Electrical impedance model

Grimnes, S., Martinsen, 0.G., 2005. Cole electrical impedance model — a critique and an alternative. IEEE Trans. Biomed. Eng. 52 (1), 132—135. 

From Eq. 19, the stress induced by the actuation input is directly related to the charge density distribution p. Therefore, as a first step in developing the aetuation model, we will derive the electrical impedance model in this section. While the latter is of interest in its own right, one also obtains the explicit expression for / as a by-product of the derivation. Consider Fig. 4, where the beam is clamped at one end z = 0 and is subject to an actuation voltage producing the tip displacement w(t) at the other end z = L. The neutral axis of the beam is denoted by x = 0, and the upper and lower surfaces are denoted hyx = h and x = —h, respectively. 

The impedance spectroscopy of steel corrosion in concentrated HC1, with and without inhibitors, exhibit relatively straightforward electrochemical phenomenology and can be represented by simple equivalent circuits involving primarily passive electrical elements. Analysis of these circuits for steel corroding in HC1 per se reveals that the heterogeneity of the surface is established rapidly and can be simulated with a simple electrical circuit model. 

This model, when applied to Nation as a function of water content, indicated a so-called quasi-percolation effect, which was verified by electrical impedance measurements. Quasi-percolation refers to the fact that the percolation threshold calculated using the single bond effective medium approximation (namely, x = 0.58, or 58% blue pore content) is quite larger than that issuing from a more accurate computer simulation. This number does not compare well with the threshold volume fraction calculated by Barkely and Meakin using their percolation approach, namely 0.10, which is less than the value for... 

Polar Cell Systems for Membrane Transport Studies Direct current electrical measurement in epithelia steady-state and transient analysis, 171, 607 impedance analysis in tight epithelia, 171, 628 electrical impedance analysis of leaky epithelia theory, techniques, and leak artifact problems, 171, 642 patch-clamp experiments in epithelia activation by hormones or neurotransmitters, 171, 663 ionic permeation mechanisms in epithelia biionic potentials, dilution potentials, conductances, and streaming potentials, 171, 678 use of ionophores in epithelia characterizing membrane properties, 171, 715 cultures as epithelial models porous-bottom culture dishes for studying transport and differentiation, 171, 736 volume regulation in epithelia experimental approaches, 171, 744 scanning electrode localization of transport pathways in epithelial tissues, 171, 792. 

Rational optimization of performance should be the main goal in development of any chemical sensor. In order to do that, we must have some quantitative tools of determination of key performance parameters. As we have seen already, for electrochemical sensors those parameters are the charge-transfer resistance and the double-layer capacitance. Particularly the former plays a critical role. Here we outline two approaches the Tafel plots, which are simple, inexpensive, but with limited applicability, and the Electrochemical Impedance Spectroscopy (EIS), based on the equivalent electrical circuit model, which is more universal, more accurate, and has a greater didactic value. 

Salter, D.C. Monitoring human skin hydration in vivo using electrical impedance — a model of skin as a solid ionic conductor. Proceedings IX International Conference on Electrical Bio-Impedance and European Community Concerted Action on Impedance Tomography, Heidelberg, September 26-30, 1995, pp. 17-20. 

Two types of electrical analogy model for the interpretation of impedance data can be used based on combinations of resistances and capacitances, or based on transmission lines. These possibilities are now described. 

EIS data analysis is commonly carried out by fitting it to an equivalent electric circuit model. An equivalent circuit model is a combination of resistances, capacitances, and/or inductances, as well as a few specialized electrochemical elements (such as Warburg diffusion elements and constant phase elements), which produces the same response as the electrochemical system does when the same excitation signal is imposed. Equivalent circuit models can be partially or completely empirical. In the model, each circuit component comes from a physical process in the electrochemical cell and has a characteristic impedance behaviour. The shape of the model s impedance spectrum is controlled by the style of electrical elements in the model and the interconnections between them (series or parallel combinations). The size of each feature in the spectrum is controlled by the circuit elements parameters. 

Figure 4.1. Nyquist plot showing the impedance spectra of an R/CPE electric circuit model [2], (Reproduced with permission from Research Solutions Resources LLC.)...

Impedance models are constructed according to the electrochemical phenomena. The total impedance of an electrochemical system can be expressed by different combinations of the electrical elements. This section covers the features of basic equivalent circuits commonly used in electrochemical systems. In Appendix D, the effect of an element parameter change on a spectrum related to a given equivalent circuit is described in detail. 

Springer TE, Raistrick ID (1989) Electrical impedance of a pore wall for the flooded-agglomerate model of porous gas-dilFusion electrodes. J Electrochem Soc 136 1594-603... 

Haake and Dual have developed models for calculating the transmitted wave as a function of the applied electric voltage and incident wave. They found application of an electrical impedance to the piezo-device to allow the reflection coefficient to be altered in amplitude and phase, and hence the characteristics of the reflected wave to be controlled [62]. 

III.l [see also Eq. (17) and Fig. 2], and that in the presence of a faradaic reaction [Section III. 2, Fig. 4(a)] are found experimentally on liquid electrodes (e.g., mercury, amalgams, and indium-gallium). On solid electrodes, deviations from the ideal behavior are often observed. On ideally polarizable solid electrodes, the electrically equivalent model usually cannot be represented (with the exception of monocrystalline electrodes in the absence of adsorption) as a smies connection of the solution resistance and double-layer capacitance. However, on solid electrodes a frequency dispersion is observed that is, the observed impedances cannot be represented by the connection of simple R-C-L elements. The impedance of such systems may be approximated by an infinite series of parallel R-C circuits, that is, a transmission line [see Section VI, Fig. 41(b), ladder circuit]. The impedances may often be represented by an equation without simple electrical representation, through distributed elements. The Warburg impedance is an example of a distributed element. 

