CNLS fitting

Other parameters, such as CNLS-fit residuals (Are and Ajm), also indicate the goodness of fit . They are defined as... 

Assuming that the Nyquist plot of the impedance does not display an ideal semicircle (e.g., it shows a depressed semicircle or a wide arc), it might be described using two or more discrete time constants or a continuous distribution of time constants. In the former case, the equivalent circuit may involve two or more parallel RCs in series. In the latter case, it may involve one or more parallel CPEs and Rs in series. As mentioned, one solution could be to use several CNLS fittings however, a more direct method would be the deconvolution of the imaginary part of the impedance data. 

Powerful data analysis programs are available (e.g., complex non-hnear least square fit or CNLS-fit). Error estimations for model parameters can be obtained. 

As both the frequency domain and the time domain methods have disadvantages, Boukamp [87] recommended that both methods be combined using the CNLS-fit procedure, data validation (Kramers-Kronig transformation), and deconvolution. The Kramers-Kronig transformation can be found in Appendix C. 

An example of porous behavior was presented by Los et al for the hydrogen evolution reaction on LaP04-bonded Ni powder electrodes in 30% NaOH. Examples of the complex plane plots are shown in Fig. 36. Using the complex nonlinear least-squares (CNLS) fit, the parameters Ret, T, and C[Pg.215]

The analysis of the results indicates that the resistance at (o = 0, Rp obtained using the correct analysis is twice that found from de Levie s equation. In addition, the plot of squared impedances produces a deformed ellipsoid instead of a perfect semicircle. It has been shown " that the CNLS fit of the simulated impedances to the de Levie equation (195) is not good, there being systematic differences between these two curves. However, when the CPE is used instead of the double-layer capacitance, the approximation is good. The values obtained for the parameter < ) are between 0.91 and 0.93. " In this case the use of the CPE only hides the inadequacy of the model. 

The choice of statistical weights in the CNLS fit is very important. Because Ihe measured impedances may vary at different frequencies over several orders of magnitude when unitary weights = 1) are used, only the largest impedances contribute to the sum of squares S. In such a case, low time constants may be overlooked. In gena-al, several repetitions of the experiment allow the standard deviation of each point and (o"j) to be determined, and the statistical weights may be obtained as wl = (ol and w" = (o". Although this is the best approach, such a procedure is... 

The internal cell resistance is approximately the sum of / , Rc i, and Rah- The values of the resistance were determined by using the complex non-linear least squares (CNLS) fitting of the impedance spectra to the equivalent circuit of Figure 10(b). As a matter of fact, whatever models one selects for the impedance spectra in Figure 10(a), the values of total internal cell resistance and relaxation time necessary for charge/discharge of all the capacitive elements remain constant (see below). 

Fig.7. Impedance plot of a polymer coated Pt-electrode in 0.1 M LiBF4/propylene carbonate. Estimated film thickness = 0.5 /im. Edc = 1.2 V vs. Ag/AgCl. Circles = experimental data. Stars = calculated data using the CNLS fit to the circuit shown in the insert, where Zd is a diffusion impedance.

Figure II. (a) Plots of initial current level lim against potential drcq) DE at various initial electrode potentials, reproduced from the cathodic current transients obtained from the Lii NiOz electrode, (b) variation in current with time up to 10 s by application of a potential drop df of 0.1 V at various initial electrode potentials, calculated using the simulation program with integrated circuit emphasis (SPICE) for the electric circuit of Figure 10 b), by taking the values of resistances and capacitances determined by CNLS fitting of the impedance spectra of Figure 10(a) to the electric circuit of Figure 10(b).

Fig. 13.3 Example of experimental data for solid electrolyte that are not Kramers-Kronig transformable, left, and residual plots (a) real to imaginary and imaginary to real, (b) relative errors of complex transformation, and (c) errors of CNLS fit to model in (a) circles real-to-imaginary squares imaginary-to-real transformations (From Ref. [575] Reproduced with permission of Electrochemical Society)...

