramers, Z. Phys. 30 (1929) 521. 15. R.L. de Kronig, J. Opt. Soc. Am. 12 (1926) 547. 16. P. Agarwal, M.E. Orazem in Application of Measurement Models to Impedance

Spectroscopy, J. Electrochem. Soc. 142 (1995) 4159. 17. M. Urguidi-Macdonald, D.D. Macdonald, J. Electrochem. Soc. 137 (1995) 3306. 18. P. Agarwal, M. E. Orazem, L.H. Garsia-Rubio, J.Electrochem. Soc. 139 (1992) 1917. 19. Z. Stoynov, B. Savova-Stoynova, J. Electroanal. Chem. 183 (1985) 133. 20. Z. Stoynov, B. Savova, J. Electroanal. Chem. 112 (1980) 157. 21. Z. Stoynov, Fourier analysis in the presence of nonstationary periodic1noise, Dissertation

1839, Swiss Federal Institute of Technology, Zuerich, 1985. 22. Z. Stoynov, B. Savova-Stoynov, T. Kossev, J. Power Sources 30 (1990) 275. 23. B. Savova-Stoynov, Z. Stoynov, Key Engineering Materials 59 & 60 (1991) 273. 24. Z.B. Stoynov, Electrochim. Acta 37 (1992) 2357. 25. Z. Stoynov, Electrochim. Acta 38 (1993)1919. 26. B. Savova-Stoynov, Z.B. Stoynov, Electrochim. Acta 37 (1992) 2353. 27. Z. Stoynov, V. Vatcheva, Bulg.Chem. Commun. 27 (1994) 588. 28. C. Gabrielli, M. Keddam, H. Takenouti, Electrochem. Acta 35 (1990) 1553. 29. C. Gabrielli, B. Tribollet, J. Electrochem. Soc. 147 (1994) 1147. 30. C. Cachet, M. Keddam, V. Mariotte, R. Wiart, Electrochim. Acta 37 (1992) 2377. 31. C. Cachet, M. Keddam, V. Mariotte, R. Wiart, Electrochim. Acta 38 (1993) 2203. 32. I. Annergren, M. Keddam, H. Takenouti, D. Thierry, Electrochim. Acta 38 (1993) 763. 33. C. Gabrielli, M. Keddam, A. Khalil, R. Rosset, M. Zidoune, Electrochim. Acta 38 (1993)

2203. 34. C. Deslouis, O. Gil, B. Tribollet, Electrochem Acta 38 (1993) 1847. 35. C. Gabrielli, M. Keddam, F. Minouflet, H. Perrot, Electrochem Acta 41 (1996) 1217. 36. I. Annergren, M. Keddam, H. Takenouti, D. Thierry, Electrochem Acta 41 (1996) 1121. 37. M.E. Orazem, M. Durbha, C. Deslouis, H. Takenouti, B. Tribollet, Electrochem Acta 44

(1999) 4403. 38. H. Gerisher, W. Mehl, Z. Electrochem. 59 (1955) 1049. 39. A.N. Frumkin, V. I. Melik-Gaikazyan, Dokl. Akad. Nauk, USSR 77 (1951) 855. 40. I. Epelboin, M. Keddam, J. Electrochem. Soc. 117 (1970) 1052. 41. I. Epelboin, M. Keddam, J.C. Lestrade, Faraday Disc of the Chem. Soc. 56 (1973) 265. 42. R.D. Armstrong and R.E. Friman, J. Electroanal. Chem. 45 (1973) 3. 43. R. de Levie, A.A. Husovsky, J. Electroanal. Chem. 22 (1969) 29. 44. R. de Levie, L. Pospisil, J. Electroanal. Chem. 22 (1969) 277. 45. C. Cao, Electrochim. Acta 35 (1990) 831. 46. C. Cao, Electrochim. Acta, 35 (1990) 837. 47. Z. Stoynov, Electrochim. Acta 35 (1990) 1499. 48. M. Sluyters-Rehnbach, Pure & Appl. Chem. 66 (1994) 1831. 49. P. Zoltowski, J. Electroanal. Chem. 260 (1989) 269. 50. P. Zoltowski, Bul. Chem., Commun. 2 (1990) 309. 51. J.R. Dygas, M.W. Breiter, Electrochim. Acta 44 (1999) 4163. 52. Z. Stoynov in:C. Julien, Z. Stoynov (Eds.), Materials for Lithium-Ion BatteriesKluwer

Academic Publishers, 3/85, 2000, 349. 53. Z. Stoynov, B. Savova-Stoynov, 166th Meeting AES, Ext. Abstr. 159, New Orleans, 1984. 54. Z. Stoynov, First Int. Symp. EIS, Ext. Abstr. PL2, Bombannes, 1989. 55. B.A. Boukamp, Sol. St. Ionics 20 (1986) 31. 56. P. Zoltowski, J. Electroanal. Chem. 178 (1984) 11. 57. E.W. Marquardt, SIAM J. Appl. Math. 11 (1963) 431.

L8-27

58. J.R. Macdonald, CNLS Immitance Fitting Program LEVM, University of North Carolina, 1988.

59. B. Boukamp, First Int. Symp. EIS, Ext. Abst. CL 1, Bombannes, 1989. 60. M.W. Kendig, E. M. Meyer, G. Lindberg, F. Mansfeld, Corros. Sci. 23 (1983) 1007. 61. B.A. Boukamp, Solid State Ionics 16 (1986) 136. 62. Equivalent Circuit Analysis Technology, Sirotech Ltd., Ness-Ziona, Izrael, 1994. 63. Zview for Windows, Impedance / Gain Phase Graphing and Analysis Software, Verson 1.4b,

Scribner Associates Inc., Charlottesville, Virginia, 1996. 64. Z. Stoynov, B. Savova-Stoynov, J. Electroanal. Chem. 170 (1984) 63. 65. E. Warburg, Ann. Phys. Chem. 67 (1899) 493. 66. T. Pajkossy, L. Nyikos, J. Electrochem. Soc. 133 (1986) 2061. 67. T. Pajkossy, J. Electroanal. Chem. 346 (1994) 111. 68. Z. Stoynov, B. Savova-Stoynova, Bulg. Chem. Communic. 23 (1990) 279. 69. Z. B. Stoynov, B. Savova-Stoynov, N. V. Stamenova, Bulg. Chem. Communic. 27 (1994)

289. 70. J. Randles, Discuss. Faraday Soc. 1 (1047) 11. 71. B.V. Ershler, Discuss. Faraday Soc. 1 (1047) 269. 72. D. Grahame, Chem. Rev. 41 (1947) 441. 73. E. Shouler, G. Giroud, M. Kleitz, J. Chim. Phys. 70 (1973) 1039. 74. J.E. Bauerle, J. Phys. Chem. Solids 30 (1969) 265. 75. B. Boukamp, Solid State Ionics 136 (2000) 75. 76. A. Schmid, Helv. Chim. Acta 1 (1933) 69. 77. S.B. Adler, J. A. Lane, B. C. Steele, J. Electrochem. Soc. 143 (1996) 3554. 78. S.B. Adler, Solid State Ionics 111 (1998) 12. 79. S.B. Adler, Solid State Ionics 135 (2000) 603.

L8-28