Equivalent Circuit Theory

The flow rates entering a given point equals the flow rates leaving the point. 

The total pressure difference (including pumps) in a closed loop of the circuit is zero. 

Volume in fluidics is equivalent to electrical charge, and mechanical elasticity of the walls of the channels (known as compliance deflned as change of volume due to change in pressure) is equivalent to electrical capacitance. More details on analogy related to channel with compliant wall will be discussed in a later section. Also the inertia of fluids has an eleetrical analog, namely, electrical inductance. 

using equivalent circuit theory, it is possible to obtain a good estimate of the total hydraulic resistance of the microfluidic network without performing complicated numerical simulations. This is extremely helpful when designing LOG systems, and if all the involved channels are long and narrow, then the result is very accurate. 

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With respect to the size and charge selectivity of paracellular pathways, equivalent pore theory has been utilized to calculate an effective radius based on the membrane transport of uncharged hydrophilic molecules, while equivalent circuit theory has been used to separate mediated from paracellular membrane transport of small ions. The term equivalent should be emphasized, as selectivity parameters are obtained from membrane transport data, so phenomenological information is used to quantitate the magnitude of aqueous pathways... 

But when considered over a wide range of frequencies, the properties of a real electrode do not match those of the equivalent circuits shown in Fig. 12.12 the actual frequency dependence of Z and a does not obey Eq. (12.21) or (12.22). In other words, the actual values of R and or R and are not constant but depend on frequency. In this sense the equivalent circuits described are simplified. In practice they are used only for recording the original experimental data. The values of R and Cj (or R and C ) found experimentally for each frequency are displayed as functions of frequency. In a subsequent analysis of these data, more complex equivalent circuits are explored which might describe the experimental frequency dependence and where the parameters of the individual elements remain constant. It is the task of theory to interpret the circuits obtained and find the physical significance of the individual elements. 

C. Paracellular Transport—Equivalent Pore and Circuit Theory... 

Additional epithelial aqueous pathways of significantly smaller radius (<3 A) have also been documented utilizing both equivalent pore and circuit theory [25], These pathways may correspond to specific channels through lipid membranes as opposed to paracellular pathways. Osmotically activated ion channels [35] and even specific water channels [36] have been characterized in renal epithelia. In intestinal epithelia, mucosal chloride channels have been studied in secreting crypt cells, and basolateral potassium channels in colonic epithelia serve cellular ion and volume homeostatic functions. 

The reduction of Cd(II) ions on DME was also investigated in 1 M perchlorate, fluoride and chloride solutions using dc, ac admittance, and demodulation methods [27]. It was found that in the perchlorate supporting electrolyte, the reduction mechanism is also CEE, and that the rate constant of the chemical step is quite close to the value characteristic for fluoride solutions. The theories available at present could not be applied to the Cd(II) reduction in chloride solution because of the inapplicability of the Randles equivalent circuit. 

If we switch-off the current after the steady state has been reached, the voltage relaxes to the initial zero-level. The electrical behavior can be taken into account by introducing in the simple equivalent circuit (eqc) (Eq. (60)) a capacitor Cl in series to Rioa or to R,.ml (if ions or electrons are blocked) In the language of system theory the equivalent circuit of the bulk represents a PDTi-element and reads3 15 e.g., for cells 3 and 4 Par (Cx. Par (Rm , Ser ( AJrai, Cl))). 

According to the superposition theorem of system theory for linear responses, this response to a step-function in the current can be employed to deduce the impedance behavior. As regards a qualitative discussion, one can adopt the above description by just replacing short/long times by high/small frequencies. Quantitatively the impedance is given by a Laplace transformation of Eq. (64) (or equivalently by applying Kirchhoff s laws to the equivalent circuit (Eq. (63))) with the result... 

Figure 11. Plots of log Z and / v.v. log / for a thiol-hexapeptide-coated mercury drop immersed in 5xlO 3M (a), I.3xlO 2M (b), 3.6xlO 2M (c), and 0.1M (d) KC1, as obtained at -1.000 V over the frequency range from 0.1 to 105 Hz. At frequencies <102 Hz all Bode plots coincide hence, only the experimental points for the lower KC1 concentration were reported. The solid curves are least-squares fits to the simple equivalent circuit of inset (1), which consists of the electrolyte resistance Ra, with in series a RSCS mesh representing the self-assembled monolayer and a further RjiCji mesh representing the diffuse layer. Rs = 0.14 Mfi cm2 C, = 11 pF cm-2 Ra = 4.53 (a), 4.17 (b), 1.27 (c) and 0.87 KO cm2 (d). CW 68 (a), 61 (b), 80 (c) and 84 pF cm 2 (d). Inset (2) shows the reciprocal, 1/Cji, of the experimental diffuse-layer capacitance vs. the l/C fajj = 0) value corresponding to the same KC1 concentration, as calculated on the basis of the Gouy-Chapman (GC) theory. The solid curves are 1 /Ca(OM) vs 1 /C,ii(ctm = 0) plots calculated from the GC theory for different charge densities afo on the metal, whose values are reported on each curve. (Reprinted from Ref.114 with permission from the Am. Chem. Soc.)...

By using small-signal theory, the equivalent circuit now becomes... 

I. A. Eshrah, A. A. Kishk, A. B. Yakovlev, A. W. Glisson, and C. E. Smith, Analysis of waveguide slot-based structures using wide-band equivalent-circuit model, IEEE Trans. Microw. Theory Tech., vol. 52, no. 12, pp. 2691-2696, Dec. 2004. doi 10.1109/TMTT.2004.837320... 

The derivation above ignores piezoelectricity (Sect. 6). The theory of the piezoelectric plate has been worked out by Tiersten [56]. Kanazawa has applied this theory rigorously to the case of a crystal loaded with a liquid or a viscoelastic film [54]. These treatments are equivalent to the treatment with equivalent circuits (Sect. 6), and we therefore defer the discussion of piezoelectricity to that section. 

These network equations differ from the Kirchhoff equations used in electrical circuit theory where the sum in the node Equation 10.5 is zero. O Keeffe (Struct. Bonding 1989, 71, 161-190) has shown that a correct mathematical correspondence requires that Kirchhoff s loop law be equivalenced with Equation 10.5 and the junction law with Equation 10.6. This requires replacing the nodes of the bond network with the loops of the equivalent Kirchhoff network and vice versa. For practical purposes it is simpler to stay with Equations 10.5 and 10.6... 

A successful equivalent circuit approach to Wagner s theory was worked out by Hoar and Price. They developed a simple voltage divider circuit which gives quantitative formulas for emf and scaling rate that are very similar to those derived more rigorously by Wagner. A linear lumped version of their proposed circuit is shown in Fig. 3. The subscripts 1, 2 and 3 refer to M cations, X anions and electrons respectively. 

In part I above, c. Wagner s theory of mixed conduction was reviewed in terms of an equivalent circuit approach. The implications of mixed conduction theory for parabolic scaling of metals in high temperature atmospheres were also detailed. It was pointed out, however, that current interest in mixed conduction theory is no longer motivated by corrosion considerations because far too few systems of practical interest conform to the conditions required for pareibolic oxidation. 

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