IEES — BAS Centre of Excellence D

IEES - BAS Centre of Excellence D. Vladikova, Institute of Electrochemistry and Energy Systems, Sofia, Bulgaria; d. vladikova@bas. bg 1

IEES - BAS Centre of Excellence ABOUT THE E-SCHOOL This e-School can be regarded as a specialized course from the group of POEMES “cd” training courses, which concerns some modern aspects in the development of the Electrochemical Impedance Spectroscopy (EIS). It introduces the technique of the Differential Impedance Analysis (DIA), developed recently in IEES. This novel and advanced method for data analysis contributes to the increase of the Electrochemical Impedance Spectroscopy’s information potential. Organized for Ph. D. students and scientists applying Impedance for energy sources studies, the course can be also interesting for researchers who use this technique in other scientific fields. The application of DIA is demonstrated on example of YSZ (single crystal and polycrystalline) solid electrolyte and Lead Acid Electrode. The DIA references are linked to the corresponding abstracts. 2

IEES - BAS Centre of Excellence 1. 2. 3. 4. 5. 6. 7. 8. 9. REFERENCES (for DIA) Z. Stoynov, D. Vladikova, “Differential Impedance Analysis”, Marin Drinov Academic Publishing House, Sofia, 2005. D. Vladikova, Z. Stoynov, J. Electroanal. Chem. 572 (2004) 377. D. Vladikova, Z. Stoynov, M. Viviani, J. Europ. Ceram. Soc. 24 (2004) 1121. D. Vladikova, http: //accessimpedance. iusi. bas. bg, Imp. Contribut. Online 1 (2003) L 3 -1; Bulg. Chem. Commun. 36 (2004) 29. G. Raikova, D. Vladikova, Z. Stoynov, http: //accessimpedance. iusi. bas. bg, Imp. Contribut. Online, 1 (2003) P 8 -1; Bulg. Chem. Commun. 36 (2004) 66. D. Vladikova, P. Zoltowski, E. Makowska, Z. Stoynov, Electrochim. Acta (2002) 2943. Z. Stoynov, Proc. 15ème Forum sur les Impedances Electrochimiques, 9 décembre 2002, Paris, France, p. 3. Z. Stoynov, H. Takenouti, M. Keddam, D. Vladikova, G. Raikova, Proc. 15ème Forum sur les Impedances Electrochimiques, 9 décembre, 2002, Paris, France, p. 235. Z. Stoynov, in: C. Julien, Z. Stoynov (Eds. ), Materials for Lithium-Ion Batteries, Kluwer Academic Publishers, 3/85, 2000, 371. 3

IEES - BAS Centre of Excellence REFERENCES (for DIA) 10. Z. Stoynov, Polish. J. Chem. 71 (1997) 1204. 11. Z. Stoynov, Electrochim. Acta 35 (1990) 1499. 12. Z. Stoynov, Electrochim. Acta 34 (1989) 1187. 13. D. Vladikova, J. A. Kilner, S. J. Skinner, G. Raikova, Z. Stoynov, Electrochim. Acta, in press 14. A. Barbucci, M. Viviani, P. Carpanese, D. Vladikova, Z. Stoynov, Electrochim. Acta, in press 4

IEES - BAS Centre of Excellence BASIC ABBREVIATIONS Ad additive term C capacitance Cdl double layer capacitance CNLS complex non-linear least squares D data set 2 -D two-dimensional CNLS complex nonlinear least squares CPE constant phase element ct charge transfer DIA differential impedance analysis dl double layer 5

IEES - BAS Centre of Excellence BASIC ABBREVIATIONS FRA frequency response analyzer gb grain boundary IEES Institute of Electrochemistry and Energy Systems Im imaginary component of the impedance L inductance LOM local operating model MRND modified Randles model Par. parameter Par. Ident. parametric identification R resistance 6

IEES - BASIC ABBREVIATIONS Centre of Excellence EIS Rct electrochemical impedance spectroscopy charge transfer resistance Re real component of the impedance Rel electrolyte resistance RND Randles model SOFC solid oxide fuel cells S. T. spectral transform Str. Ident. structural identification T time-constant W Warburg impedance YSZ yttria stabilized zirconia Z impedance 7

IEES - BAS Centre of Excellence CONTENTS 1. WHY DIA? ? (Introduction) 2. 2. PRINCIPLE OF DIA 2. 1. Structural Identification 2. 2. Local Operating Model (LOM) 2. 3. DIA Procedure 2. 4. Noise Immunity 3. SECONDARY DIA 5. 1. Recognition of CPE 5. 2. Recognition of Randles Model 5. 3. Recognition of a Simple Faradaic Reaction with Capacitive CPE 4. DIA APPLICATION ON YSZ Solid Electrolyte 8

IEES - BAS Centre of Excellence 1. WHY DIA? ? (Introduction) To answer on this question, we should have to make a small promenade in the classical EIS (more info about the classical impedance data analysis can be found in the prerequisite e-school “abc” Impedance). The power of the EIS lies in its unique capability for separation of the kinetics of different steps comprising the overall electrochemical process as well as in the detailed information obtained about the surface and the bulk properties of a wide spectrum of electrochemical systems and phenomena, including power sources. Since the method does not ensure the direct measurement of a physical phenomenon, the interpretation of the experimental data, however, demands the construction of an impedance model. 9

IEES - BAS Centre of Excellence WHY DIA? ? (Introduction) The identification of the impedance model follows some obligatory steps, which are presented schematically on page 12: • The model structure is assumed a priori , i. e. one or more hypothetical models are chosen; • The most probable combination of the model parameters is evolved by parametric identification; • The degree of proximity between the model and the object’s impedance is a measure for the model validity. 10

IEES - BAS 1. WHY DIA? ? Centre of Excellence (Introduction) There are many works devoted to improvements in the procedures for parametric identification and model validation (see “abc Impedance”), BUT the principal subjectivity, set in the “choice of a hypothetical model”, can not be avoided. The Differential Impedance Analysis is a novel and advanced technique for impedance data processing, which increases the information potential of the EIS, introducing the more powerful and objective “structural identification”. It does not require an initial working hypothesis and the information about the model structure is extracted from the experimental data: 11

IEES-BAS Parametric Identification Centre of Excellence Choice of a Hypothetical Model STRUCTURAL A D I APPROACH Parametric Identification Model Validation: • Model simulation: • Choice of criterion of proximity between the initial and estimated data: • Analysis of the residuals: 12

