Electric network models

As the theory of LPS provides a fairly general and physically sound basis to develop models for the dielectric behavior of sorbent-sorbate systems, we here will present some of its main results. These then are used to develop an electric network model which includes the well-known Debye model of dielectric materials as a special case [6.24]. Another approach to model dielectric properties of solid sorption systems has been discussed by Coelho in [6.29], which however will not be considered here. 

Instead we want to emphasize that simple electric network models of LPS may include three different elemental systems capacitors, resistances, and inductances [6.12]. The basic physical relations, admittance functions, elements of the representation theorem (6.55) and corresponding static and optical permittivity are collected in Table 6.1 below. These elements can be combined by series or parallel connections in may different ways. For the admittance functions of the electric network generated in this way, the simple rules hold that... 

Table 6.1 Elements of electric network models for the (complex) admittance function Y=Y(p) and the frequency dependent permittivity t, = of a sorption system.

J. Jamnik, J. Maier and S. Pejovnik A Powerful Electrical network Model for the Impedance of Mixed Conductors Electrochim. Acta, 44 (1999) 4139. The figure is reprinted from this reference. Copyright 1989, with permission from Elsevier. 

The Maxwell and Voigt models of the last two sections have been investigated in all sorts of combinations. For our purposes, it is sufficient that they provide us with a way of thinking about relaxation and creep experiments. Probably one of the reasons that the various combinations of springs and dash-pots have been so popular as a way of representing viscoelastic phenomena is the fact that simple and direct comparison is possible between mechanical and electrical networks, as shown in Table 3.3. In this parallel, the compliance of a spring is equivalent to the capacitance of a condenser and the viscosity of a dashpot is equivalent to the resistance of a resistor. The analogy is complete... 

The second main category of neural networks is the feedforward type. In this type of network, the signals go in only one direction there are no loops in the system as shown in Fig. 3. The earliest neural network models were linear feed forward. In 1972, two simultaneous articles independently proposed the same model for an associative memory, the linear associator. J. A. Anderson [17], neurophysiologist, and Teuvo Kohonen [18], an electrical engineer, were unaware of each other s work. Today, the most commonly used neural networks are nonlinear feed-forward models. 

Inasmuch as the nature of pipeline elements sets these networks apart from electrical networks (more commonly referred to as electrical circuits) we shall review briefly the modeling of these elements. We shall, however, limit ourselves to the correlations developed for single-phase fluid flow the modeling of two-phase flow is a subject of sufficient diversity and complexity to merit a separate review. 

In general, percolation is one of the principal tools to analyze disordered media. It has been used extensively to study, for example, random electrical networks, diffusion in disordered media, or phase transitions. Percolation models usually require approximate solution methods such as Monte Carlo simulations, series expansions, and phenomenological renormalization [16]. While some exact results are known (for the Bethe lattice, for instance), they are very rare because of the complexity of the problem. Monte Carlo simulations are very versatile but lack the accuracy of the other methods. The above solution methods were employed in determining the critical exponents given in the following section. 

The classical theory predicts values for the dynamic exponents of s = 0 and z = 3. Since s = 0, the viscosity diverges at most logarithmically at the gel point. Using Eq. 1-14, a relaxation exponent of n = 1 can be attributed to classical theory [34], Dynamic scaling based on percolation theory [34,40] does not yield unique results for the dynamic exponents as it does for the static exponents. Several models can be found that result in different values for n, s and z. These models use either Rouse and Zimm limits of hydrodynamic interactions or Electrical Network analogies. The following values were reported [34,39] (Rouse, no hydrodynamic interactions) n = 0.66, s = 1.35, and z = 2.7, (Zimm, hydrodynamic interactions accounted for) n = 1, s = 0, and z = 2.7, and (Electrical Network) n = 0.71, s = 0.75 and z = 1.94. 

