Equivalent circuit transmission-line model

Fig. 24 A simple equivalent circuit-transmission line model.

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.

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 frequency dispersion of porous electrodes can be described based on the finding that a transmission line equivalent circuit can simulate the frequency response in a pore. The assumptions of de Levi s model (transmission line model) include cylindrical pore shape, equal radius and length for all pores, electrolyte conductivity, and interfacial impedance, which are not the function of the location in a pore, and no curvature of the equipotential surface in a pore is considered to exist. The latter assumption is not applicable to a rough surface with shallow pores. It has been shown that the impedance of a porous electrode in the absence of faradaic reactions follows the linear line with the phase angle of 45° at high frequency and then... 

There are two electrical equivalent circuits in common usage, the transmission line model (TLM) and a lumped element model (LEM) commonly referred to as the Butterworth-van Dyke (BvD) model these are illustrated in Figs. 2(a and b), respectively. In the TLM, there are two acoustic ports that represent the two crystal faces one is exposed to air (i.e. is stress-free, indicated by the electrical short) and the other carries the mechanical loading (here, a film and the electrolyte solution, represented below by the mechanical loading Zs). These acoustic ports are coimected by a transmission line, which is in turn connected to the electrical circuitry by a transformer representing the piezoelectric coupling. For the TLM, one can show [18, 19] that the motional impedance (Zj ) associated with the surface loading can be related to the mechanical impedances of... 

Fig. 2 Electrical equivalent circuit models for a TSM resonator (a) transmission line model (TLM) and (b) Butterworth-vanDyke lumped element model (LEM). Circuit elements are defined in the main text.

The cathode of a modem Ni-Cd battery consists of controlled particle size spherical NiO(OH)2 particles, mixed with a conductive additive (Zn or acetylene black) and binder and pressed onto a Ni-foam current collector. Nickel hydroxide cathode kinetics is determined by a sohd state proton insertion reaction (Huggins et al. [1994]). Its impedance can therefore be treated as that of intercalation material, e.g. considering H+ diffusion toward the center of sohd-state particles and specific conductivity of the porous material itself. The porous nature of the electrode can be accounted for by using the transmission line model (D.D. Macdonald et al. [1990]). The equivalent circuit considering both diffusion within particles and layer porosity is given in Figure 4.5.9. Using the diffusion equations derived for spherical boundary conditions, as in Eq. (30), appears most appropriate. 

Generally, electroacoustical resonators can be described by mechanical and electrical equivalent circuits. For the quartz, two electrical models are often used the transmission line model and the Butterworth-van Dyke circuit (BVD circuit). These models were made in order to describe the propagation of the acoustic wave in analogy with the electrical waves. More detailed descriptions of electrical equivalent circuits can be found, for example, in [4, 11, 26,48,49]. 

Equivalent circuits for supercapacitor containing both double-layer and pseudocapacitances, (a) Alternative model to Figure 7.10a. (b) Transmission line model, (c) Transmission line model with pure capacitance replaced by constant phase constant. 

Piezoelectric MIcrodlspenser, Figure 1 Equivalent circuit modeling of (a) planar, thickness-polarized, thickness-vibrating piezoelectric transducers using (b) a simple lumped-element Van Dyke model and (c) a more complex transmission-line model. A model of a piezoelectric microdispenser (d) may be formed by connecting the output terminals to the equivalent circuit model of the remaining components (e)... 

A simple but intuitive way to illustrate potential distributions and current fluxes in a porous electrode is the transmission line model (TLM) that was developed by R. De Levie in the 1960s (de Levie, 1964 Levie, 1963). Figure 1.10 shows the transmission-line equivalent circuit for the CCL under steady-state current flux. Resistances due to electron transport in the metal phase, Rm, proton transport in... 

Fig. 19.5. Equivalent circuit based on the transmission line model, including both a Faradaic charge-transfer reaction and double-layer charging in the honeycomb diamond electrode...

The impedance of a porous electrode can be simulated with the transmission line model, and the penetration depth can be evaluated [24]. For the non-porous Pt-modified as-deposited surface, the methanol oxidation reaction can be simulated as a simple Randles equivalent circuit comprising a parallel combination of a double layer capacitance and a semi-infinite Warburg impedance in series with a charge transfer resistance. 

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]. 

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. 

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. 

Fig. 11. Transmission line equivalent RC circuit model for a porous carbon [25], Reprinted with permission from D. Qu, H. Shi, J. Power Sourc., 74 (1998) 99.

The Mason equivalent circuit may be derived directly from Eq. 19. It is sometimes called a transmission-line circuit model since the transcendental terms in the matrix appear in the same way when modeling power transmission lines. Most importantly, the circuit represents more than one resonance with these transcendental terms. Consider first an element that does not have piezoelectricity, implying the piezoelectric stress coefficient e = 0. The force-velocity relationships in the nonpiezoelectric element would then be... 

The waveguide discontinuities shown in Fig. 4.23(a) to Fig. 4.23(f) illustrate most clearly the use of E and H field disturbances to realize capacitive and inductive components. An E-plane discontinuity (Fig. 4.23(a)) can be modeled approximately by a frequency-dependent capacitor. H-plane discontinuities (Fig. 4.23(b) and Fig. 4.23(c)) resemble inductors as does the circular his of Fig. 4.23(d). The resonant waveguide iris of Fig. 4.23(e) disturbs both the E and H fields and can be modeled by a parallel LC resonant circuit near the frequency of resonance. Posts in waveguide are used both as reactive elements (Fig. 4.23(f)) and to mount active devices (Fig. 4.23(g)). The equivalent chcuits of microstrip discontinuities (Fig. 4.23(k) to Fig. 4.23(o)) are again modeled by capacitive elements if the E field is interrupted and by inductive elements if the H field (or current) is interrupted. The stub shown in Fig. 4.23(j) presents a short chcuit to the through transmission line when the length of the stub is A. /4. When the stubs are electrically short (shunt capacitances in the through transmission Hne. 

The equivalent circuit of printed resistors is shown in Figure 9.52 [53]. Depending on the sheet resistivity of fhe paste, the model will be simplified by omitting some elements. The capacitors C represent the influence of the ground plane. They are equal to the capacitance of a transmission line of identical dimensions (width, length, height above groxmd, etc.). Therefore, C reflects the equivalent line type (microstrip-like and stripline-like). 

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