Basics of the Kramers-Kronig- and the Z-Hit algorithms Lecture at the Kronach Impedance Days 2015 Dr. Werner Strunz

Transcription

1 Theoretical Aspects for Testing Impedance Data Basics of the Kramers-Kronig- and the Z-Hit algorithms Lecture at the Kronach Impedance Days 215 Dr. Werner Strunz 1

2 Motivation The entire process of measurement, interpretation and analysis of EIS data usually aims toward winning a set of these characteristic parameters. We want to see results like: Flatband potential = V Effective gas diffusion length =.2.1 m Corrosion rate =.3.1 mol m -2 a -1 The importance of an information grows with the accuracy of the results. An unknown uncertainty invalidates the results. 2

3 Problems of Daily Life 3

4 The most important uncertainty contributions: Uncertainties Caused by the Analysis Procedure Imperfectness of the modeling: Idealizing theory, approximating theory, ambiguous impedance network representations, unconsidered spectral contributions. Imperfectness of the fitting procedure: Sensitivity on the starting parameter values, sticking on local minima, early fitting abort. Both the quality of the experimental data as well as the uncertainties coming from the analysis procedure affect the numeric accuracy of the finally output parameters! Most EIS simulation and fitting programs consider only the fitting quality but are unable to take the quality of the experimental data into account unrealistic parameter accuracy estimation. 4

5 EIS-Measurement at a Single Frequency 5

6 response The basics of Weighted Harmonics Autocorrelation Monochromatic Oversampling Excitation-signal: sinusoidal One sharp frequency line Response-signal under ideal conditions: as well exactly one line in the spectrum time 6

7 The basics of Weighted Harmonics Autocorrelation Monochromatic Oversampling Under realistic conditions (in the presence of any disturbance) - - the response signal contains noise! The additional lines, appearing in the frequency spectrum can be unambiguously assigned to unwanted distortions! 7

8 Uncertainty estimation based on harmonic analysis response The appearance of isolated frequency lines does not mean, that the accuracy is impaired. Isolated frequency lines appear, if a distortion component matches exactly a harmonic of the signal time + no uncompensated residual The uncompensated residuals vanish in the case of periodicity. In that case the Fourier transform filter is able to remove the distortion components perfectly. 8

9 Uncertainty estimation based on harmonic analysis response response High frequencies contribute less to errors than low frequencies uncompensated residual uncompensated residual time Distortion residual at an interference frequency of F signal * 3.5 time Smaller distortion residual at an interference frequency of F signal * 7.5 Blue: Signal Red: Interference 9

10 Results of process simulations Actual Deviation Estimate Deviation Digital simulation and calibration of the weighted harmonics autocorrelation procedure Sine-shaped distortion signal with comparable amplitude under variation of the frequency ratio. The time window is one period of the signal frequency Actual deviation (blue) vs. predicted deviation provided by the weighted harmonics autocorrelation method (red) The estimation correlates with the real deviation, except for distortion frequencies close to or identical with the signal frequency. This is a fundamental problem Frequency Ratio F inter / F meas 1.5 1

11 Weighted Harmonics Autocorrelation (WHA) On-line error determination and processing for electrochemical impedance spectroscopy measurement data based on weighted harmonics autocorrelation C. A. Schiller, R. Kaus; Bulgarian Chemical Communications, Volume 41, Number 2 (pp ) 29 Consistent Discussion of the Uncertainty of Physical Parameters Evaluated by EIS, Based on an Automatic Measurement Error Determination C.A. Schiller, R. Kaus; ECS Transactions, 25 (32) (21) 11

12 From Measurement to Parameters Raw Data Z * i, H i,o WHA Z * i, i,o original data smoothed data ZHIT data generally not equidistant in log(f) NOISE SPIKES equidistant in log(f) DRIFT equidistant in log(f) CNLRS fitter model P 1,P 2,...,P. n 12

13 The Validation of Experimental Impedance Data Detection and reconstruction (!!) of non- steady and/or disturbed systems (see Wkipedia ZHIT (currently:de)) 13

