New CNLS software for analysis of impedance spectra Outline

New CNLS software for analysis of impedance spectra Outline Motivation Impedance spectroscopy on thermal barrier coatings (TBC) Insertion: RQ-element thermally activated conductivity New CNLS software MVCNLS Introduction Comparison with standard CNLS software fit problem from practice Summary and Outlook Source: MVCNLS-Slides.PPT, Folie: 1, 04.02.2004

Motivation Measurement setup for impedance spectroscopy (IS) I( ω) U ˆ sin( ω t ) ~ U( ω) Z( ω) U ω = ( ) I( ω) IS IS measurementsn between 200 200 C C and and500 C C from from10 10mHz mhztoto 1 MHz MHz impedance analyzer furnace Source: IWE, Thilo Hilpert MVCNLS-Slides.PPT, Folie: 2, 04.02.2004

Motivation temperatur dependance of TBC impedance spectra 6,50 6,00 200 C R decreases R decreases 76M3 300 76M3 250 76M3 200 log( Z ) /Ohm 5,50 5,00 4,50 250 C 300 C 4,00 3,50-1,00 0,00 1,00 2,00 3,00 4,00 5,00 6,00 log(f) /Hz MVCNLS-Slides.PPT, Folie: 3, 04.02.2004

Insertion difference between RC- and RQ-elements C1 C2 Q1 Q Q2 Q R1 R2 R1 R2 Z R = + 1+ jωrc 1+ 1 2 R jωr C 1 1 2 2 Z R R = + 1 + ( jω) RQ 1 + ( j ) R Q 1 2 n1 n2 1 1 ω 2 2 A Constant Phase Phase Element (Q) (Q) is is artificial. For For n=1 n=1 it it is is an an ideal ideal capacitor, for for n=0 n=0 an an resistance. Index Index n is is a measure for for the the broadness of of the the distribution of of the the electrical properties, thus thus a measure for for the the (electrical) inhomogeneity. an alternative expression of our ignorance of the mechanism (Cole 1968) MVCNLS-Slides.PPT, Folie: 4, 04.02.2004

Insertion Constant Phase Element temperature dependence of Q 1,4E-08 1,2E-08 6,0E-10 Q shows exponential temperature behaviour 5,5E-10 1,0E-08 for YSZ not feasible Q* 8,0E-09 6,0E-09 4,0E-09 Q 5,0E-10 4,5E-10 C* equivalent capacity shows linear or 1/T behaviour => feasible 2,0E-09 4,0E-10 200 250 300 350 T / C MVCNLS-Slides.PPT, Folie: 5, 04.02.2004

Insertion Conversion of a RQ-element to an equivalent RC-element log( Z ) /Ohm 6,0 5,0 4,0 3,0 infinite number of RC-elements 0,0 1,0 2,0 3,0 4,0 5,0 6,0 7,0 log(f) /Hz... RC-element with the highest weight 1 n 1/ n C* = ( R Q) absolute value RC* RQ Q R Im(Z) / Ohm 0,E+00 1,E+05 2,E+05 3,E+05 4,E+05 5,E+05 6,E+05 7,E+05 8,E+05 9,E+05 1,E+06 0,E+00-1,E+05-2,E+05-3,E+05-4,E+05-5,E+05 C* R phi / Grad 0-10 -20-30 -40-50 -60-70 -80-90 nyquist plot RQ phase RC* Re(Z) / Ohm RC* 0,0 1,0 2,0 3,0 4,0 5,0 6,0 7,0 log(f) /Hz MVCNLS-Slides.PPT, RQ Folie: 6, 04.02.2004

Motivation temperatur dependance of TBC impedance spectra 6,50 6,00 200 C R decreases R decreases 76M3 300 76M3 250 76M3 200 log( Z ) /Ohm 5,50 5,00 4,50 250 C 300 C 4,00 3,50-1,00 0,00 1,00 2,00 3,00 4,00 5,00 6,00 log(f) /Hz MVCNLS-Slides.PPT, Folie: 7, 04.02.2004

Motivation typical nyquist plot and fit model 0 Re(Z) / Ohm 0 25000 50000 Q RQ1 Im(Z) / Ohm -10000 Q Q RQ2 RQ3 model model of ofmicrostructure Source: IWE, Thilo Hilpert MVCNLS-Slides.PPT, Folie: 8, 04.02.2004

Motivation thermally activated conductivity 623K 573K 523K 473K 1 1,12 ev σ 1 σ 2 arrhenius plot σ T = A exp( E ) kt σt / (SK/m) 0,1 0,01 1,6 1,7 1,8 1,9 2,0 2,1 1000K / T 0,96 ev but... MVCNLS-Slides.PPT, Folie: 9, 04.02.2004

