Equivalent transmission line

However, what we might do is to lower the impedance Zi relative to Zi+, where Zi denotes the input impedance of the equivalent transmission line looking left see insert at the top of Fig. 6.6. If such a manipulation would leave... 

A key element of PEM fuel cell is the CCL, which contains a double layer capacitance connected to resistivities due to proton and oxygen transport. To understand measured CCL spectra, many works employ the equivalent circuit method (ECM). This method is based on the construction of an equivalent transmission line, which gives an EIS spectrum close to the spectrum of the system of interest. The components of the resulting circuit are then attributed to CL physical parameters. For example, in a recent work, Nara et al. (2011) used the ECM to study the CCL degradation mechanisms in the PEFC. 

In studies of these and other items, the impedance method is often invoked because of the diagnostic value of complex impedance or admittance plots, determined in an extremely wide frequency range (typically from 104 Hz down to 10 2 or 10 3 Hz). The data contained in these plots are analyzed by fitting them to equivalent circuits constructed of simple elements like resistances, capacitors, Warburg impedances or transmission line networks [101, 102]. Frequently, the complete equivalent circuit is a network made of sub-circuits, each with its own characteristic relaxation time or its own frequency spectrum. 

Figure 8. Equivalent circuit of a transmission line network representing the ion migration into the pores. R and C denote the resistance of the electrolyte inside a pore and the doublelayer capacitance of the electrode/electrolyte interface, respectively, both of which are taken per unit length.

In the case of viscoelastic loaded QCM two approaches have been followed one methodology is to treat the device as an acoustic transmission line with one driven piezo-electric quartz layer and one or more surface mechanical load (TLM) [50, 51]. A simpler approach is to use a lumped-element model (LEM) that represents mechanical inter-actions by their equivalent electrical BVD circuit components [52, 53]. 

Quantitatively, we proceed via the use of equivalent circuit models. The most general model is the distributed transmission line model of Fig. 

Fig. 13.8. Equivalent circuit models for crystal impedance responses (a) transmission line model (b) lumped clement (modified Butterworth van Dyke) model.

Transmission line — This term is related to a more general concept of electric -> equivalent circuits used frequently for interpretation of experimental data for complex impedance spectra (-> electrochemical impedance spectroscopy). While the complex -> impedance, Z, at a fixed frequency can always by obtained as a series or parallel combinations of two basic elements, a resistance and a capacitance, it is a much more compli-... 

Figure 4.37. Equivalent circuit of conducting polymers [10]. (Albery WJ, Mount AR. Dual transmission line with charge-transfer resistance for conducting polymers. J Chem Soc Faraday Trans 1994 90 1115-9. Reproduced by permission of The Royal Society of...

Figure 5.30. Schematic of the catalyst layer geometry and its composition, exhibiting the different functional parts, a A sketch of the layer, used to construct a continuous model, b A one-dimensional transmission-line equivalent circuit where the elementary unit with protonic resistivity Rp, the charge transfer resistivity Rch and the double-layer capacitance Cj are highlighted [34], (Reprinted from Journal of Electroanalytical Chemistry, 475, Eikerling M, Komyshev AA. Electrochemical impedance of the cathode catalyst layer in polymer electrolyte fuel cells, 107-23, 1999, with permission from Elsevier.)...

When discussing the ionic conductivity of catalyst layers, one must mention the finite transmission-line equivalent circuit, which is widely used to model porous electrodes and was shown as Figure 4.33 in Chapter 4. For ease of discussion, the circuit is re-plotted here as Figure 6.23. 

Figure 6.23. Finite transmission-line equivalent circuit describing the impedance behaviour of a PEMFC electrode [24], (Reprinted from Electrochimica Acta, 50(12), Easton EB, Pickup PG. An electrochemical impedance spectroscopy study of fuel cell electrodes. Electrochim Acta, 2469-74, 2005, with permission from Elsevier and the authors.)...

We may write a lumped equivalent circuit for the resonator and coupled transmission lines following the prescription in RLS-8 (Montgomery et al., 1948) as shown in Fig. 6. At resonance, the power Pi into the load... 

From this physical model, an electrical model of the interface can be given. Free corrosion is the association of an anodic process (iron dissolution) and a cathodic process (electrolyte reduction). Ther ore, as discussed in Section 9.2.1, the total impedance of the system near the corrosion potential is equivalent to an anodic impedance Za in parallel with a cathodic impedance Zc with a solution resistance Re added in series as shoxvn in Figure 13.13(a). The anodic impedance Za is simply depicted by a double-layer capacitance in parallel with a charge-transfer resistance (Figure 13.13(b)). The cathodic branch is described, following the method of de Levie, by a distributed impedance in space as a transmission line in the conducting macropore (Figure 13.12). The interfacial impedance of the microporous... 

Here, the impedance response is independent of the working point, and the frequency dependence is determined solely by the material parameters of the composite. For / <linear branch appears only at frequencies co > a/Cfr). Doublelayer charging and proton transport dominate the overall electrode response in this regime, whereas Faradaic processes are insignificant due to the high frequencies. An equivalent representation of this system is an RC-transmission line [130], Since no fractality or branching of the network is assumed, the response resembles that of a Warburg impedance with a characteristic proportionality Z a where... 

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. 

In the schematic shown in Figure 4.2.10, the RF path is visible between the two signal sources (RF ports) used for extracting the S parameters, and is composed of a length of microstrip transmission line from each port connected to a model for a series-switch plate . Driven by the 6 mechanical wires at each side, which control its position, the switch plate is internally modeled as an equivalent circuit including transmission line, frequency-dependent resistance, and variable capacitance between the conductor on the plate and the underlap of the ends of the microstrip lines separated by the gap for the switch isolation. As with the beams, this model is defined by a complete set of parameters, such as the dimensions and material properties. Parameters can be adjusted quickly to achieve the desired RF performance for different closing states of the electromechanical structure. 

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 coaxial transmission line is basically an insertion loss measurement working from 0.5 MHz to 1 GHz. The dimensions of the 50 ohm line are calculated in order to easily deduce the equivalent free space properties of the material. 

Greszczuk et al. [252] employed the a.c. impedance measurements to study the ionic transport during PAn oxidation. Equivalent circuits of the conducting polymer-electrolyte interfaces are made of resistance R, capacitance C, and various distributed circuit elements. The latter consist of a constant phase element Q, a finite transmission line T, and a Warburg element W. The general expression for the admittance response of the CPE, Tcpr, is [253]... 

A very important issue to consider when working with porous electrodes is that the capacitance is only accessible through a distribution of ohmic resistances, due to the finite resistance of flie supporting electrolyte inside the pores. These situations can be roughly represented by an equivalent circuit, as shown in Fig. 11, where the porous electrode is described by a truncated RC transmission line of R and C elements representing the double Ityer capadtance and the electrolyte resistance in a particular pore size. 

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