14 Motivation Development and/or improvement of important technical products Fuel cells Batteries Rechargeable batteries Solar cells Coatings NON-STATIONARY Under CONDITIONS load (may) result in Under illumination Water NON-STATIONARY uptake SPECTRA 14

15 Spectrum of a Fuel Cell Under Load (I) 3 Z / m phase / o artifact m 1m K 1K 1K frequency / Hz Reliable detection of artifacts 15

16 functions E(t) & I(t) EIS-Principle at a Single Frequency Excitation at constant frequency 5µ 1m 1.5m 2m 2.5m 3m time /s Z How to validate EIS-spectra? What s the specific property? 16

17 17 The Kramers-Kronig Relations 2 2 ) ( Im 2 - () = Re ) ( Re d H PV H H 2 2 ) ( Re 2 = ) ( Im d H PV H BUT WHERE ARE THE PROBLEMS?

18 function y(t) function y(t) function y(t) Mathematical Relations (Real-Valued Quantity) y( t) e t y( t) at m time /s 5µ 1m 1.5m 2m 2.5m 3m time /s y( t) time /s sin t Unequivocal relationship between dependent and independent variables => y(t) is determined / measured with a distinct accuracy 18

19 functions E(t) & I(t) Mathematical Relations (EIS) (Complex-Valued Quantity) Excitation at constant frequency 5µ 1m 1.5m 2m 2.5m 3m time /s Z - Z & : measured independently with different accuracy and sensitivity - Z & : strongly correlated (in theory) BUT IN PRACTICE? 19

20 The Sensitivity of Objects (Z & ) - Excellent Examples: Sensors! - Temperature Dependent Resistor (NTC, PTC) Pt 1, Pt 1, KTY 81, - Light Dependent Resistor (LDR) - Magnetic Dependent Resistor (MDR) - Humidity Dependent Capacity -.. 2

21 Phase / Phase / The Course of Phase and Impedance when Heating NTC/PTC LOG Impedance / 4, 3,9 3,8 3,7 3,6 3,5 KTY 1k (NTC) LOG Impedance / 3,15 3,1 3,5 Pt 1 (PTC) , LOG frequency / Hz LOG frequency / Hz Z & : Phase is more stable than impedance Z 21

22 The Z-HIT Approximation (evaluation of impedance modulus from the phase angle ln H( ) const.+ 2 ( ) d S ln d( O dln ) Detection of artifacts Detection of instationarities (drift) Reconstruction of causal spectra => Reliable interpretation of spectra 22

23 Deduction of the Z-HIT (IV): - Relationship of elementary two-poles log Z R Z ' = Z ' = 1 L C Z ' = -1 W Z ' = H j const j d ln d H ln j log f Z' 23

24 ln Z Deduction of the Z-HIT (V) - refinement Randle circuit E [A] ln ( O ) S Phase / rad, -,2 -,4 -,6 -,8-1, -1,2-1, [B] Integral (shifted) ln Z ,5,4,3,2,1, -,1 -,2 -,3 -,4 -,5 -,6 -,7 [C] ln Z - Integral () d / dln ,5,4,3,2,1, -,1 -,2 -,3 -,4 -,5 -,6 -,7 [D] ln Z - Integral () /6 * d / d ln

25 25 The Z-HIT Approximation - frequency boundaries ln ) ( ln ) ( 2 onst.+ ) ( ln d d d c H O S Considering Kramers Kronig relations 2 2 ) ( Re 2 = ) ( Im d H PV H

26 The Limited Bandwidth Problem (I) Simulation of a coating during water up-take Measured frequency range 1 KHz 5 mhz :? :? 26

27 Implementation of the Z-HIT Algorithm in the THALES Analysis Software Package 1) The experimental data are filtered by a smoothing algorithm. The result is a set of continuous samples equidistant in log f. 2) The integral term is calculated by numerical integration. 3) The first derivate is taken from the smoothing function. 4) The integration constant is determined by a least squares fit. ln 2 d( O) H( ) ( ) d ln const. d ln S 27