Motivation thermally activated conductivity, fit problems 623K 573K 523K 473K arrhenius plot σ T = A exp( E ) kt 1 σ 1 σ 2 σt / (SK/m) 0,1 permutation of RQ1 and RQ2 The fit software does not know anything about the thermally activated conductivity! 0,01 difficult to analyze 1,6 1,7 1,8 1,9 2,0 2,1 1000K / T MVCNLS-Slides.PPT, Folie: 10, 04.02.2004

Motivation Comparison with standard CNLS software merit function: relative weighting: Complex Nonlinear Least Squares (CNLS) fit N 1 S = Z ˆ j Z w j = 1 N j 1 S = Z ˆ 2 j Z j = 1 Z j j 2 j 2 w j : weight factors 2 ideal: w = 1 σ j Z j : measured impedance Zˆ j : modelled impedance j standard fit software LEVM (Macdonald): difficult input and output files, many equivalent circuits possible, flexible weighting, many impedance representations EquivCrt (Boukamp): grafical and interactive, good for orientation, for mass fit to slowly to operate, any equivalent circuit through CDC (Circuit Description Code) possible, only relative weighting, many impedance representations MVCNLS-Slides.PPT, Folie: 11, 04.02.2004

new fit software approach new approach simultaneous fit of many impedance spectra consideration of assumptions regarding temperature dependance of equivalent circuit elements R, C, Q, n better boundary conditions for fit requirements impedance spectra at all relevant temperatures can be described by the same equivalent circuit model impedance spectra can be modelled by (infinite) RQ-elements in series MVCNLS-Slides.PPT, Folie: 12, 04.02.2004

new fit software common temperature equation common equation for R, C, n xt ( ) = A T exp( E ) + M T+ B+ D T kt N thermally activated conductivity linear, constant square root laws, 1/T, etc. superposed effects E xt ( ) = A T exp( ) + M T+ B + D T + 1 N1 1 kt 1 1 1 E2 N2 2 exp( ) 2 2 2 + A T + M T + B + D T + kt MVCNLS-Slides.PPT, Folie: 13, 04.02.2004

new fit software boundary conditions common temperature equation xt ( ) = A T exp( E ) + M T+ B+ D T kt example: thermally activated conductivity RT ( ) = A T exp( E ) kt N limit on equiv. crt. value R R( T) R u o f u limit on any parameter combined limit Au A Ao 1 f E 1/ n o u E Eo 2 π ( RQ) no other fit software supports these necessary multitude of boundary conditions MVCNLS-Slides.PPT, Folie: 14, 04.02.2004

new fit software command file (easy) readable command file the fit problem is defined by a file, changes in starting values of boundaries are easily possible [FITPREFS] GNUPlotCmd="..\gnuplot\wgnupl32.exe" FitName="P222M07_1" R1=f(A1,E1) A1>=1e-12 A1<=1e-5 E1>=0.8 E1<=1.3 C1=f(F) definition of temperature n1=f(f) dependance of RQ1 R2=f(A1,E1) A1>=1e-12 A1<=1e-5 E1>=0.8 E1<=1.3 C2=f(F) definition of temperature n2=f(f) dependance of RQ2 R3=f(A1,E1) A1>=1e-12 A1<=1e-5 E1>=0.8 E1<=1.3 C3=f(F) definition of temperature n3=f(f) dependance of RQ3 f1=1 start and end frequence of the fit f2=5e5 [ISFILE] P222M07.258 4 250 [RQPREFS] R1!=6,5021e+005 Q1!=6,9201e-010 n1!=0,741 >=0.5 R2!=4,1961e+005 Q2!=7,2801e-009 n2!=0,853 >=0.5 R3!=7,6072e+007 Q3!=1,0384e-008 n3!=0,739 >=0.5 [ISFILE] P222M07.307 4 300 [RQPREFS] R1!=7,2287e+004 Q1!=1,8819e-010...... n1!=0,867 >=0.5 R2!=3,015e+004 Q2!=1,8891e-010 n2!=0,9 >=0.5 R3!=1,6852e+007 Q3!=2,2945e-008 n3!=0,674 >=0.5 [ISFILE] P222M07.355 4 350 [RQPREFS] R1!=2e+004 Q1!=4e-009 n1!=0,69 >=0.5 R2!=1e4 Q2!=4e-008 n2!=0,62 >=0.5 R3!=3e+006 Q3!=4e-8 n3!=0,70 >=0.5 MVCNLS-Slides.PPT, Folie: 15, 04.02.2004

