Theoretical Aspects for Testing Impedance Data Basics of the Kramers-Kronig- and the Z-Hit algorithms

Transcription

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

2 The validation of experimental impedance data Detection and reconstruction (!! of non- steady and/or disturbed systems Fuel cells Batteries Rechargeable batteries Solar cells Coatings Motivation Development and/or improvement of important technical products NN-STATINARY UnderCNDITINS load (may result in Under illumination Water NN-STATINARY uptake SPECTRA 1 Spectrum of a fuel cell under load (I EIS-principle at a single frequency Z / mω 3 1 phase / o 3 artifact functions E(t & I(t Excitation at constant frequency µ 1m 1.m m.m 3m ϕ -3 1m 1m K 1K 1K frequency / Hz Reliable detection of artifacts How to validate EIS-spectra? What s the specific property? 3 Re Im The Kramers-Kronig Kronig Relations { H ( } = Re{ H (} { H ( } = PV - PV { H ( } Im { H ( } Re d d BUT WHERE ARE THE PRBLEMS? function y(t function y(t Mathematical relations (real real-valued quantity 3 1 λ y( t = e t y( t = m y( t = sin( t µ 1m 1.m m.m 3m function y(t a t Unequivocal relationship between dependent and independent variables => y(t is determined / measured with a distinct accuracy

3 Mathematical relations (EIS (complex complex-valued quantity The Sensitivity of bjects (Z & ϕ - Excellent Examples: : Sensors! functions E(t & I(t Excitation at constant frequency µ 1m 1.m m.m 3m ϕ - Z & ϕ : measured independently with different accuracy and sensitivity - Z & ϕ : strongly correlated (in theory BUT IN PRACTICE? Z - Temperature Dependent Resistor (NTC, PTC Pt 1, Pt 1, KTY 81, - Light Dependent Resistor (LDR - Magnetic Dependent Resistor (MDR - Humidity Dependent Capacity The course of Phase and Impedance when heating NTC/PTC The approximation (evaluation of impedance modulus from the phase angle ln H ( const. + ϕ( ln γ d + dln S Detection of artifacts Detection of instationarities (drift Reconstruction of causal spectra Z & ϕ : Phase ϕ is more stable than impedance Z => Reliable interpretation of spectra 9 1 Deduction of the (I Deduction of the (II Find a general relationship between impedance and phase Based on an empirical observation (Bode plot ( Integral / Hilbert transformation log H R = 1 Ω Phase Impedance of R H (R f (ln( = const Phase shift of R ϕ(r f (ln(= const = ln H ( ~ ϕ( d ln +? Integral = = > Integral ϕ ( = Must hold for all impedance elements (-pole ln H ( ~ ϕ( d ln+ const. 11 1

4 Deduction of the (III Impedance of C 1 pf H (C = f (ln( ~ (C -1 log H Phase Phase shift of C ϕ(c f (ln( = const = -9 = > Integral ϕ ( s = Integral log Z j Deduction of the (IV: - Relationship of elementary two-poles R L C W Z ' = - 1 Z ' = Z ' = 1 Z ' = -1 H ( j = const( j ϕ = ( dln H = α dln α α s ln H ( ~ ϕ( d ln+ const. Z' = α 13 1 Randle circuit Deduction of the (V ln Z E [A] - refinement Phase / rad S (, 1 -, -, -, 1 -,8 13-1, -1, 1-1, [B] Integral (shifted ln Z ln The approximation ln H ( const. + - frequency boundaries ϕ( ln γ d + dln S,,,3,,1, -,1 -, -,3 -, -, -, [C] ln Z - Integral (ϕ dϕ / dln -, ,,,3,,1, -,1 -, -,3 -, -, -, -,7 [D] ln Z - Integral (ϕ / * dϕ / d ln Considering Kramers Kronig relations Im { H ( } = PV { H ( } Re d 1 The limited bandwidth problem (I Simulation of a coating during water up-take The limited bandwidth problem (II 1 pf log H Phase Changing frequency boundary Measured frequency range 1 KHz mhz :? :? s s s to s nly Shift of const LCAL relationship between impedance and phase Not affected by the limited bandwidth 17 18

5 Implementation of the Z algorithm in the THALES analysis software package Spectrum of a fuel cell under load 1 The experimental data are filtered by a smoothing algorithm. The result is a set of continuous samples equidistant in. Z / mω 3 phase / o 1. Z / mω 3 phase / o The integral term is calculated by numerical integration The first derivate is taken from the smoothing function. The integration constant is determined by a least squares fit m 1m K 1K 1K frequency / Hz. FIT 1-3 1m 1m K 1K 1K frequency / Hz ln H ( ϕ( d ln + γ d ln + c S onst. 19 Fuel cell under C poisoning (I Fuel cell under C poisoning (II cell voltage / mv 7 3 Series measurement ~ 1 minutes per spectrum Strong influence No. Rapid changes,, 1, 1,, time / h - imaginary part / hm m 1m m -m -1m 1, m m 3m m real part / hm imaginary part / Ω,1, -,1 -, 1K Raw data.1 Hz DATA FIT Hz 3 Hz imaginary part / Ω,1, -,1 -, Refined data.1 Hz 1 KHz DATA FIT Hz 3 Hz Relaxation impedance as a model for the deactivation mechanism of fuel cells due to C poisoning C. A. Schiller, F. Richter, E. Gülzow, N. Wagner; J. Phys. Chem. Chem. Phys. 3 (1 113,,1,,3,, real part / Ω,,1,,3,, real part / Ω 1 Batteries: optimising wiring Water uptake of coatings (I Series measurement ~ minutes / spectrum log(impedance / Ω phase shift / log(impedance / Ω phase shift / spec 1. spec. spec ( min.. spec ( min. 8. spec ( h 33. spec (11 h requency /Hz requency /Hz => Water uptake: a very slow process 3

6 Water uptake of coatings (II log(impedance / Ω 1, 1, 1, 9,8 9, spec Data -1,8-1, -1, -1, -1, -,8 -, 1. spec complete Data 1st zoomed 9, -1,8-1, -1, -1, -1, -,8 -, log(frequency / Hz nly the lowest frequencies are affected nly at the earliest spectra 1, 1, 1, 9,8 9,. spec Data Coating A (11 µm nd zoomed capacity / F p p 3p 3p p p p 1p Water uptake of coatings (III C CTRW- C C 111 ± 1 pf 1 time / h Evaluation of series Wide frequency range Confirmation of model (dielectric & pores = > only pores affected Prerequisite: Reliable handling of drift affected data Conclusion approximation ln H ( const. + ϕ( d ln γ + dln S 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 Thank you for your attention 7 8