28 Applications 28

29 Application Spectrum of a fuel cell under load 3 25 Z / m phase / o 6 1. Z-HIT 3 25 Z / m phase / o m 1m K 1K 1K frequency / Hz 2. FIT 1-3 1m 1m K 1K 1K frequency / Hz 29

30 cell voltage / mv Application Fuel Cell under CO Poisoning (I) Series measurement ~ 1 minutes per spectrum Strong influence No. Rapid 6 changes 3 2,,5 1, 1,5 2, time / h Relaxation impedance as a model for the deactivation mechanism of fuel cells due to CO poisoning C. A. Schiller, F. Richter, E. Gülzow, N. Wagner; J. Phys. Chem. Chem. Phys. 3 (21)

31 cell voltage / mv Application Fuel Cell under CO Poisoning (II) - imaginary part / Ohm Series measurement ~ 1 minutes per spectrum 7 6 Strong influence 15m 1m 5 4 No. Rapid 6 changes 5m -5m 1, m 2,,5 1, 1,5 2, time / h 1m 2m 3m 4m real part / Ohm Relaxation impedance as a model for the deactivation mechanism of fuel cells due to CO poisoning C. A. Schiller, F. Richter, E. Gülzow, N. Wagner; J. Phys. Chem. Chem. Phys. 3 (21)

32 Application Fuel Cell under CO Poisoning (II) Raw data Refined data,1.1 Hz DATA FIT,1.1 Hz DATA FIT imaginary part /, -,1 1K 2 Hz 3 Hz imaginary part /, -,1 1 KHz 2 Hz 3 Hz -,2 -,2,,1,2,3,4,5 real part /,,1,2,3,4,5 real part / 32

33 Application Water Uptake of Coatings (I) Series measurement ~ 2 minutes / spectrum log(impedance) / 1. spec 2. spec (2 min.) 84. spec (26 h) log frequency /Hz phase shift / log(impedance) / spec 2. spec (2 min.) 33. spec (11 h) phase shift / log frequency /Hz => Water uptake: a very slow process

34 Application Water Uptake of Coatings (II) 1,4 1,4 1,2 1st zoomed 1,2 2nd zoomed 1, 1, 9,8 1. spec 9,8 2. spec 9,6 Data Z-HIT -1,8-1,6-1,4-1,2-1, -,8 -,6 9,6 Data Z-HIT 9,4-1,8-1,6-1,4-1,2-1, -,8 -,6 log(impedance) / spec complete Data Z-HIT Coating A (11 µm) log(frequency) / Hz Only the lowest frequencies are affected Only at the earliest spectra 34

35 Water Uptake - Waterborne Coating Series measurement Z / phase / o 1G 1M 1M 1M 1K a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a st..4 th spectrum a a a 1K a a a a a a a a a 5 th spectrum 1K a K 3K 1K 1K frequency / Hz With kind permission of U. Christ, Fraunhofer-Institut für Produktionstechnik und Automatisierung IPA, Stuttgart 35

36 Water Uptake - Waterborne Coating With kind permission of U. Christ, Fraunhofer-Institut für Produktionstechnik und Automatisierung IPA, Stuttgart 36

37 Battery under Load - Mutual Inductance & Drift High-frequency Data (inductance) With kind permission of R. Gross, bno-consult, Würzburg 37

38 Lithium-Ironphosphate Check for Drift AND (if possible) Elimination Part of daily live 38

39 Estimate of Accuracy (I) = 1 =.5 39

40 Estimate of Accuracy (II) ZSIM ZZ HIT rel. Error 1 Z SIM = 1 =.5 4

41 Conclusion Z-HIT Approximation ln H( ) const.+ 2 ( ) d S ln d( O dln ) Local relationship between impedance and phase => Not affected by the limited bandwidth problem => Reliable detection of artifacts and instationarities (drift) => Reconstruction (!!) of causal spectra => Reliable interpretation of spectra 41

42 Thank you for your attention 42