IEES-BAS Centre of Excellence 2. PRINCIPLE OF DIA 2. 1. Structural Identification The structural identification approach utilizes the impedance data to identify both the structure S and the parameters P of the model M and thus to eliminate the necessity for initial working hypothesis : The operator Str. Ident. symbolizes the structural identification, which includes the parametric one. The procedures of the DIA start with the initial 3 D set of experimental data (angular frequency, real and imaginary components of the impedance): D 3 [ Rei, Imi, ωi] 13

IEES-BAS 2. PRINCIPLE OF DIA Centre of Excellence Scanning Local Analysis The kernel of the DIA is the scanning local analysis - a combination of a scanning procedure with a local parametric analysis. The procedure is based on a consecutive analysis of the experimental data by applying a local estimator moving along the analytical coordinate. In the Impedance Spectroscopy a preliminary selected model, called local operating model (LOM), is used as a local estimator, while the frequency is recognized as an analytical coordinate. Scanning Local Analysis Scanning with a Local estimator - Local Operating Model (LOM) (scanning axis - w) + Local Parametric Identification 14

IEES-BAS 2. PRINCIPLE OF DIA Centre of Excellence Local Operating Model DIA operates with a LOM, represented by a simple first order inertial system, extended with an additive term. Its transfer function follows the First Cauer Form: H(p) = A + B/(1 + pt) The simplest electrical equivalent is a mesh of resistance R and capacitance C in parallel, extended with additive term Ad connected in series. Since in electrochemical systems the resistance of the electrolyte is observed very often at high frequencies, this additive term may be accepted to be a resistance Rad. The effective time-constant T = RC is also introduced as a parameter. CPE C Rad R 2 R 15

IEES-BAS Centre of Excellence 2. PRINCIPLE OF DIA Procedure I. Scanning with the LOM throughout the whole frequency range with a scanning window of a single frequency point; II. Parametric identification of the LOM at every working frequency; III. Frequency analysis of the LOM parameters’ estimates. I Scanning with the LOM in the whole ω range 16

2. PRINCIPLE OF DIA IEES-BAS Centre of Excellence DIA Procedure II. LOM Parametric Identification Since the scanning identification window width is equal to one frequency point, the application of a deterministic approach is necessary. The number of the independent data available inside the observation window is smaller than the number of the unknown parameters and the initial 3 D set of impedance data is extended with two additional terms - the derivatives of the real and imaginary components of the impedance with respect to the frequency. As a result a new five dimensional set of data is obtained and the LOM parameters’ estimates are determined at every frequency. ˆ PLOM = Par. Id [ D 3 (ωi, Re i, Imi, ) d Rei /d wi d Imi/d wi ˆ P LOM ˆ Þ R ad, ˆ C ˆ , R ˆ , T ] 17

IEES-BAS 2. PRINCIPLE OF DIA - DIA Procedure Centre of Excellence II. LOM Parametric Identification Impedance of the LOM (1) R (2) Rad C (3) V is the vector of the parameters (4) (5) (6) (7) New data set : 18

2. PRINCIPLE OF DIA IEES-BAS Centre of Excellence DIA Procedure III. Frequency analysis of the LOM parameters’ estimates. The analysis of the LOM parameters’ estimates frequency behaviour , which can be performed in different ways, provides for the structural and parametric identification of the impedance model. Every form of representation emphasizes specific parts of the model structure and thus their combination ensures a more complete and precise identification. III Frequency Analysis of the LOM Parameters Estimates ˆ P LOM f (w) Determines the model structure 19

2. PRINCIPLE OF DIA IEES-BAS Centre of Excellence DIA Procedure III. Frequency analysis of the LOM parameters’ estimates. 1. Temporal Analysis – direct recognition of lumped parameters (8) where Tf = 1/f and f is the frequency As far as Tf represents the time interval of perturbation, the function is a temporal one and the corresponding analysis is termed temporal analysis. For its further implementation the following notation will be used: (9) where the vector ; contains the decimal logarithms of the parameters : ; The new experimentaldata set is: ; and (10) 20

IEES-BAS Centre of Excellence 2. PRINCIPLE OF DIA - DIA Procedure III. Frequency analysis of the LOM parameters’ estimates. Temporal Analysis – graphical presentation The temporal analysis displays the functional dependencies of Eqn. (8). in two types of plots – temporal plots and spectral plots. Temporal plots In the frequency regions, where the LOM corresponds to the object’s behaviour, i. e. to a sub-model with a structure of a time-constant (R and C in parallel connection – Example 1), the dependencies described by Eqn. 8 are frequency invariant and their temporal plots exhibit plateaus. In Examples 2 - 4 the object is a Faradaic reaction involving one adsorbed species, i. e. the reaction has two steps and the observed plateaus are also two. They are separated by regions of frequency dispersion, which in this case correspond to the transition between the two reaction steps. To summarize : the presence of plateaus ensures the recognition of the model, while their position enables the parametric estimation. 21

2. PRINCIPLE OF DIA - DIA Procedure IEES-BAS Centre of Excellence III. Frequency analysis of the LOM parameters’ estimates. Temporal Analysis – graphical presentation Temporal plots ( α , r, c, τ ) Object Impedance Temporal plot Example 1 – simple Faradaic Reaction C CPE R 1 R 2 Example 2– Faradaic Reaction involving one adsorbed species (I) C R R 22

2. PRINCIPLE OF DIA - DIA Procedure IEES-BAS Centre of Excellence III. Frequency analysis of the LOM parameters’ estimates. Temporal Analysis – graphical presentation Temporal plots ( r, c, τ ) Object Impedance Temporal plot Example 3– Faradaic Reaction involving one adsorbed species (II) C R R Example 4– Faradaic Reaction involving one adsorbed species (III) C R R 23

IEES-BAS Centre of Excellence 2. PRINCIPLE OF DIA - DIA Procedure III. Frequency analysis of the LOM parameters’ estimates. Temporal Analysis – graphical presentation The temporal analysis displays the functional dependencies of Eqn. (8). in two types of plots – temporal plots and spectral plots. Spectral plots The Spectral presentation (analysis) offers an enhanced performance of the results obtained by the temporal analysis. It displays the dependencies of Eqn. 8 in the more informative spectral form Sj, which reveals the presence of a different number of time-constants: (11) where j denotes the corresponding parameter and S. T. symbolizes the spectral transformation procedure, which converts the plateaus of the temporal plots into spectral lines. 24

IEES-BAS Centre of Excellence 2. PRINCIPLE OF DIA - DIA Procedure III. Frequency analysis of the LOM parameters’ estimates. Temporal Analysis – graphical presentation Spectral plots The simplest spectral transform can be regarded as a construction of an ordinary histogram. The parametric spectra are obtained by accumulating frequency bands with approximately equal values of the parameter . The intensity of the individual spectral peaks is estimated by introducing a d. B grid, where an intensity of 10 d. B corresponds to a frequency band of one decade. Since the spectral representation takes into account the frequency spacing of the real data, the amplitude of the spectral line is proportional to the length of the frequency interval where a given phenomenon is well pronounced. To summarize : The spectral presentation transforms the temporal plateaus into spectral peaks. The number of spectral lines gives the number of timeconstants (R/C meshes) in the model. The position of the spectral lines defines the value of the corresponding 25 parameter.