Experiments carried out on monocrystalline Au(lll) and Au(lOO) electrodes in the absence of specific adsorption did not show any fre-quency dispersion. Dispersion was observed, however, in the presence of specific adsorption of halide ions. It was attributed to slow adsorption and diffusion of these ions and phase transitions (reconstructions). In their analysis these authors expressed the electrode impedance as = R, + (jco iJ- where is a complex electrode capacitance. In the case of a simple CPE circuit, this parameter is = T(Jcaif. However, an analysis of the ac impedance spectra in the presence of specific adsorption revealed that the complex plane capacitance plots (C t vs. Cjnt) show the formation of deformed semicircles. Consequently, Pajkossy et al. proposed the electrical equivalent model shown in Fig. 29, in which instead of the CPE there is a double-layer capacitance in parallel with a series connection of the adsorption resistance and capacitance, / ad and Cad, and the semi-infinite Warburg impedance coimected with the diffusion of the adsorbing species. A comparison of the measured and calculated capacitances (using the model in Fig. 29) for Au(lll) in 0.1 M HCIO4 in ths presence of 0.15 mM NaBr is shown in Fig. 30. 

Electrochemical impedance spectroscopy (EIS) simplest electrical-circuit model... 

With reference to the EIS method, prove that for the electrical-circuit model of Fig. 6.18, the equivalent circuit impedance is given by Eq 6.64. 

The first commercially successful automated blood cell counter, the Model A Coulter Counter, was introduced in 1956. The Model A counted cells by using electrical impedance properties, and the Coulter Counter quickly became the instrument of choice for counting red and white blood cells. When it was later demonstrated that cell volume was roughly proportional to the electrical impedance signal amplitude, the Coulter Counter was modified to provide MCV as the mean of the individually measured red cell volumes. Impedance counters calculated HCT as (RBC x MCV)/10 and, by the early 1970s, these instruments also counted platelets. By the mid-1970s, impedance counters combined with photometric hemoglobinometers to produce CBCs. 

For discussion of the consequences of acoustic wave velocity and coupUng factor for sensor apphcations we recall the electrical impedance, Z, as it can be derived from the one-dimensional transmission Une model [22,23] ... 

Naturally, electrical engineers have designed equivalent circuits for nonelectrical wave phenomena. The waves may or may not be confined to cables. For simple propagating waves, the equivalent circuits are often called transmission line models. The transmission line has two ports representing input and output. The input-output relation can be predicted by applying the Kirchhoff laws to the set of elements located in between. The circuit elements may be simple resistors or capacitors, but their electrical impedance may also be a more complicated function of frequency (see, for instance. Fig. 6)... 

The valves are straightforward and are often implemented as ideal diodes (for electrical circuit models) or as IF-THEN-ELSE statements (for algorithmic models) to keep all flows nonnegative. Defects in the valves can be added to simulate heart defects (e.g., leaky diodes for regurgitation). Other types of heart defects are just as easily simulated. For example, Blackstone et al. [1976] placed an impedance between the atrial chambers to simulate a septal opening. 

Brown BH, Barber DC, Wang W, Lu L, Leathard AD, Smallwood RH, Hampshire AR, Mackay R, Hatzigalanis K. i994a. Multi-frequency imaging and modeling of respiratory related electrical impedance changes. Physiol Meas. 15(suppl.), 1-12. 

For developing the MRR equation for EMM, the following resistances and impedances are to be considered (1) double-layer capacitance (2) Warburg impedance, (3) charge transfer resistance, and (4) electrolyte resistance [12]. Let us consider the double-layer electrical equivalent model circuit for EMM as shown in Fig. 3.6. It consists of an active electrolyte resistance along shorter path, Rshort. in series with the parallel combination of the double-layer capacitance, Cj, and an impedance of a faradaic reaction. The faradaic reaction consists of an active charge transfer resistance R and Warburg resistance Rw-... 

An almost identical electric conduction model is proposed by W. Gopel. The electric diagrams eqrrivalent to such models are displayed in Figure 7.3 they show the presence of a capacitive process at the depletion area. This effect was discovered using complex impedance spectrometry. 

For the dynamic lung impedance model to be useable in Finite Difference Method or Finite Element Method impedance signal simulations, the dynamic tissue sample model is discretized into volume data. At first 3D data with 35 x 35 x 35 voxel resolution is prepared from each of the 40 time frames. This allows for easy import into MATLAB or COMSOL based calculation. The volume data includes percentage of blood vessels (blood) for each of the 35 X 35 X 35 X 40 voxels. It can readily be transformed into electric/dielectric properties for each voxel with tissue data available on the internet. But data can also be exported with arbitrary resolution depending on calculation-simulation requirements. The simulations are run separately for each of the 40 time-frames to get full frequency characteristic of impedance measurement across the tissue sample. Finally we can get 40 frequency characteristics—one for each time-frame and to see a dynamic electrical impedance signal on a certain frequency, we just need to plot the impedance value at the chosen frequency from the 40 time-frames. 

The usability for this model in FDM was tested with a 2D version of the same model in previous article [12]. The similarity of the arterial network structure to actual arterial networks have been estimated by the creators of the CCO algorithm [5], but its apphcability in final electric impedance simulation is yet to be tested. At the moment the model is not incorporated into a dynamic chest or full body model. The authors are investigating possibilities to add the features of this dynamic model into existing full body dynamic anatomy models for sophisticated and user-friendly modeling methods. 

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