Werner Strunz, Zahner-elektrik.

Transcription

1 Werner Strunz, Zahner-elektrik

2 Outline

3 1. Resistor [ R ] Spannung Strom

4 2. Inductance [ L ] Spannung Strom

5 3. Capacitor [ C ] Spannung Strom

6 4. Warburg Impedance [ W ] Spannung Strom

7 4. Warburg Impedance [ C ] - Example?

8 5. Nernst-Diffusion[ N ]

9 5. Nernst-Diffusion[ N ] - Example (FC)

10 6. Finite Diffusion [ FD ]

11 6. Finite Diffusion [ FD ] - Example

12 Z / Z'' / 7. Constant Phase Element [ CPE ] 100m 10m 1m 1 1 Z CPE Y j phase / o K frequency / Hz Z' /

13 CPE Flexible Element

14 Constant Phase Element [ CPE ] Impedance of an Electrolyte Capacitor HF : Z ~ (like L) 1K Z / phase / o 90 MF : Z ~ const (like R) LF : Z ~ 1 / (like C) j > -90 => CPE m 0 100m K 3K 10K 100K frequency / Hz

15 Constant Phase Element [ CPE ] a = - 1 a = - 0,5 a = 0 a = + 1 Z ( w) = 1 C 1 j w Z ( w) = "W" 1 ( j w) Z ( w) = R Z ( w) = L j w Capacitor Diffusion (Warburg) Ohmic Inductance 0 > a - 1 : Z ( w) = 1 Y o 1 ( j w) a [Y O ] = f ()

16 CPE a frequency dependent capacity log 1 C log Y 0 log C norm Y 0 norm Without precaution f norm = (2) -1 (natural = mathematical normalization)

17 Normalization of CPE (R CPE!) K.S. Cole, R.H. Cole; J. Chem. Phys. 9 (1941) K.S. Cole, R.H. Cole; J. Chem. Phys. 10 (1942) G. J. Brug, A. L. G. van den Eeden, M. Sluyters-Rehbach, J. H. Sluyters; Journal of Electroanalytic Chemistry 176 (1984) C.H. Hsu, F. Mansfeld; Corrosion 57/ No. 9 (2001) M.R. Shoar Abouzari, F. Berkemeier, G. Schmitz, D. Wilmer; Solid State Ionics 180 (2009) B. Hirschorn, M. Orazem, B. Tribollet, V. Vivier, I. Frateur, M. Musiani; El. Acta 55 (2010)

18 Constant Phase Element - Normalization Z 1 1 C Z CPE C Y0 1 1 C Y 0 OR C norm Y 0 norm

19 Normalization of CPE (R CPE)! - simplified derivation Normalized capacity is independent of the exponent Y R C Y R with j R Z j C R R Z CPE R C R

20 Constant Phase Element [ R CPE ] Z( w) = R R Y 0 ( jw) = R a RC jw G( t)dt with G( t)dt = 1 -

21 Constant Phase Element [ CPE ] Distribution Function (G()) G( t) = 1 2p sin( p ( 1 - a) ) cosh( a x) - cos( p ( 1 - a) ) with x = ln t t 0

22 Constant Phase Element [ CPE ] Distribution Function (G()) G( t) = 1 2p sin( p ( 1 - a) ) cosh( a x) - cos( p ( 1 - a) ) with x = ln t t 0

23 Constant Phase Element [ R CPE ]

24 Constant Phase Element [ R CPE ] R n : equidistand spacing in log(/ 0 )

25 Validation of Spectra

26 Problems of Daily Life

27 functions E(t) & I(t) EIS-Principle at a Single Frequency Excitation at constant frequency 0 500µ 1m 1.5m 2m 2.5m 3m j time /s Z How to validate EIS-spectra? What s the specific property?

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

29 Motivation : what we need 30 Z / m phase / o artifact Reliable detection of artifacts m 100m K 10K 100K frequency / Hz

30 The Kramers-Kronig Relations ) ( Im 2 - (0) = Re ) ( Re d H PV H H ) ( Re 2 = ) ( Im d H PV H BUT WHERE ARE THE PROBLEMS?

31 The Limited Bandwidth Problem Simulation of a coating during water up-take Measured frequency range 100 KHz 50 mhz 0 :? :?