2. PRINCIPLE OF DIA - DIA Procedure IEES-BAS Centre of Excellence III. Frequency analysis of the LOM parameters’ estimates. Spectral Transform ( α , r, c, τ ) Object Temporal plot Spectral plot Example 1 – simple Faradaic Reaction C CPE R 1 R 2 Example 2– Faradaic Reaction involving one adsorbed species (I) C R R 26

2. PRINCIPLE OF DIA - DIA Procedure IEES-BAS Centre of Excellence III. Frequency analysis of the LOM parameters’ estimates. Spectral Transform ( r, c, τ ) Object Temporal plot Spectral plot Example 3– Faradaic Reaction involving one adsorbed species (II) C R R Example 4– Faradaic Reaction involving one adsorbed species (III) 27

2. PRINCIPLE OF DIA - DIA Procedure IEES-BAS Centre of Excellence Spectral Transform – Noise Immunity The spectral transform plays the role of a consecutive integration, since the intensity of an individual spectral peak is proportional to the length of the frequency interval where the corresponding parameter’s estimate has close values. Thus the spectral transform provides for data stratification and for efficient filtration of non-statistical noise, because the presence of an outlier introduces only a fuzzy low intensity line, located away from the spectral kernel of the basic phenomenon. Example on synthetic model of Faradaic Reaction involving one adsorbed species: C 1 R C 2 R 1 R 2 T 1 = R 1 C 1 • Simulation of noise – by truncation; • Simulation of non-Gaussian noise – T 1 = R 2 C 2 by introduction of a “wild point” via q = T 2/T 1 = 0. 001 10% increase of Re. 28

2. PRINCIPLE OF DIA - DIA Procedure IEES-BAS Centre of Excellence Spectral Transform – Noise Immunity Impedance diagrams 7 digits q = T 2/T 1 = 0. 001 Truncation to 4 digits +wild point 29

IEES-BAS Centre of Excellence q = T 2/T 1 = 0. 001 7 digits 2. PRINCIPLE OF DIA - DIA Procedure Spectral Transform – Noise Immunity Spectral Plots 4 digits +wild point • DIA recognizes the model with q = 0. 001 (CNLS recognizes the model up to q = 0. 01); • DIA recognizes the model with q = 0. 001 up to 4 digits. The procedure of truncation increases only the number of low intensity “fuzzy” lines in the vicinity of the 2 spectral peaks; • The introduction of a “wild point” does not influence the selectivity of DIA, because its 30 presence only gives additional “fuzzy line” located far away from the 2 basic peaks.

3. SECONDARY DIA - Introduction IEES-BAS Centre of Excellence When the LOM does not correspond to the object’s structure, frequency dependence is observed in the temporal plots. It may be due to: • mixing of two neighbouring phenomena or • presence of frequency dispersion (CPE, W etc). C R CPE The regions of mixing cannot produce spectral peaks. They form ”tails” in the spectral plots. R CPE 31

IEES-BAS Centre of Excellence 3. SECONDARY DIA - Introduction Although introduced to the identification of models with lumped elements, the applied LOM can be used for the recognition of frequency distribution. This property increases the analytical power of the DIA since it ensures structural and parametric identification within a wider frequency range, including regions where the LOM does not correspond to the structure of the investigated object. Thus DIA can be applied for investigation of a wide variety of real samples, containing rough and inhomogeneous surfaces or inhomogeneous volume properties. The identification of models with frequency dispersion behaviour is based on the so-called Secondary Differential Impedance Analysis. 32

IEES-BAS Centre of Excellence 3. SECONDARY DIA Differential Temporal Analysis The Secondary DIA is performed by differentiation of the logarithmic LOM parameters’ estimates with respect to the logarithm of the frequency. It presents their frequency dependence in the form: (12) where j denotes the corresponding LOM’s parameter estimate or . Thus a new set of functions is obtained, which forms a new estimated data set : (13) Since the Secondary Analysis describes the differentiation of the temporal functions, it is called Differential Temporal Analysis. 33

IEES-BAS Centre of Excellence 3. SECONDARY DIA Differential Temporal Analysis Secondary DIA for the “lumped parts of a model” The derivatives of the parameters’ estimates become equal to 0 in the frequency range where the LOM corresponds to the nature of the impedance object, i. e for a model of a simple Faradaic reaction as well as for the plateaus of a lumped (homogeneous) model involving one or more adsorbed species The two frequency (14) bands, where is equal to 0 are separated by a bell-shaped region R R R of transition, which C C corresponds to the mixing of the two time-constant type sub-models. 1 3 2 1 2 34

IEES-BAS Centre of Excellence 3. SECONDARY DIA Differential Spectral Analysis Secondary DIA for the “lumped parts of a model” The Differential Spectral Analysis represents the results obtained by the differential temporal analysis in a spectral form. The spectral transform procedure follows that introduced in the spectral transform of the temporal functions. In this case, however, . The set of differential spectra in a position 0 focuses on the presence of timeconstant sub-models. R 1 C 1 R 2 R 3 C 2 The appearance of additional “fuzzy lines” which form small spectral “tails” next to the main pure reaction peak at visualizes the mixing of two uniform submodels. 35

IEES-BAS Centre of Excellence 3. SECONDARY DIA Recognition of CPE Dispersion When the dispersion curve in the temporal plot has a constant slope, the functions also acquire a constant value, different from 0, which is characteristic of this dispersion. In such cases the Secondary DIA is obligatory, since it can identify the dispersion zone in a more explicit way. The irregular electrode surface, due to surface roughness or non-uniformly distributed properties, leads to a dispersion of the parameters. Very often this behaviour is successfully approximated by an empirical relationship, known as constant phase element (CPE). (15) CPE Impedance Temporal plot The dispersion is described by straight lines, i. e. it has a constant 36 slope.

