Electrical Analog Models

By making use of these analogies, electrical analog models can be constructed that can be used to determine the pressure and flow distribution in a porous medium from measurements of voltage and current distribution in a conducting medium, for example. The process becomes more complex, however, when the local permeability varies with position within the medium, which is often the case. 

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. 

Fig. 4. Simplified electric analog model of CO2 fluxes in plants.

Claudy et al. (14, 15) used an electrical analog model of a heat flux DSC apparatus with the numerical values of the resistors computed using a Mettler TA 2000B DSC apparatus. 

Electric analog models are a class of lumped models and are often used to simulate flow through Ae network of blood vessels. These models are useful in assessmg Ae overall performance of a system or a subsystem. Integration of Ae fluid momentum equation (longitudinal direction, in cy-... 

FIGURE 1.8 Electrical analog model to simulate atrial and ventricular pumping. Variable capacitances simulate the muscle contractions, and the filling and emptying through the ventricle can be simulated by a series of inductance and... 

Models are the handiwork of theoreticians and may be mechanical or electrical analogs, pictorial representations, or purely mathematical constructs. 

This model leads to A = 0.67 at the gel point, using the zero-frequency values for s and u. Use of the values for s and u calculated by treating the gelation phenomena as a three-dimensional percolation model of a supra-conductor/resistor network (electrical analogy), gives A = 0.72 0.02. 

Figure 2. Electrical analog of transport and reaction resistances in the Kunni-Levenspiel model using the data of Massimillia and Johnstone (9)...

We notice that the elements in series in the mechanical model are transformed in parallel in the electrical analogy. The converse is true for the Kelvin-Voigt model. The electrical analog of a ladder model is thus an electrical filter. 

An electrochemical cell is a type of electrical circuit. As such, it may be modeled with an electrical analog circuit. The potentiometric cell can be considered to be an electrical potential applied to a capacitor and a resistor in series. The capacitor represents the interface between the electrode and the solution, the applied potential is the solution Eh, and the resistor represents the heterogeneous kinetics of the aqueous redox species. The term "heterogeneous kinetics" denotes electron transfer between different phases, in this case aqueous species and the noble-metal electrode. The time required for the capacitor to equilibrate with the applied potential depends on the size of the capacitor and the electrical current. 

Here, the discussion of the viscoelastic exponent n in relation to the assumed gelation model (e.g. electrical analogy percolation or Rouse model), as well as the fractal dimension dfof the critical gel (Muthukumar and Winter 1986 Muthukumar 1989) will be ignored. The reader is referred e.g. to (Adam and Lairez 1996 Martin and Adolf 1991). It has been also shown, that stoichiometry, molecular weight and concentration have an impact on the critical gel properties (Winter and Mours... 

Electric networks have been used to describe radiation heat transfer. Because electric networks have commonly available solutions, this analogy is useful. It also permits the use of an analog computer for solving complex problems. Similarly, conduction systems have been studied using small analog models made of various materials, including conducting paper. 

Nevertheless, the simplicity of Equation 63.2 and the analogy with the transmission line have made the 1-D model popular. We note that there is interest in utilizing the principles in an analog model built on a silicon chip, because of the high performance of the actual cochlea. Watts [ 1993 ] reports on the first model with an electrical analog of 2-D fluid in the scab. An interesting observation is that the transmission line hardware models are sensitive to failure of one component, while the 2-D model is not. In experimental models, Bekesy found that a hole at one point in the membrane had little effect on the response at other points. 

Impedance data may be analyzed directly from impedance plots (Bode and Nyquist plots). Alternatively, the capacitances and the resistances are varied until the model predicted ac behavior matches the experimental data using a circuit analog model. These electrical properties are then interpreted as the properties of the system. 

In addition to capacitors and resistors, equivalent circuit models include elements that do not have electrical analogs, i.e., as the Warburg (W) element and the constant phase element (CPE). These elements can explain the deviations from theoretical predictions of the models. The Warburg element is frequency-dependent, and its impedance may be represented by following equation ... 

The electric analogy is one of the most extensively used methods for flow and species transport modeling in channel-based microfluidic systems. A microfluidic network is equivalent to an electric circuit, of which each component can be individually described by resistors, ccmductors, and inductors. Equations 1 and 2 show the RLC circuit models in electric and fluidic domain ... 

EIS data are analysed by fitting them to an equivalent electrical circuit model consisting of resistors, capacitors, and inductors. The working electrode interface undergoing an electrochemical reaction is analogous to an electronic circuit and can be characterised as an electrochemical system in terms of equivalent circuit. Typical circuits are shown in Figs. 1.10, 1.11, 1.12 and 1.13 where is admittance (ohm-cm ),Cf is double-layer capacitance and a is the exponents [114]. (/ .E Reference Electrode and W Working Electrode)... 

It was suggested earlier that the electrical analog of an isotropic, homogeneous, ion-ically conducting solid is a pure resistance in parallel with a high-frequency ideal capacitor. This model assumes the absence of electrode polarization and of relaxation processes within the crystal that would lead to additional parallel branches in the equivalent circuit. This model is generally accepted, and several studies of singlecrystal materials have demonstrated its validity. 

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