Berggren, K.-F., and A.F. Sadreev. Chaos in quantum billiards and similarities with pure-tone random models in acoustics, microwave cavities and electric networks. Mathematical modelling in physics, engineering and cognitive sciences. Proc. of the conf. Mathematical Modelling of Wave Phenomena , 7 229, 2002. 

As mentioned in Section 2.4, in the ionic model the chemical bond is an electrical capacitor. It is therefore possible to replace the bond network by an equivalent electric circuit consisting of links which contain capacitors as shown in Fig. 2.6. The appropriate Kirchhoff equations for this electrical network are eqns (2.7) and (2.11). It is thus possible in principle to determine the bond fluxes for a bond network in exactly the same way as one solves for the charges on the capacitors of an electrical network. While solving these equations is simple in principle providing the capacitances are known, the calculation itself can be... 

Figure 1 A distributed resistor network models approximately how the apphed potential is distributed across a DSSC under steady-state conditions. For various values of the interparticle resistance, fiT,o2, and the interfacial charge transfer resistance, Rc the voltage is calculated for each node of the Ti02 network, labeled Vj through V . This is purely an electrical model that does not take mobile electrolytes into account and, therefore, potentials at the nodes are electrical potentials, whereas in a DSSC, all internal potentials are electrochemical in nature.

Fig. 15. Cluster network model for highly cation-permselective Nafion membranes126). Counterions are largely concentrated in the high-charge shaded regions which provide somewhat tortuous, but continuous (low activation energy), diffusion pathways. Coions are largely confined to the central cluster regions and must, therefore, overcome a high electrical barrier, in order to diffuse from one cluster to the next...

This result is identically that for the SSR model as obtained previously in Eq. (5-128 ). This equation is also valid for Mr > 1 as long as Mh = 2. The electrical network analog methodology can be generalized for enclosures having M > 3. 

To model the microstructure and evaluate the thermoelectric properties, we used following simple equivalent electric circuit model shown in Figure 2. We considered the two phase composite as a cluster pararrel network circuit. Setting for each cluster the characteristic single phase physical property, and settle the material composition to the cluster number ratio, we can simulate the total thermopower of the system by Millman s theorem of d.c. circuit. 

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. 

The AT-cut quartz resonator can be modeled mechanically as a body containing mass, compliance, and resistance. Figurel-a) shows the mechanical vibration motion depicting the vibration of the quartz resonator. An electrical network called an equivalent electrical circuit consisting of inductive, capacitive and resistive components can represent this mechanical model. Figure... 

Except in high vacuum the contribution 2(a) may be neglected. Figure 11.7.a-4 represents this model by means of an electrical network. By expressing each of these contributions by means of the basic formulas for heat transfer and combining... 

Tsitlik J.E., Halperin H.R., Popel A.S., et al. 1992. Modeling the circulation with three-terminal electrical networks containing special nonlinear capacitors. Ann. Biomed. Eng. 20 595. 

Many systems in nature and technology exhibit typical structures which can be used for modeling considerations via analogy. One of those structures are networks. Examples include the vascular system of animals, rivers or streets in a given region, or electrical networks. The features and related scientific questions are... 

We will now give a very general description of the black box and how to characterize it electrically, irrespective of the box content. The black box may be considered to contain the real tissue with electrodes for excitation and response measurement, or our model in the form of an electric network as a combination of lumped (discrete) electrical components. The network may be with two, three, or four external terminals (compare the number of electrodes used). A pair of terminals for excitation or recording is called a port. The treatment is so general that the content can be characterized with global variables not particularly linked with electrophysiology. 

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-... 

Fig, 1. Sketched cross section of a ceramic sample with electrodes applied (a), corresponding simplified brick wall model (b), and the equivalent electrical network... 

Consider the simple linear electrical network depicted in Fig. 4.19. It can be viewed as an electrical analogue of the coupled hydraulic tank system considered in Section 4.4.1. A bond graph of the direct model with the two inputs I(t) and E(t) and the two outputs e and /2 appears in Fig. 4.20. There is one set of two disjoint input-output causal paths... 

---