IEES-BAS 3. SECONDARY DIA Centre of Excellence Recognition of CPE Dispersion The secondary frequency analysis in the case of CPE operates with the projection of the CPE impedance in the internal LOM’s space, represented in coordinates R, C and T: lg. R = r = - lg A + lg [cos(pn/2)] – n lgω (16) lg. C = c = (17) lg. A – lg[sin(πn/2) – (1 – n)lgω lg. T = τ = - lg w – lg [tg (pn/2) (18) lg. A = -lg. R + lg[cos(pn/2)] – n lg w = lg C + lg[(pn/2)] + (1 -n) lg w (19) The derivatives of equations (16) – (18) form a new set of interdependent “δ” functions with definite for the CPE behaviour, determined by the behaviour constant slope of the dispersion curves in the temporal plot: (20) 37

IEES-BAS Centre of Excellence 3. SECONDARY DIA Example for Recognition of CPE Dispersion Differential Temporal Plot 0, 3 0, 7 1 Differential Spectral Plot The combined differential spectral plot of , and , which is characterized by three distinct spectral peaks, can be regarded as a “fingerprint” of the CPE behaviour. 38

IEES-BAS Centre of Excellence Differential Temporal Plot Differential Spectral Plot 3. SECONDARY DIA TEST Example: Find the value of the CPE coefficient using the given Differential plots (click on the correct value) The CPE coefficient is: 0 0. 10 0. 20 0. 30 0. 40 0. 55 0. 60 0. 70 0. 80 0. 90 1 39

IEES-BAS 3. SECONDARY DIA Centre of Excellence Recognition of Randles Model The application of the Secondary Differential Analysis promotes the capability of the DIA for model recognition. It can be applied to the identification of electrochemical elements or more complicated models with CPE in their structure. This section introduces some examples for the application of the DIA on synthetic models often used in electrochemical systems. Randles model (RND) is widely applied to electrochemical systems for modelling of a simple Faradaic reaction with a mass transport towards the electrode’s surface, limited by a linear semi-infinite diffusion. The deviations from Fick’s law can be presented by modification of the Warburg impedance with CPE. The model is known as Modified Randles Model (MRND). R 1 C R 2 R 1 W C R 2 CPE BW 40

IEES-BAS Centre of Excellence DIA Results: 3. SECONDARY DIA Recognition of Randles Model Temporal Analysis The applied estimates data set is: (21) The temporal analysis discovers three segments with differing frequency behaviour. The high frequency region I is frequency invariant. The low frequency parts II and III, however, show two types of frequency dispersion, which require a Secondary DIA. The lack of frequency dispersion in segment I confirms the correspondence between the LOM and the object in this region, i. e. the recognized structure is a parallel connection between Rct (R) and Cdl (C). RND MRND 41

IEES-BAS 3. SECONDARY DIA Centre of Excellence Recognition of Randles Model DIA Results: Temporal Analysis - Spectral presentation The spectral plots sharply distinguish too types of behaviour – frequency independent, which is represented by a well defined spectral line, and frequency dependent, which forms a spectral “tail”. This result demonstrates the high sensitivity of the spectral analysis for the identification of time-constants. The plateaus in the temporal plots and the spectral images ensure the parametric identification of the Rct/Cdl sub-model. RND MRND 42

IEES-BAS Centre of Excellence DIA Results: 3. SECONDARY DIA Recognition of Randles Model Secondary DIA - Differential Temporal Analysis The Secondary DIA identifies the model in the regions of frequency dispersion (segments II and III) in the temporal plots. It operates with the estimates data set (21): The Differential Temeporal Analysis separates well three segments and ensures the information needed for their structural identification. RND MRND 43

IEES-BAS Centre of Excellence 3. SECONDARY DIA Recognition of Randles Model Secondary DIA - Differential Temporal Analysis Segment I: The dependencies obtained for segment I confirm the structure of resistance and capacitance in parallel, since Segment II For RND the obtained dependencies recognize Warburg Impedance, since For MRND the obtained dependencies recognize CPE (n = 0. 4), since Segment III: The bell-shaped region III defines the region of mixing between the two sub-models in the Randles structure. 44

IEES-BAS Centre of Excellence 3. SECONDARY DIA Recognition of Randles Model Secondary DIA - Differential Spectral Analysis The combined Differential Spectral Plot ( see page 46) represents in an explicit 46) way the structural and parametric identification, emphasizing the two kernels of that structure. The spectral lines in zero position (segment I) manifest the presence of the timeconstant sub-model, corresponding to the Faradaic reaction. The combination of the other three spectral peaks (segment II) visualizes the transport limitation: • For values 0. 5, 0. 5 and 1 - Warburg impedance (RND); • For values 0. 3, 0. 7, 1 - CPE (MRND). 45

IEES-BAS 3. SECONDARY DIA Centre of Excellence Recognition of Randles Model Secondary DIA - Differential Spectral Analysis R 1 R 2 R 1 C R 2 W C CPE BW 46

IEES-BAS Centre of Excellence 3. SECONDARY DIA Recognition of Simple Faradaic Reaction with Capacitive CPE The impedance diagram of a modified time-constant model with a capacitive CPE is characterized by depressed semicircle, which is often observed in real systems. In electrochemical systems this structure is known as modified polarizable electrode. R 1 CPE R 2 47

IEES-BAS 3. SECONDARY DIA Centre of Excellence Recognition of Simple Faradaic Reaction with Capacitive CPE DIA Results: Temporal Analysis The applied estimates data set is: The behaviour of the temporal plots is surprising to some extent, since the dispersion concerns not only c and τ, but also r within the whole frequency range. Formally the temporal plots can be divided into a high frequency segment I and a low frequency segment II. Temporal plot 48

IEES-BAS Centre of Excellence 3. SECONDARY DIA Recognition of Simple Faradaic Reaction with Capacitive CPE DIA Results: Temporal Analysis In the r – temporal plot the two segments (I and II) can be regarded as separated by a region of pseudo-mixing (segment III), which has a shape close to a plateau and creates the small intensity spectral peak in the spectral plots. It also determines the approximation to the estimate of the model’s resistance R (Rct), i. e. the temporal analysis of segment III in the temporal plot ensures its parametric identification. Temporal plot Spectral plot 49