Comparison with standard CNLS software synthetic data, 2 RQ elements, same initial values equivalent circurit parameters Q Q R1 = 3,11e6 R2 = 5,06e6 Q1 = 9,73e-10 Q2 = 7,45e-8 n1 = 0,9 n2 = 0,7 noise Re nyquist plot initial values Im 0 5 5 6 6 6 + R1 = 3e6 Q1 = 9e-10 n1 = 0,88 R2 = 4,5e6 Q2 = 7e-8 n2 = 0,72 fit software MVCNLS-Slides.PPT, Folie: 16, 04.02.2004

Comparison with standard CNLS software: nyquist plots synthetic data, 2 RQ elements, same initial values Im(Z) / Ohm Im(Z) / Ohm 0,0E+00 1,0E+06 2,0E+06 3,0E+06 4,0E+06 5,0E+06 6,0E+06 7,0E+06 8,0E+06 9,0E+06 0,0E+00-4,0E+05-8,0E+05-1,2E+06-1,6E+06-2,0E+06 Original EquivCrt LEVM MVCNLS Re(Z) / Ohm 1% noise 200 C 0,0E+00 1,0E+06 2,0E+06 3,0E+06 4,0E+06 5,0E+06 6,0E+06 7,0E+06 8,0E+06 9,0E+06 0,0E+00-4,0E+05-8,0E+05-1,2E+06-1,6E+06-2,0E+06 Original EquivCrt LEVM MVCNLS Re(Z) / Ohm 5% noise 200 C MVCNLS-Slides.PPT, Folie: 17, 04.02.2004

Comparison with standard CNLS software: fit results synthetic data, 2 RQ elements, same initial values 1,015 1% noise 1,040 1,010 1,010 1,005 1,020 1,005 0,995 0,990 R1* R2* 0,980 0,960 Q1* Q2* 0,995 n1* n2* 0,985 0,940 0,990 1,100 5% noise 1,100 1,100 1,050 1,050 0,900 0,800 0,950 0,700 Q1* Q2* 0,950 R1* R2* n1* n2* 0,900 0,600 0,900 MVCNLS-Slides.PPT, Folie: 18, 04.02.2004

Comparison with standard CNLS software: fit results synthetic data, 3 RQ elements, same initial values Im Re 1% noise Re 3. RQ element only partial in measured frequency range Im Original EquivCrt LEVM MVCNLS fit is ok, but the errors... MVCNLS-Slides.PPT, Folie: 19, 04.02.2004

Comparison with standard CNLS software: fit results synthetic data, 3 RQ elements, same initial values 1,100 1,050 1% noise R1* R2* 1,500 1,400 1,300 1,200 Q1* Q2* 1,050 1,100 0,900 0,800 n1* n2* 0,950 0,700 0,950 1,200 1,150 1,100 1,050 0,950 0,900 0,850 0,800 0,750 0,700 5% noise R1* R2* 2,000 1,800 1,600 1,400 1,200 0,800 0,600 0,400 0,200 0,000 Q1* Q2* 1,300 1,250 1,200 1,150 1,100 1,050 0,950 0,900 n1* n2* MVCNLS-Slides.PPT, Folie: 20, 04.02.2004

thermally cycled TBC-Systems: fit problem from practice fit results equivalent circuit of 3 RQ elements same initial values for both LEVM: each temperature alone MVCNLS: 3 temperatures at once MVCNLS 623K 573K 523K Q Q Q LEVM 623K 573K 523K 1 1 σt / (SK/m) 0,1 0,01 1E-3 S1T S2T S3T S1T S2T S3T σt / (SK/m) 0,1 0,01 1E-3? S1T S2T S3T S1T S2T S3T 1E-4 1,6 1,7 1,8 1,9 1000K / T 1E-4 1,6 1,7 1,8 1,9 1000K / T MVCNLS-Slides.PPT, Folie: 21, 04.02.2004

thermally cycled TBC-Systems: fit problem from practice fit results 1 MVCNLS 1 LEVM C* / (F/µm) 0,1 0,01 C1 C2 C3 C1 C2 C3 C* / (F/µm) 0,1 0,01 C1 C2 C3 C1 C2 C3 1E-3 1,0 0,9 0,8 1E-3 523 573 623 Q 1 n 1/ n C* = ( R Q) 1,0 0,8 523 573 623 n 0,7 R C* n 0,6 n1 n2 n3 n1 n2 n3 0,6 n1 n2 n3 n1 n2 n3 0,5 523 573 623 R 523 573 623 MVCNLS-Slides.PPT, Folie: 22, 04.02.2004

Summary and Outlook fit software for analysis of temperature depended impedance spectra standard CNLS fit with boundary conditions possible direct determination of activiation energies expertise can be used graphical interface MVCNLS-Slides.PPT, Folie: 23, 04.02.2004