Equivalent circuit equations

All the circuits in this chapter are with ideal components, that is, frequency independent resistance, conductance, and capacitance. Derived parameters, however, are often 

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Approximating Transducer Behavior with Equivalent Circuit Equations... 

Ion transport across membranes can be evaluated by using mucosal and serosal electrodes to read transepithelial current (I) and potential difference OP). With these parameters, equivalent circuit analysis can be utilized to account for the relative contributions of transcellular and paracellular pathways. Ionic flux (J) is defined by the Nernst-Planck equation,... 

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

Commercial impedance analyzers offer equivalent circuit interpretation software that greatly simplifies the interpretation of results. In this Appendix we show two simple steps that were encountered in Chapters 3 and 4 and that illustrate the approach to the solution of equivalent electrical circuits. First is the conversion of parallel to series resistor/capacitor combination (Fig. D.l). This is a very useful procedure that can be used to simplify complex RC networks. Second is the step for separation of real and imaginary parts of the complex equations. 

The reaction of our investigator to the puzzle presented by the black box will differ according to whether he is a mathematician, electrical engineer, physicist or chemist. The mathematician will be satisfied by a description in terms of differential equations and the engineer by an equivalent circuit. However the physicist or chemist will want an interpretation in terms of the structure of the material whose response can be represented by the black box. The materials scientists will often be disappointed. 

To be more specific, we now refer to Equations 8.86 and 8.88, which represent the impedance of the equivalent circuit corresponding to the electrochemical reaction at the electrode-electrolyte interface. If in Equation 8.86 we calculate the limit for co —> 0, then the intercept of the plot in the real axis is Zr = Rs + Rp on the other hand, if the limit for co —> is calculated, then Zr = Rs. Besides, at the frequency where a maximum of X, (co) is detected, we have Rp Cdl = l/comax = x, where the time constant, x, indicates how fast the electrochemical reaction is. Finally knowing Rp Cdl = l/comax, it is possible to calculate Cdi [129,130],... 

In these equations a = Rs + Rp,b = RSRPC, t is time, v = sweep rate, and x = 1/RSC + 1 IRVC. Time in Eq. (19) can be equated to the sweep rate since t = A pP/v. All three equations include a term that is independent of voltage scan rate and a second term that depends on voltage scan rate. The scan-rate-dependent term becomes negligible at low scan rates. Macdonald (31) and Townley (33) separately derived the current response of the standard three-element electrical equivalent circuit (Fig. 3a) to a small-amplitude triangular voltage excitation... 

The interpretation of measured data for Z(oi) is carried out by their comparison with predictions of a theoretical model based either on the (analytical or numerical) integration of coupled charge-transport equations in bulk phases, relations for the interfacial charging and the charge transfer across interfaces, balance equations, etc. Another way of interpretation is to use an -> equivalent circuit, whose choice is mostly heuristic. Then, its parameters are determined from the best fitting of theoretically calculated impedance plots to experimental ones and the results of this analysis are accepted if the deviation is sufficiently small. This analysis is performed for each set of impedance data, Z(co), measured for different values of external parameters of the system bias potentials, bulk concentrations, temperature... The equivalent circuit is considered as appropriate for this system if the parameters of the elements of the circuit show the expected dependencies on the external parameters. 

Equivalent circuits for the catalyst layer are similar to those for porous electrodes, where charge-transfer resistance, capacitance, and Warburg resistance should be considered. The catalyst layer can be conceived of as a whole uniform unit or as a non-uniform circuit. In the case of a uniform unit, the equivalent circuits are similar to the modified ones discussed in Section 4.2.2 2, and the equations in that section apply. In many cases, such as in the presence of adsorbents, the surface is covered by the adsorbed species. For example, in direct methanol fuel cells and in H2/air fuel cells, CO adsorption should be considered. One example is illustrated in Ciureanu s work [7], as shown in Figure 4.31. 

Figure 4.31. Equivalent circuit with adsorption-related electrode surface The resistance of this equivalent circuit is given by the following equation ...

By using the combined parameters, the equivalent circuit is simplified to Figure 6.69b. The catalyst layer is a porous structure and the impedance of each pore is given by the following equations ... 

The electrical characteristics of the TSM resonator with a generalized surface perturbation can be described by the equivalent-circuit model of Figure 3.7b. Measurements can be made on a dry TSM resonator to determine C o,L, Ci, and R. Fixing these parameters and fitting the equivalent-circuit model to data measured on an immersed device determines R2 and L2. Equations 3.21 can then be used to determine the components of from L2 and R2. 

The equivalent circuit model of Figure 3.7 can be used to describe the near-resonant electrical characteristics of the quartz resonator coated by a viscoelastic film. The surface film causes an increase in the motional impedance, denoted by the complex element Zg. From Equation 3.19, this element is proportional to the ratio of the surface mechanical impedance Zj contributed by the film to the characteristic shear wave impedance Zq of the quartz. 

Turning now to the case in which diffusion control must be considered, we have already seen that the equivalent circuit takes the form shown in Fig. 2K, in which the symbol -W- represents the so-called Warburg impedance which accounts for diffusion limitation. The diffusion equations have been solved for the case of a low amplitude sine wave applied to the interphase. The Warburg impedance is given by... 

Consider the equivalent circuit shown in Fig. 2K, ignoring, for the moment, the Warburg impedance. When a galvanostatic pulse is applied to such a circuit, the response is that shown in Fig. 5M. The equation describing the change of overpotential with time during the transient is... 

It should be remembered that the curves shown in Fig. 13L are all simulated and therefore "ideal" in the sense that they follow exactly the equations derived for the given equivalent circuit. In practice, the points are always scattered as a result of experimental error. Also, the frequency range over which reliable data can be collected does not necessarily correspond to the time constant which one wishes to measure. For the case shown in Fig. 13L(a) the semicircle can be constructed from measurements in the range of 1 > o) > 20. In Fig. 13N(b) one would have to use data in the range of about 10 > to 200 to evaluate the numerical values of the circuit elements. From the Bode magnitude plots, can be evaluated from high-frequency measurements (to 100), while R can be obtained from low frequency data (to < 1). The capacitance can be obtained approximately as = l/co Z at the inflection point (which coincides with the maximum on the Bode angle plot), but this would be correct only if (p - 90 that is, if the... 

Once we have an equivalent circuit, we may manipulate the circuit equations to explore the effect on spectrometer performance of changing the values of circuit elements. It would be useful to have a variable coupling scheme in order to tune the spectrometer for optimum performance, just as in the microwave spectrometer case. Such a scheme is described in Section X. 

It has to be mentioned that such equivalent circuits as circuits (Cl) or (C2) above, which can represent the kinetic behavior of electrode reactions in terms of the electrical response to a modulation or discontinuity of potential or current, do not necessarily uniquely represent this behavior that is other equivalent circuits with different arrangements and different values of the components can also represent the frequency-response behavior, especially for the cases of more complex multistep reactions, for example, as represented above in circuit (C2). In such cases, it is preferable to make a mathematical or numerical analysis of the frequency response, based on a supposed mechanism of the reaction and its kinetic equations. This was the basis of the important paper of Armstrong and Henderson (108) and later developments by Bai and Conway (113), and by McDonald (114) and MacDonald (115). In these cases, the real (Z ) and imaginary (Z") components of the overall impedance vector (Z) can be evaluated as a function of frequency and are often plotted against one another in a so-called complex-plane or Argand diagram (110). The procedures follow closely those developed earlier for the representation of dielectric relaxation and dielectric loss in dielectric materials and solutions [e.g., the Cole and Cole plots (116) ]. 

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