IEES-BAS 3. SECONDARY DIA Centre of Excellence Recognition of Simple Faradaic Reaction with Capacitive CPE Secondary DIA The Secondary DIA identifies the model in the frequency dispersion regions (segments I and II) in the temporal plots. It operates with the estimates data set (21): The Differential Temeporal Analysis of Segment I distinguishes dependencies of CPE-type, which ensure identification of the CPE exponent n (n = 0. 8). (22) Knowing n, the parameter A can be also calculated. Differential Temporal Plot Differential Spectral Plot 50

IEES-BAS Centre of Excellence 3. SECONDARY DIA Recognition of Simple Faradaic Reaction with Capacitive CPE Secondary DIA The Differential Temeporal Analysis of Segment I performs the parametric identification, since it recognizes the CPE exponent n. The Differential Temporal Analysis of Segment II, however, ensures the structural II identification of the model, since the results, obtained for are characteristic only for this model: (23) Differential Temporal Plot Differential Spectral Plot 51

IEES-BAS Centre of Excellence 3. SECONDARY DIA Recognition of Simple Faradaic Reaction with Capacitive CPE Secondary DIA The Structural and Parametric Identification of a Faradaic Reaction with Capacitive CPE is ensured by simultaneous Secondary DIA of both segments I and II from the temporal plots, following the correlated dependencies (22) and (23). The illustrative Differential Spectral Plot can be regarded as a “fingerprint” of the model. Differential Temporal Plot Differential Spectral Plot 52

IEES - BAS TEST: Recognize the model using the Differential Temporal Plots Centre of Excellence Model A or Circuit 1 R 1 C 1 R 2 Model B R 3 C 2 Circuit 1 Circuit 2 R 1 CPE R 1 R 2 C 1 R 2 or R 3 C 2 Circuit 2 R 1 CPE R 2 53

IEES-BAS Centre of Excellence MAIN ADVANTAGES of DIA: NO NEED of a preliminary working Hypothesis (the model is extracted from the data) VERY HIGH: · Selectivity · Noise immunity · Robustness DISTRIBUTION Recognition & Characterization 54

IEES - BAS Centre of Excellence 4. DIA APPLICATION of YSZ Solid Electrolyte About the object: high conductivity YSZ What is Known: SOFC Oxygen separators • High ionic conductivity - due to the high concentration of Y 2 O 3; • Ionic Conductivity – takes place by vacancy (hopping) mechanism; • The substitution of Y 3+ for Zr 4+ results in the formation of oxygen vacancies as compensating defects; • The concentration of oxygen vacancies and Y 3+ ions at zirconia sites is high enough for the formation of dopant-vacancy associates, which serve as vacancy trapping centres (Since the amount of Y 2 O 3 is large - about 9 wt%): 53 (Y’Zr V. . O)- (mainly) + (Y’Zr V. . O)x (less)

IEES - BAS Centre of Excellence 4. DIA of YSZ What is Uncertain and Unknown: 1. Conductivity: • The temperature dependence of mobility and concentration of “free” vacancies able to take part in the transport process; • Possible changes of the activation energy Ea with the temperature? • Possibility formation of vacancy ordered micro-domain structures due to the high concentration of the acceptors and thus to possibility for complex defect interactions. 2. Surface oxygen exchange: The current trends for reduction of SOFC operating temperature below 1000 o. C concern not only the increase of the electrolyte conductivity and the electrodes catalytic activity, but also the surface oxygen exchange, which may become the rate limiting step in the oxygen transfer process. 56

IEES - BAS Centre of Excellence 4. DIA of YSZ Advantages of the DIA Studies : The application of techniques for common investigation of both the oxygen exchange and ionic conductivity of the bulk oxide will provide for better reevaluation of YSZ transport properties. • One possibility is the application of isotope exchange and secondary ion mass spectrometry (SIMS). The small number of publications is due to the difficulties in performing tracer experiments on solid electrolytes; • EIS can investigate both the ionic conductivity and oxygen surface exchange, since it ensures clear distinction between the bulk and grain boundary resistance and the electrode reaction. • The application of DIA eliminates the need of preliminary hypotheses for the description of the complicated electrode reaction, which should be divided into individual steps, involving charge transfer across the interface, as well as noncharge transfer processes such as adsorption, solid state diffusion, gas phase diffusion etc. 57

IEES - BAS 4. DIA of YSZ Centre of Excellence Experimental: YSZ Single crystal (9. 5 mol % Y 2 O 3) Polycrystalline (8. 5 mol % Y 2 O 3) Electrodes: (Platinum) Instrument • Solartron FRA 1260 Conditions: • Amplitude = 50 m. V; • Frequency range: 13 MHz - 0, 1 Hz; • density: 9 points/decade; • temperatures: room to 1000 o C • parasitic inductance elimination 58

IEES - BAS Centre of Excellence 4. DIA of YSZ Experimental: Impedance diagrams of YSZ single crystal: • frequency range: 13 MHz – 0. 1 Hz; • temperature interval: room – 1000 o C; • data after L-correction. 59

IEES - BAS 4. DIA of YSZ Centre of Excellence Typical impedance diagrams of YSZ in different temperature zones T > 500°C T < 500°C YSZ – single crystal YSZ – poly 60

IEES - BAS 4. DIA of YSZ Centre of Excellence Typical impedance diagrams of YSZ in different temperature zones YSZ – single crystal T < 500°C & T > 500°C The high frequency depressed semicircle of the single crystal sample, denoted as Segment I, corresponds to the bulk properties. YSZ – poly T < 500°C & T > 500°C The expected arcs for the bulk and for the grain boundaries of the polycrystalline material are strongly mixed and define only one segment (Segment I) with a similar shape. YSZ – single crystal & YSZ – polycrystalline The lower frequency Segment II corresponds to the electrode reaction. At T < 500 o C it is presented with a straight line. At high temperatures its shape takes the form of a distorted semicircle for both single crystal and polycrystalline sample. 61

IEES - BAS Centre of Excellence 4. DIA of YSZ General Remarks The examples based on single crystal and polycrystalline samples of YSZ demonstrate the effect of the combined application of the temporal and the differential temporal analyses to the complete identification of the corresponding models. For DIA investigation of both the electrode and the reaction (oxygen exchange) conductivities, which have response in different frequency regions, precise frequency segmentation is performed in the R - temporal plot. In this study the R -temporal plot is mainly used since it is the basis for parametric identification of the resistance, which ensures quantitative characterization of the phenomena under investigation. 62

