Equivalent circuit modeling

W. J. Eleming, Zirconia Ouygen Sensor—-An Equivalent Circuit Model, SAE 800020, Society of Automotive Engineers, Warrendale, Pa., 1980. 

Figure 53. Idealized half-cell response of a thin solid electrolyte cell, (a) Cell geometry including working electrodes A and B and reference electrode (s). (b) Equivalent circuit model for the cell in a, where the electrolyte and two electrodes have area-specific resistances and capacitances as indicated, (c) Total cell and half-cell impedance responses, calculated assuming the reference electrode remains equipotential with a planar surface located somewhere in the middle of the active region, halfway between the two working electrodes, as shown in a.

EIS data is generally interpreted based on defining an appropriate equivalent circuit model that best fits the acquired data. The elements of the circuit model involve a specific arrangement of resistors, capacitors, and inductors that tacitly represent the physicochemical reality of the device under test. Under these circumstances the numerical value for chemical properties of the system can be extracted by fitting the data to the equivalent circuit model. Impedance measurements are typically described by one of two models ... 

Fig. 2 Equivalent circuit models (Hubrecht J (1998) Metals as Biomaterials, Helsen J, Breme H (eds) John WUey Sons Limited. Reproduced with permission)...

Fig. 12 Single diode equivalent circuit model commonly employed in estimating solar cell losses...

Modeling and optimization of chemical sensors can be assisted by creating equivalent electrical circuits in which an ordinary electrical element, such as a resistor, capacitor, diode, and so on, can represent an equivalent nonelectrical physical parameter. The analysis of the electrical circuit then greatly facilitates understanding of the complex behavior of the physical system that it represents. This is a particularly valuable approach in the analysis and interpretation of mass and electrochemical sensors, as shown in subsequent chapters. The basic rules of equivalent circuit analysis are summarized in Appendix D. Table 3.1 shows the equivalency of electrical and thermal parameters that can be used in such equivalent circuit modeling of chemical thermal sensors. 

The objective of the equivalent circuit modeling is to identify four unknowns, L, C, R, and Co. Therefore, we need four equations. They are ... 

Quantitatively, we proceed via the use of equivalent circuit models. The most general model is the distributed transmission line model of Fig. 

Fig. 13.8. Equivalent circuit models for crystal impedance responses (a) transmission line model (b) lumped clement (modified Butterworth van Dyke) model.

Figure 3 Electrical equivalent circuit model commonly used to represent an electrochemical interface undergoing corrosion. Rp is the polarization resistance, Cd] is the double layer capacitance, Rct is the charge transfer resistance in the absence of mass transport and reaction intermediates, RD is the diffusional resistance, and Rs is the solution resistance, (a) Rp = Rct when there are no mass transport limitations and electrochemical reactions involve no absorbed intermediates and nearly instantaneous charge transfer control prevails, (b) Rp = Rd + Rct in the case of mass transport limitations.

Figure 5 Nyquist, Bode magnitude and Bode phase angle plots for hypothetical corroding interfaces with Rp = 10, 100, or 1,000 ohms, Cd, = 100 tF, and Rs = 10 ohms using the electrical equivalent circuit model of Fig. 3a.

For these reasons, EIS has been explored as an alternative proof test for evaluation of conversion coatings. In these tests, conversion coated surfaces are exposed to an aggressive electrolyte for some period of time during which coating damage will accumulate. An impedance spectrum is collected and evaluated using a suitable equivalent circuit model and complex nonlinear last-squares fitting. 

Equivalent circuit analysis is well suited for analysis of EIS measurements of conversion coatings and is the primary method for interpreting EIS spectra from conversion coated metal surfaces. A widely accepted generalized equivalent circuit model for the EIS response of pitted conversion coatings is shown in Fig. 22a (66,67). Several related models discussed below are also shown. In the gener-... 

Figure 22 Equivalent circuit models for conversion coatings, (a) Generalized model for pitting conversion coated surfaces, adapted from Ref. (b) Model for a barrier conversion coating, (c) Model for the early stages of CCC breakdown.

This method of estimating Rc is useful when it can be applied, since the determination is not based on any presumed model of the corrosion damage process or any of the assumptions that come with assignment of an equivalent circuit model. This method is particularly helpful when there is more than one time constant in the spectrum, or the impedance spectrum is particularly complicated. Caution is warranted however. This method of estimation can be in serious error for samples with large capacitance-dominated low-frequency impedances. As a general rule, for this estimation method to be reasonably accurate, the impedance function must exhibit a clear DC limit, or a diffusional response that can be modeled by a constant phase element in equivalent circuit analysis (75). 

Using the generalized equivalent circuit model for conversion coated surfaces shown in Fig. 22, it is possible to track the time-dependent changes in the resistances and capacitances of the intact coating and evolving pits. Figure 25 shows representative Bode plots for CeCl3-passivated and bare A1 7075-T6 immersed in 0.5 M NaCl solution (81). Spectra like these were collected over 35... 

Figure 25 (a) Bode plot for CeCl3-treated 7075-T6 during immersion in 0.5 M NaCl solution, (b) Results of the equivalent circuit modeling of the EIS data from Fig. 25 using the generalized equivalent circuit model in Figure 22a. (From F. Mansfeld, S. Lin. S. Kim. H. Shih. Corrosion 45, 615 (1989).)... 

Figure 29 (a) Explicit equivalent circuit model for an unsealed porous anodized film... 

Figure 31 Time-dependent changes in equivalent circuit model element values due to interrupted sealing, (a) Solution resistance, Rs, (b) porous layer resistance, Ra, (c) porous layer capacitance, CH, (d) barrier layer resistance, Rb, (e) barrier layer capacitance, Cb. (From J. L. Dawson, G. E. Thompson, M. B. H. Ahmadun. p. 255, ASTM STP 1188, ASTM, Philadelphia, PA (1993).)...

Breakdown of anodic films is yet another phenomenon for which EIS is well suited. Equivalent circuit analysis has been used to analyze EIS spectra from corroding anodized surfaces. While changes in anodic films due to sealing are detected at higher frequencies, pitting is detected at lower frequencies. Film breakdown leads to substrate dissolution, and equivalent circuit models must be amended to account for the faradaic processes associated with localized corrosion. 

Figure 32 (a) The equivalent circuit model and physical model for a passive pit in a... 

Figure 40 Equivalent circuit models for analyzing impedance data from degraded polymer coated metals, (a) General model, (b) Model I for coatings with defects corroding under activation control, (c) Model II for coatings with defects corroding under diffusion control. (From F. Mansfeld, M. W. Kendig, S. Tsai. Corrosion 38, 478 (1982) and M. Kendig, J. Scully. Corrosion 46, 22 (1990).)...

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