(1) Nonlinear system


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1 Liner vs. nonliner systems in impednce mesurements I INTRODUCTION Electrochemicl Impednce Spectroscopy (EIS) is n interesting tool devoted to the study of liner systems. However, electrochemicl systems re often nonliner. Before explining the different wys to del with this issue, we will point out the min differences etween liner nd nonliner systems. These differences of ehvior re shown in Tle I. The impednce mesurement ws performed using the potentiosttic mode. The potentil is defined y: ( ) sin ( ) E t = E + V ft () WE where EWE is the sttionry potentil for the impednce mesurement, V is the potentil mplitude of the sine signl, f is the frequency, nd t is the time. Tle I: Differences etween liner nd nonliner systems. Stedystte I vs. EWE curve Impednce vs. EWE Impednce vs. V Liner system Stright line I = f(ewe) Invrint with EWE Invrint with V for ll V vlues Nonliner system I = f(ewe) Vrint with EWE: Z(EWE) Invrint with V only for low V vlues Three test circuits hve een designed in order to highlight the differences in ehvior etween liner nd nonliner systems. II TEST BOX3 The experiments descried on this pper were performed with test ox specificlly designed for the user to lern how to chieve impednce mesurements on liner nd nonliner electrochemicl systems. There re three different electricl circuits inside the test ox simulting rel electrochemicl systems. With Test Box3, it is possile to study generl electrochemistry protocols like Cyclic Voltmmetry or corrosion protocols such s Liner Polriztion nd Generlized Corrosion. III TEST CIRCUIT # Figure shows the electricl circuit of the test circuit #, the corresponding stedystte curve (I vs. EWE), nd the Nyquist digrm of the impednce mesured t points nd of the stedystte curve. The impednce digrms do not depend on the stedystte potentil EWE or on the mplitude V. The Nyquist impednce digrm displys two semicircles. Only one impednce mesurement is needed to chrcterize the equivlent test circuit #. It cn e good exercise to determine the different frequencies chrcterizing circuit test # nd compre them with the experimentl vlues shown on the impednce digrm of Fig. 3. Impednce is mesured t stedystte points nd (Fig., Fig. 3). The rrow indictes BioLogic Science Instruments, Rue de Vucnson, 387 SeyssinetPriset, FRANCE Tel: Fx:
2 incresing frequencies. R = 5, R = k, R = 3.56 k, C = nf, C =.3 µf. Figure : Test circuit #. CV_circuit.mpr..8.6 PEIS_circuit.mpr # PEIS_circuit_zfit.mpp Figure : I vs. EWE stedystte curve..7 khz PEIS_circuit.mpr # Figure 3: Nyquist digrm for the impednce. Vlues of the vrious prmeters of circuit # re otined using ZFit, ville in ECL nd ECL Express softwre. Result of the fit is given Figure.. PEIS_V_circuit.mpr 3. Hz Figure : ZFit result on circuit # of Test Box3. To verify these results with test circuit #, you cn perform the following experiments: ) Plot the stedystte I vs. E WE curve using cyclic voltmmetry technique (in ECL softwre, lod the CV_circuit.mps file, ccept, nd run the experiment). ) Select the PEIS technique. Apply constnt potentil E WE = V nd perform n impednce mesurement from f i = khz to f f = Hz frequency with low mplitude V = mv (it is lso possile to lod the PEIS_circuit.mps file). 3) In the PEIS technique, pply constnt potentil E WE =. V nd perform n impednce mesurement from f i = khz to f f = Hz frequency with low mplitude V = mv (it is lso possile to lod the PEIS_V_circuit.mps file). ) In the PEIS technique, pply constnt potentil E WE =. V nd perform n impednce mesurement from f i = khz to f f = Hz frequency with high mplitude V = mv (it is BioLogic Science Instruments, Rue de Vucnson, 387 SeyssinetPriset, FRANCE Tel: Fx:
3 Im(Z)/Ohm lso possile to lod the PEIS_mplmV_circuit.mps file). PEIS_5V_circuit.mpr # PEIS_V_circuit.mpr 5) Overly the three impednce mesurement curves in the Nyquist plot mode. As test circuit # is liner system, the three impednce digrms should e identicl khz 86.3 Hz IV TEST CIRCUIT # Test circuit # is mde minly of two semiconductor diodes. This is model for exponentil nonlinerity. Test circuit # results from circuit studied in []. The I vs. EWE stedystte curve is not stright line for test circuit #. Therefore the test circuit # is nonliner circuit. The impednce of this circuit depends on the stedystte vlue of the working electrode potentil EWE nd on the mplitude V. The Nyquist digrm of the impednce mesurement performed on test circuit # is semicircle whose dimeter chnges long with the electrode potentil EWE. (Fig. 6). The impednce of nonliner system is potentil dependnt. CV_circuit.mpr Re(Z)/Ohm Figure 6: Nyquist digrm for the impednce mesured t point nd. Arrows indicte incresing frequencies. Figure 8 shows the impednce chnge versus the mplitude V for given Ewe vlue. We cn clerly see tht the semicircle dimeter decreses when V vlues increse. The impednce does not depend on the mplitude of the excittion signl for low vlues of mplitude (Fig. 8). In tht cse, this system s ehvior is similr to tht of liner system. Therefore, one impednce mesurement is not sufficient to chrcterize nonliner system. It cn e good exercise to try nd find n equivlent electricl circuit for test circuit #. The user cn lso determine the electricl components vlues when its ehvior cn e compred to one of liner system..5.5 PEIS_5V_xxmV_circuit.mpr.6 Hz.6 Hz Figure 5: I vs. EWE stedystte curve..3.6 Hz Figure 7: Nyquist impednce digrm mesured for different vlues of potentil (EWE =.5 V). BioLogic Science Instruments, Rue de Vucnson, 387 SeyssinetPriset, FRANCE Tel: Fx:
4 RLF (kohms) log(v/mv) Figure 8: digrm displying the inphse impednce (evluted for limiting vlue in low frequencies) versus log(v). It is possile to verify these results with test circuit #: ) Plot the stedystte I vs. EWE curve using cyclic voltmmetry technique (in ECL softwre, lod the CV_circuit.mps file, ccept, nd run the experiment). ) Select PEIS technique. Apply constnt potentil EWE =. V nd perform n impednce mesurement from frequency fi = khz to ff = Hz with n mplitude V = mv (it is lso possile to lod the PEIS_V_circuit.mps file). 3) In the PEIS technique, pply constnt potentil EWE =.5 V nd perform n impednce mesurement from frequency fi = khz to ff = Hz with n mplitude V = mv (it is lso possile to lod the PEIS_5V_circuit.mps file). ) In the PEIS technique, pply constnt potentil EWE =.5 V nd perform severl impednce mesurements from frequency fi = khz to ff = Hz with n mplitude V/mV=.5, 5,, 5, 5, (it is lso possile to lod the PEIS_5V_xxmV_circuit.mps files for the different vlues of Vpp)..5 The shpe of the impednce digrm evolves long with the electrode potentil EWE s it cn e oserved in Figs. 9, nd. Let us consider the stedystte I vs. EWE curve showing pek (s in the cse of metl pssivtion, Fig. 9). For low frequencies, the inphse impednce is negtive. Therefore, prt of the impednce digrm is on the left side of the complex plot, i.e. Re(Z(w)) < CV_circuit3.mpr Figure 9: I vs. EWE stedystte curve. Figure : Nyquist digrm for the impednce which hs een mesured t point. The rrow indictes incresing frequencies khz PEIS_3V_circuit3.mpr PEIS_8V_circuit3.mpr 66.5 Hz 66.5 Hz 3.9 khz V TEST CIRCUIT #3 Test circuit #3 minly consists of two trnsistors. It is model for metl pssivtion which hs een extrcted from []. This circuit hs lso een studied in [].. Figure : Nyquist digrm for the impednce which hs een mesured t point. The rrow indictes incresing frequencies...3 BioLogic Science Instruments, Rue de Vucnson, 387 SeyssinetPriset, FRANCE Tel: Fx:
5 It is possile to verify these results with test circuit #3 y performing the following experiments: ) Plot the stedystte I vs. EWE curve using cyclic voltmmetry technique (in ECL softwre, lod the CV_circuit3.mps file, ccept, nd run the experiment). ) To plot digrm on Fig. : select the PEIS technique nd pply constnt potentil EWE =.8 V nd perform n impednce mesurement from frequency fi = khz to ff = Hz with n mplitude V = mv (it is lso possile to lod the PEIS_ 8V_circuit3.mps file). 3) To plot the digrm (Fig. ): in the PEIS technique pply constnt potentil EWE =.3 V nd perform n impednce mesurement from frequency fi = khz to ff = Hz with n mplitude V = mv (it is lso possile to lod the PEIS_3V_circuit3.mps file). VI CONCLUSION Liner systems re simpler to study. Only one impednce mesurement is sufficient to chrcterize their ehvior. On the other hnd, it ecomes quite complicted when deling with nonliner systems. Severl impednce mesurements re necessry to chrcterize their ehvior. In order to study nonliner system with impednce mesurement y ssimilting its ehvior with liner system ehvior, it is necessry to use low modultion mplitude, V. Dt files cn e found in : C:\Users\xxx\Documents\EC L\Dt\Smples\Corrosion\Appliction Note 9 REFERENCES ) J.P. Dird, B. Le Gorrec, nd C. Montell, J. Electronl. Chem., 3 (997) 7. ) K. Mhdevn, nd Y. Gopl, Electronic Engineering, (973). 3) J.P. Dird, nd B. Le Gorrec, J. Electronl. Chem. 3 (979) 363. ) J.P. Dird, P. Lndud, B. Le Gorrec, nd C. Montell, Electronl. Chem., (3). Exercise nswers:  circuit test #: the chrcteristic frequency, i.e. the frequency t the top of the semicircle of n RC circuit, is given y fc = RC The two chrcteristic frequencies for circuit test # re: f f = 6 khz = Hz c 3 8 c 3 6 These vlues re close to the frequency vlues given t the top of the semicircles present on Fig Circuit test #: the impednce digrm represents semicircle. The equivlent electricl circuit cn e ssimilted to n RC circuit. It is possile to determine the circuit resistnce vlues on the Im(Z) vs. Re(Z) plot. They correspond to the inphse impednce for the lowest frequencies (Im(Z) ). R vlue vries with Ewe. For point of the stedystte curve, we find: R = Ω nd for point : R = 7 Ω. With the resistnce vlues determined ove, the user cn determine the cpcitnce using the following eqution: C = π fr c where fc is the frequency t the top of the semicircle. The results should e similr for R nd R. We find C~9 nf. Revised in 7/8 BioLogic Science Instruments, Rue de Vucnson, 387 SeyssinetPriset, FRANCE Tel: Fx:
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