IEES - BAS 4. DIA of YSZ Centre of Excellence YSZ Single Crystal : r- Temporal plot segmentation There are two well distinguished regions with different temporal behavior. The high frequency Segment I forms a plateau, i. e. it is frequency independent and determines the bulk resistance of the single crystal. The r, c and τ spectral images give a single spectrum, which confirms the timeconstant model, i. e. a parallel connection of resistance and capacitance. The low frequency Segment II is frequency dependent and a Secondary DIA has to be performed for its identification. Spectral plot of Segment I Temporal plot r c τ C R 63

IEES - BAS Centre of Excellence 4. DIA of YSZ Polycrystalline : r- Temporal plot segmentation Segment I combines two regions – IA and IB. Both of them are frequency invariant and thus they recognize the characteristic for this system two timeconstant model with Voigt’s structure, applied to solid state. Sub-segment I A determines the bulk resistance Rb and sub-segment I B - the grain boundary resistance Rgb. The two spectral lines in the r, c and τ spectral plots confirm the identification of a two time-constant model. The low frequency Segment II is frequency dependent and a Secondary DIA has to be performed for its identification. C C R R 64

IEES - BAS Centre of Excellence 4. DIA ON YSZ Single Crystal : Secondary DIA (below 500 o. C) The investigation of the electrode reaction concerns the analysis of Segment II, which requires Secondary DIA for both the single crystal and the polycrystalline samples. Since the approach is the same and the results are similar, only the example with the single crystal sample is included. The Secondary DIA of Segment II at temperatures below 500 o C shows, that the parameters , and obey the mutual relationships (20), which identify CPE of Warburg type, since the values of n vary in the interval between 0. 5 and 0. 6 for that temperature range. Temporal plot of r Temporal plot of Differential Spectral Plot 65

IEES - BAS Centre of Excellence 4. DIA ON YSZ Single Crystal : Secondary DIA (above 500 o. C) At temperatures above 500 o C Segment II splits into two sub-regions IIA and IIB. The Differential Temporal Analysis confirms the existence of the correlations given by Eqns. (22) and (23). It recognizes a modified time-constant model with capacitive CPE (the estimate of the CPE coefficient n is about 0. 8). CPE (n ~ 0. 8) Temporal Plot Recognized model: R Differential Spectral Plot Differential Temporal Plot 66

IEES - BAS 4. DIA ON YSZ Centre of Excellence Arrhenius Plots and Activation Energy Ea (bulk and grain boundary) The parametric identification of the resistivity corresponding to the bulk, to the grain boundaries and to the electrode reaction processes ensures the building of the Arrhenius plots and thus the calculation of their activation energies Ea. The Arrhenius plots for both the single crystal and the polycrystalline sample are characterized with a well pronounced kink in the range about 650 o. C. The activation energies Ea of the bulk and the grain boundaries are smaller at higher temperatures. YSZ – polycrystalline YSZ – single crystal 7 6 lg r 5 4 Ea = 1. 10; 0. 31 e. V Ea = 1, 14 e. V; (g. b) (bulk) Ea = 1. 13; 0. 29 e. V (bulk) 3 2 1 0 0, 5 67 1, 0 1, 5 -1 2, 0 2, 5

IEES - BAS 4. DIA ON YSZ Centre of Excellence Arrhenius plots and calculation of Ea (for the bulk and electrode reaction of single crystal) There is a kink in the Arrhenius plot of the electrode reaction. It marks a change in the activation energy from 0. 66 e. V at temperatures below 650 o. C to 1, 21 e. V at temperatures above 650 o. C. The observed changes of the activation energy of the bulk conductivity and the oxygen exchange reaction at one and the same temperature (650 o. C) are an evidence for the correlation between the bulk and surface properties. Ea = 1, 12; 0, 66 e. V reaction YSZ bulk 68

IEES Centre of Excellence The technique of the DIA was developed for improvement of the model identification, eliminating the subjective step of an a priori selected hypothetical model(s) (the weak point in the classical (parametric) identification). Thus it is expected, that the combination of the two procedures would increase the analytical power and information capability of the Electrochemical Impedance Spectroscopy. 69

IEES Centre of Excellence It is my pleasure to Acknowledge with gratitude: • EC – EESD - Part B: Energy Program (project NNE 5/2002/18) and • The ROYAL SOCIETY for the financial support; • UNESCO – UVO ROISTE and EICIS for the professional discussions environment; • MY COLLEAGUES: Zdravko Stoynov, who invented and advances the DIA technique, Geri Raikova from IEES, John Kilner and Stephen Skinner from Imperial College – London for the wonderful partnership and fruitful collaboration. 70

Journal of Electroanalytical Chemistry Vol. 572, 2004, pp 377 -387 SECONDARY DIFFERENTIAL IMPEDANCE ANALVSIS – A TOOL FOR RECOGNITION OF CPE BEHAVIOR D. Vladikova, Z. Stoynov Central Laboratory of Electrochemical Power Sources, Bulgarian Academy of Sciences Abstract The procedure of a secondary DIA, which analyses the frequency dependence of the local operating model parameters’ estimates in the zones where there is no adequacy between the object and the local estimator. The new algorithm is successfully examined on synthetic models of CPE, Warburg, Randles with CPE modification, simple Faradaic reaction with CPE capacitance, as well as on real data obtained on yttrium iron garnet single crystals and yttria stabilized zirconia single crystals. Keywords: ac impedance; Conductivity; Diffusion; CPE; Differential impedance analysis; Model recognition 71