Although a more complicated nonlinear least squares procedure has been described by Tsai and Whitmore [1982] which allows analysis of two arcs with some overlap, approximate analysis of two or more arcs without much overlap does not require this approach and CNLS fitting is more appropriate for one or more arcs with or without appreciable overlap when accurate results are needed. In this section we have discussed some simple methods of obtaining approximate estimates of some equivalent circuit parameters, particularly those related to the common symmetrical depressed arc, the ZARC. An important aspect of material-electrode characterization is the identification of derived parameters with specific physicochemical processes in the system. This matter is discussed in detail in Sections 2.2 and 3.3 and will not be repeated here. Until such identification has been made, however, one cannot relate the parameter estimates, such as Rr, Cr, and y/zc, to specific microscopic quantities of interest such as mobilities, reaction rates, and activation energies. It is this final step, however, yielding estimates of parameters immediately involved in the elemental processes occurring in the electrode-material system, which is the heart of characterization and an important part of IS. 

Franceschetti model involves relatively general boundary conditions at the electrodes and so includes the possibility of charge transfer reactions and specific adsorption. Because of its generality, however, the model prediction for Z,((o) is very complicated and, in general, cannot be well represented by even a complicated equivalent circuit. The Z,(m) expression, may, however, be used directly in CNLS fitting. Here, for simplicity, we shall consider only those specific situations where an approximate equivalent circuit is applicable. Idealizations involved in the model include the usual assumption of diffusion coefficients independent of field and position, the use of the simplified Chang-Jaff6 [1952] boundary conditions, and the omission of all inner layer and finite-ion-size effects. Some rectification of the latter two idealizations will be discussed later. 

Theoretical Results for Various Cases of Interest. Thus far we have only considered some exact 0 results and typical adsorption-reaction interface frequency response for a half-cell or full cell. Let us now turn to further predictions of the complete full-cell model (Macdonald and Franceschetti [1978]), predictions derived from its specific analytical results in several simplified cases and from a large amount of CNLS fitting of various equivalent circuits to the exact model pre-... 

It has been customary in much past work to plot -Z" or M" vs. v and either not give estimates of parameters, such as R and C, leading to the response or to estimate them roughly by graphical means. It is often found that the frequency at the peak, is at least approximately thermally activated. But this frequency involves both R and C, quantities which may be separately and differently thermally activated. Thus instead of using the composite quantity (0, which may be hard to interpret properly, we believe it to be far preferable to find estimates of all the parameters entering into an equivalent circuit for the situation. As we shall see, such estimates can best be obtained from CNLS fitting. 

We shall illustrate the above by showing results for IS impedance data for a single-crystal Na )3-alumina with blocking gold electrodes, measured at T = 83K. These data (Almond and West [1981], Bruce et al. [1982]) were kindly provided by Dr. West [1983]. This set, and data for eight higher temperatures, have been reanalyzed with CNLS fitting (Macdonald and Cook [1984, 1985]). 

What should one do about anomalies of this kind, points which do not seem to lie close to a smooth curve If the experiment can be repeated, that should be done, and averaged or best data used. In the present instance, measurement at a few more frequencies between the present lowest and next lowest point would yield intermediate points which would help clarify whether the last point is badly off or not. If the experiment cannot be repeated, then outliers of appreciable magnitude, such as the lowest-frequency point in the present plot, should be omitted (or weighted very low) in subsequent CNLS fitting. 

Z"(ft)) data to find an estimate of the distribution function of time constants implicit in the data (Franklin and de Bruin [1983], Colonomos and Gordon [1979]). Such a distribution, if sufficiently accurate, will separate out the various time constants present, even if they are completely invisible in 3-D plots, and by the width of the individual relaxations apparent in the distribution suggest whether they may be best described by discrete circuit elements or by continuous distributions in the frequency domain. From the values of relaxation time t where relaxation peaks occur, one may also calculate the approximate frequency region (Op = Tp where the relaxation produces its maximum effect. These results may then be used to construct an appropriate equivalent circuit and estimate initial values of the parameters for subsequent CNLS fitting. 

As a first example of CNLS fitting, the circuit shown in Figure 3.3.5 was constructed with lumped elements whose values were measured on an impedance bridge (top figures) (Macdonald, Schoomnan, and Lehnen [1982]). This circuit leads to very little structure in either the Z or T 3-D plots shown in Figures 3.3.6 and 3.3.7. The... 

---