32 The measurement model

33 log impedance Z / Ohm Phase angle / Measurement Model Simulation log frequency / Hz

34 Measurement Model Z RC ( ) = R - Drawback: 1+RC RC is not linear! Z RC ( ) = R 1+RC = Z real ( ) + Z imag ( ) = R 1+ ( RC ) 2 j R2 C 1+ ( RC ) 2 Z RC ( ) = R 1+ = Z real ( ) + Z imag ( ) = R 1+ ( ) 2 j R 1+ ( ) 2 Complex number Z = a + j b

35 Z RC ( ) = Measurement Model R 1+RC = Z real ( ) + Z imag ( ) = R 1+ ( RC ) 2 j R2 C 1+ ( RC - Solution: RC Replacement : RC= Z RC ( ) = R 1+ = Z real ( ) + Z imag ( ) = R 1+ ( ) 2 j R 1+ ( ) 2 Complex number Z = a + j b Z real

36 1+RC 1+ ( RC ) 1+ ( RC Linear-KK Check Z RC ( ) = R 1+ = Z real ( ) + Z imag ( ) = R 1+ ( ) 2 j R 1+ ( ) 2 Complex number Z = a + j b Complex Matrix Entry Z real = a b Z imag b a

37 Linear-KK Tool from KIT Karlsruhe

38 Linear-KK Battery (I)

39 Linear-KK Supercap (Sub m-range)

40 Linear-KK Coating (Huge-Z-Range)

41 Linear-KK Battery (II)

42 Battery voltage / V Drift in Batteries 3,26 Discharge 4A (0.1 C) Relaxation 0A 3,25 3,24 3,23 3,22 3,21 3,20 2,0 2,2 2,4 2,6 2,8 3,0 3,2 3,4 3,6 3,8 4,0 4,2 time of experiment / h Measurement time ~ 6 h

43 The Z-HIT Approximation (evaluation of impedance modulus from the phase angle ln H ( 0 ) const S j( ) d ln d j( dln O ) Detection of artifacts Detection of instationarities (drift) History (time) preserving Reconstruction of causal spectra => Reliable interpretation of spectra

44 Validation of Spectra Z-HIT

45 ln Z Deduction of the Z-HIT Randle circuit E ( O ) S Phase / rad 0,0 16-0,2 15-0,4-0,6 14 Integral (shifted) ln Z [A] ln -0,8-1,0-1,2-1, [B] ,5 0,4 0,3 0,2 0,1 0,0-0,1-0,2-0,3-0,4-0,5-0,6-0,7 [C] ln Z - Integral (j) dj / dln ,5 0,4 0,3 0,2 0,1 0,0-0,1-0,2-0,3-0,4-0,5-0,6-0,7 [D] ln Z - Integral (j) /6 * dj / d ln

46 The Sensitivity of Objects (Z & j) - Excellent Examples: Sensors! - Temperature Dependent Resistor (NTC, PTC) Pt 100, Pt 1000, KTY 81, - Light Dependent Resistor (LDR) - Magnetic Dependent Resistor (MDR) - Humidity Dependent Capacity -..

47 The Course of Phase and Impedance when Heating NTC/PTC Phase / Phase / LOG Impedance / 4,0 3,9 3,8 3,7 3,6 3,5 KTY 10k (NTC) LOG Impedance / 3,15 3,10 3,05 Pt 1000 (PTC) , LOG frequency / Hz LOG frequency / Hz 0 Z & j : Phase j is more stable than impedance Z

48 Implementation of the Z-HIT 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 0 2 dj( O) H( 0) j( ) d ln const. d ln S

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

50 Z-HIT Examples Battery (I)

51 Z-HIT Examples Supercap

52 Z-HIT Examples Battery (II)

53 Restriction (2-Gate) Z-HIT What is History (Time) Preserving? Considering Kramers Kronig relations ) ( Re 2 = ) ( Im d H PV H j j ln ) ( ln ) ( 2 onst.+ ) ( ln 0 0 d d d c H O S Integral-Term preserved integration along the frequency axis leads to weighting (measuring time)

54 History (Time) Preserving Randle circuit with NTC as Charge Transfer Resistance

55 History (Time) Preserving Randle circuit with NTC as Charge Transfer Resistance Only Smoothing Z-HIT refinement

56 History (Time) Preserving Randle circuit with NTC as Charge Transfer Resistance Only Smoothing Z-HIT refinement Dangerous: expanding the model without physical justification

57 Water Uptake - Waterborne Coating Z / Series measurement phase / o 1G 100M 10M 1M 100K 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 10K 1K a a a K 3K 10K 100K frequency / Hz a a a a a a a 5 th spectrum With kind permission of U. Christ, Fraunhofer-Institut für Produktionstechnik und Automatisierung IPA, Stuttgart

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

59 Z-HIT: Estimate of Accuracy (I) = 1 = 0.5

60 Z-HIT: Estimate of Accuracy (II) ZSIM ZZ HIT rel. Error 100 Z SIM = 1 = 0.5

61