Journal of the European Ceramic Society Vol. 24, 2004, pp 1121 -1127 APPLICATION OF THE DIFFERENTIAL IMPEDANCE ANALYSIS FOR INVESTIGATION OF ELECTROCERAMICS D. Vladikova 1, Z. Stoynov 1, M. Viviani 2 1 Central Laboratory of Electrochemical Power Sources, Bulgarian Academy of Sciences 2 National Research Council, Institute for Energetics and Interphases, Genoa Department Abstract This work shows the enhanced performance of Impedance Spectroscopy for investigation and characterization of electroceramics by applying a new structural approach for data analysis, called Differential Impedance Analysis (DIA). The main advantage of this technique is the possibility for model identification directly from the experimental data, i. e. without the use of a preliminary working hypothesis. DIA provides for separation and phenomenological characterization of the different steps involved in the investigated object. The capabilities of the method are demonstrated in conductivity studies of yttrium iron garnet (YIG) single crystal and Er doped PTCR Ba. Ti. O 3. The analysis ensures a more detailed investigation of the bulk properties of YIG, including the separation and characterization of the hopping conductivity. The application of DIA on Ba. Ti. O 3 enriches the information about the role of the ferroelectric domain structure on the PTCR. Keywords: Ba. Ti. O 3 and titanates; Differential impedance analysis; Electrical conductivity; 72 Ferrites; Ferroelectric properties

Bulgarian Chemical Communications, Volume 36, Number 1, 2004, pp 29 -40 CONDUCTIVITY STUDIES OF SOLID OXIDE MATERIALS FOR ELECTRICAL APPLICATIONS D. Vladikova Institute of Electrochemistry and Energy Systems, Bulgarian Academy of Sciences Abstract Conductivity studies of solid oxide electroceramics. The method can distinguish between the bulk and the grain boundary contributions, as well as the electrode reaction. Both advantages and problems are discussed. In addition some recent improvements of the impedance data analysis, based on the technique of the Differential Impedance Analysis, are presented. They are illustrated with examples of conductivity studies on yttrium iron garnet single crystal with ferrimagnetic properties and Er doped barium titanate with positive temperature coefficient of the resistivity. Key words: conductivity, electroceramics, impedance spectroscopy, yttrium iron garnet, Er doped barium Key words: 73

Bulgarian Chemical Communications, Volume 36, Number 1, 2004, pp 66 -71 DIFFERENTIAL IMPEDANCE ANALYSIS OF BOUNDED CONSTANT PHASE ELEMENT G. Raikova, D. Vladikova, Z. Stoynov Institute of Electrochemistry and Energy Systems, Bulgarian Academy of Sciences Abstract The technique of the Differential Impedance Analysis (DIA) is applied for the identification of the so called bounded constant phase element (BCP), which describes the impedance of a bounded homogeneous layer with finite thickness and conductivity following constant phase element (CPE) behaviour. The enhanced identification capability of DIA is approbated on two models – a BCP one and a model of a simple Faradaic reaction with similar shapes of the impedance diagrams, presented with part of a semi-circle. DIA ensures a clear distinction between the two models. Key words: Electrochemical Impedance Spectroscopy, Constant Phase Element, Bounded Constant Phase Element, Differential Impedance Analysis 74

Electrochimica Acta Vol. 47, pp 2943 -2951, 2002 SELECTIVITY STUDY OF THE DIFFERENTIAL IMPEDANCE ANALYSIS - COMPARISON WITH THE COMPLEX NON-LINEAR LEAST-SQUARES METHOD D. Vladikovaa, P. Zoltowskib, E. Makowskab, Z. Stoynova a. Central Laboratory of Electrochemical Power Sources, Bulgarian Academy of Sciences b. Institute of Physical Chemistry, Polish Academy of Sciences Abstract The analytical power of the differential impedance analysis (DIA) as a new approach for model identification is investigated. The minimum of the ratio q of the two time-constants T 1 and T 2 (q = T 2/T 1) describing a two step reaction with ladder structure, at which the second time-constant T 2 is recognized, is accepted as a measure for the selectivity evaluation. The investigation is performed on synthetic models with variation of q in the interval from 30 to 0. 001. The influence of data quality is studied by applying data truncation from seven to two digits, as well as by introduction of a wild point. Experimental data measured on two actual dummy cells with q equal to 30 and 0. 03, respectively, are also analyzed. The results are compared with those obtained by applying the algorithm of the wide-spread complex non-linear least-squares method, performed on Zview software product. The comparative study demonstrates a higher selectivity of the DIA, combined with an additional noise immunity and increased robustness of the analysis. 75 Keywords: Electrochemical impedance spectroscopy; Differential impedance analysis; Complex non-linear least-squares; System identification; Selectivity

15ème Forum sur les Impedances Electrochimiques, 9 décembre 2002, Paris, France, pp 3 -14 DIFFERENTIAL IMPEDANCE ANALYSIS – PRINCIPLE AND APPLICATION Z. Stoynov Central Laboratory of Electrochemical Power Sources, Bulgarian Academy of Sciences Abstract The development of the modern instrumentation for Impedance Spectroscopy has led to its intensive application for investigation of large variety of practical objects. The classical identification of appropriate models for those objects meets strong limitations caused by the lack of theoretical developments as well as by the fuzzy nature of the studied processes. The aim of the Differential Impedance Analysis is the overcome these limitations and to serve as powerful analytical tool for investigation of unknown objects or objects with distributed parameters. Key words: Impedance Spectroscopy, Differential Analysis, Fuzzy Modeling, Spectral Transform. 76

15ème Forum sur les Impedances Electrochimiques, 9 décembre 2002, Paris, France, pp 3 -14 ANALYSE SPECTRALE D'IMPÉDANCE DIFFÉRENTTIELLE DES MODÈLES CINTÉTIQUES FOUNDAMENTAUX Z. Stoynov 1, H. Takenouti 2, M. Keddam 2, D. Vladikova 1, G. Raikova 1 1 Central Laboratory of Electrochemical Power Sources, Bulgarian Academy of Sciences 2 UPR 15 du CNRS, Physique des Liauides et Electrochimie, Universite Pierre et Marie Curie Abstract L’Analyse d’impédance différentielle (DIA) est appliquée pour les différents modèles cinétiques fondamentaux afin d’établir reconnaissance de modèle mécanistique par une nouvelle approche structurale. La comparaison des résultats expérimentaux et la structure prédéfinie permettra alors de déterminer sans hypothèse préliminaire, le mécanisme réactionnel adéquat. Les modèles de bases utilisent les schémas électriques équivalents déduits de réaction électrochimique; comme exemple, le transfert de charge en une seule étape et ceux impliquant une ou deux espèces intermédiaires de réaction. Nous illustrerons notamment dans le cas où la constante de temps de relaxation d’espèce intermédiaire est proche de celle définie par la capacité de double couche et la résistance de transfer! De charge. Nous présenterons également comment le modèle avec une capacité en parallèle avec une resistance permet d’appréhender l’impédance faradique sous forme inductive. 77

Materials for Lithium-Ion Batterie, Eds. C. Julien and Z. Stoynov NATO Science Series 3. High Technology –Vol. 85 Kluwer Academic Publishers pp 371 -380, 2000 ADVANCED IMPEDANCE TECHNIQUES FOR LITHIUM BATTERIES STUDY Part IV: Differential Impedance Analysis Z. Stoynov Central Laboratory of Electrochemical Power Sources, Bulgarian Academy of Sciences Abstract The aim of the experimental analysis is the interpretation of impedance data. It can follow two different pathways: the confirmation of a preliminary stated hypothetical model, or alternatively the derivation of the working model from the experimental result itself. The method of frequency scanning analysis based on a moving point to point estimator is summarized. The essentials of the local differential analysis are presented. The important properties of this new method are discussed. 78

Polish J. Chem. , Vol. 71, 1997, pp 1204 -1210 DIFFERENTIAL IMPEDANCE ANALYSIS – AN INSIGHT INTO THE EXPERIMENTAL DATA Z. Stoynov Central Laboratory of Electrochemical Power Sources, Bulgarian Academy of Sciences Abstract A new method for analysis of the experimental data of the impedance spectroscopy is presented. The method is based on the assumption of a local operating model. Using the derivatives of the impedance with respect to frequency, the model parameters and their effective time-constants are calculated for each frequency. The analytical equations are used as a moving estimator over the entire frequency range. Important properties of the studied system can be derived. Key words: impedance analysis, impedance spectroscopy, splines 79

Electrochimica Acta Vol. 35, No. 10, 1990, pp 1493 -1499 IMPEDANCE MODELLING AND DATA PROCESSING: STRUCTURAL AND PARAMFTRICAL ESTIMATION Z. Stoynov Central Laboratory of Electrochemical Power Sources, Bulgarian Academy of Sciences Abstract Identification of the appropriate model is the aim of experimental data analysis. Some principal problems of the classical approach for parametrical identification are discussed. A new method for structural identification where the working model is derived from the experimental results is developed. Some recent results which expand the applicability of the method arc also given. Key words: electrochemical impedance, identification, structural spectroscopy. 80

Electrochimica Acta Vol. 34, No. 8, 1989, pp 1187 -1192 STRUCTURAL SPECTRAL ANALYSIS OF ELECTROCHEMICAL IMPEDANCE Z. Stoynov Central Laboratory of Electrochemical Power Sources, Bulgarian Academy of Sciences Abstract The interpretation of impedance data can follow two different pathways: the confirmation of a preliminary stated hypothetical model, or alternatively the derivation of the working model from the experimental result itself. The solution of the second problem is the goal of the new method described in this paper. The method recognizes the presence of the basic phenomena (time constants or steps of the reaction, diffusion limitations, adsorption, distribution of the parameters) and their configuration. After the estimation of their parameters the main results (basic structure and dominant parameters) are presented in temporal and spectral forms. 81

Electrochimica Acta Vol. XX, No. XX, 2005 DIFFERENTIAL IMPEDANCE ANALYSIS OF SINGLE CRYSTAL AND POLYCRYSTALLINE YTTRIA STABILIZED ZIRCONIA D. Vladikovaa, J. A. Kilnerb, S. J. Skinnerb, G. Raikovaa, Z. Stoynova a. Institute of Electrochemistry and Energy Systems– Bulgarian Academy of Sciences b. Centre for Ion Conducting Membranes, Department of Materials, Imperial College of Science, Technology and Medicine, London Abstract This study aims at a more detailed simultaneous investigation of the bulk, grain boundary and electrode reaction behaviour of YSZ by the technique of the Differential Impedance Analysis. The experiments are performed on both single crystal and polycrystalline samples in a wide temperature interval – from 200 o. C up to 950 o. C. For improvement of the measurement accuracy at high temperatures, a procedure for correction of the parasitic inductance is introduced. Key words: Electrochemical impedance spectroscopy, Yttria stabilized zirconia, Differential impedance analysis, Bulk conductivity, Grain boundary conductivity, Electrode reaction 82

Electrochimica Acta Vol. XX, No. XX, 2005 IMPEDANCE ANALYSIS OF OXYGEN REDUCTION IN SOFC COMPOSITE ELECTRODES A. Barbucci, a M. Viviani, b P. Carpanese, a D. Vladikova, c Z. Stoynovc a. DIChe. P, Università di Genova, Italy b. CNR, IENI, Genova, Italy c. IEES – BAS, Sofia, Bulgaria Abstract The electrochemistry of oxygen reduction porous composite electrodes consisting of La(1 -x)Srx. Mn. O 3 -d (LSM) and Y-stabilised Zirconia (YSZ) has been analysed for better understanding of the mechanism of oxygen reduction and improvement of the cathode’s performance. Half cells consisting of YSZ electrolyte pellets and slurry coated cathodes were tested with a three electrodes configuration. The composite cathodes considered in this study have a fixed volume ratio LSM/YSZ equal to 1 and variable apparent surfaces, ranging from 0. 022 to 0. 23 cm 2. The dependence of electrode resistance on size indicates that the process takes place in the bulk of the cathodes. Part of the impedance data is additionally analyzed by the technique of the Differential Impedance Analysis (DIA), which does not need a preliminary working hypothesis. The application of DIA gives additional information about the dominant phenomena, based on comparative study of the cathode behaviour of LSM and the composite materials. 83 Key words: Electrochemical impedance spectroscopy, Solid oxide fuel cells, LSM/YSZ Composite cathode materials, Differential impedance analysis

IEES-BAS Centre of Excellence Differential Temporal Plot 3. SECONDARY DIA TEST Example : Find the value of the CPE coefficient using the given Differential plots. NO Differential Spectral Plot The value is not correct! 84

IEES-BAS Centre of Excellence Differential Temporal Plot 3. SECONDARY DIA TEST Example: Find the value of the CPE coefficient using the given Differential plots 0, 5 1 Warburg Differential Spectral Plot The value is correct! You have recognized Warburg Impedance, Impedance since n = (1 – n) = 0. 5 85

IEES - BAS Centre of Excellence YES! R 1 CPE R 2 R 1 C 1 Model A Circuit 2 Model B R 2 R 3 C 2 Circuit 1 86

IEES - BAS Centre of Excellence N O !!! R 1 C 1 R 2 R 3 C 2 R 1 CPE Model A Circuit 1 Model B R 2 Circuit 2 87