Ladder Circuit Model

Fadder circuits have been used to model double-layer capacitive behavior in pulse load and slow discharge applications and have demonstrated success in modeling nickel-carbon fiber electrodes. Through the use of software employing various statistical techniques, the parameters for several ladder circuits (Fi to FJ can be assessed. 

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Fig. 10.4 Modeling of the electric circuit for the cell. A ladder circuit model is employed. Eo is the cell operating voltage Nernst potential R, the ohmic resistance tj/, the overpotential and / , the electric current.

For simple calculations of the electric current, in the simplest case, a ladder circuit model is sufficient for the simulation. A discrete anode/electrolyte/cathode unit shown in Figure 10.4 is employed for the modeling. The direction of the electric current path in the electrolyte is vertical to the interfaces between the electrolyte and electrodes, and the in-plane path is treated as negligible. The electric current I, in the /th discrete cell is determined by the operating cell voltage Voeii as follows ... 

Predictive operating dynamics of a complex supercapacitor device can be achieved through the implementation of one of several model circuit analogies. Distributed parameter models are often used the most common is the classical equivalent model described in Figure 6.9. The ladder circuit models represent extended distributions of capacitances and resistances in reference to the classic equivalent and can be expanded to include several resistance and capacitance elements in parallel for the consideration of non-uniform pore charging in highly porous materials. 

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. 

The circuits most often used in measuranent modeling are the Voigt, ladder, and Maxwell circuits, as presented in Fig. 45. Zoltowski proposed using ladder circuits for measurement modeling, substituting circuit resistances and capacitances by the CPE elements. 

Figure 12.5 The equivalent circuit of a porous carbon electrode. It consists of a single vertical ladder network in series with an flC parallel network. The ladder network models the response of pores in the body of...

Since all three of the Figure 2.2.4 circuits are equivalent as far as fitting is concerned, how does one choose between them, particularly in cases when element estimates for the different circuits are quite different First, one may use continuity and knowledge of the physical processes involved, as in the above brief discussion of inductive-like effects in adsorption. Second, one may be able to compare the circuits with the predictions of a physical model—one which yields simpler expressions for the elements of one of the circuits than for the others. This has been done for the unsupported conduction case (Franceschetti and Macdonald [1977]) and the work showed that in the case of charge of a single-sign mobile the ladder circuit was much superior to the others (see Section 2.2.2.3). 

However, in various papers by Miller [1972], he has been able to model the impedance spectrum and power-density (related to the distribution of RC time-constant values) of practical carbon double-layer capacitor devices by a five-element ladder circuit of C and R elements as illustrated in Figure 4.5.27. The resulting impedance behavior is exemplified in Figure 4.5.28. 

Figure 4.5.27. Five-element ladder RC circuit modeling the impedance behavior of a porous, electrochemical capacitor, designed for memory back-up (from Miller [1972]).

This is an iterative technique used to solve linear electric networks of the ladder type. Since most radial distribution systems can be represented as ladder circuits, this method is effective in voltage analysis. An example of a distribution feeder and its equivalent ladder representation are shown in Fig. 10.116(a) and Fig. 10.116(b), respectively. It should be mentioned that Fig. 10.116(b) is a linear circuit since the loads are modeled as constant admittances. In such a linear circuit, the analysis starts with an initial guess of the voltage at node n. The current I is calculated as... 

The greatest physiological application for the equivalent circuit model is where myocardial cells are represented as a ladder network. In the case of the heart, myocytes are represented as being connected end to end (which they are within the heart) with an internal resistance (/ ) between two cells. Within the context of the heart, it must be realised that myocytes are connected to adjacent cells and therefore the network most accurately reflects the physiological scenario when considered in two or even three dimensions. 

A number of models describing supercapacitor resistor and capacitor behaviors used to mimic their performances in power systems have been reported and include classical equivalent, ladder circuit, and lumped or distributed parameter electrical and Debye polarization cell models [6]. An established design of a dynamic model of the often-used polymer electrolyte membrane fuel cell (PEMFC) is included in MATLAB and Simulink software to simulate performance under varying conditions specific to applications. 

A model by Rafik et al. [9] proposed using a limited number of variables to account for the dependencies of capacitance on frequency, voltage, and temperature, avoiding the complex RC element determinations necessary for ladder circuits. The model (Figure 6.12a) shows three incorporated circuits, two of which are similar to those described in the previous two model discussions. 

Tatarchuk, B. J. 2003. Modeling double-layer ladder circuits. IEEE Transactions on Power Electronics, 39, 430-438. 

FIGURE 3-5 Equivalent circuit models with two time constants for R = S kohm, = 50 = 10 kohm, = 0.5 A. Maxwell B. Ladder C. Voigt... 

Figure 45. Typical circuits used in ac modeling they are experimentally indistinguishable (a) Voigt, (b) Maxwell, and (c) ladder. ...

Recently, a new equivalent circuit was proposed for porous carbon electrodes (Figure 12.5). Naively, one might suppose that this would involve multiple ladder networks in parallel, in order to model the response of multiple pores in parallel. However, the somewhat surprising result is that the circuit in Figure 12.5 is able to capture the complete multipore behavior [37]. 

FIGURE 1.82. Schematic representation of the equivalent circuit ladder network corresponding to Fletcher porous electrode model for electronically conducting polymers (see Refs. 68, 69). The specific equivalent circuit representation of the interfacial impedance element is also illustrated. 

Figure 1.82 shows the model circuit which takes the form of a diagonally connected discrete ladder network or in simple terms, a dual-rail transmission line of finite dimension. The essential problem is to replace the general impedance elements x, y, and z by suitably arranging such passive circuit elements as resistors and capacitors that adequately represent the microscopic physics occurring within an electronically conducting polymer. 

We extend the analysis and consider the entire ladder network in terms of distinct R and C circuit elements. The impedance x can be represented by a resistance Ri, which defined the resistance of counterions in the pore electrolyte. Furthermore the impedance element z, which is that of the solid polymer, is replaced by a Randles equivalent circuit (see Fig. 1.84), where there is a parallel arrangement of a resistor Rj. and a capacitor Q in series with a resistor Ra- Hence we see that the pore solution is modeled in terms of a simple resistor, whereas the solid polymer is a binary composite medium. TTie latter assumption can be justified as follows. From a macroscopic viewpoint (and this has been demonstrated experimentally), the electronic resistance of the polymer is due to two contributions the first, Ra, from regions of high structural order the second, R, from regions of low structural order. Hence Ra is smaller than R. From a microscopic point of view, the polymer may exhibit two fundamentally different types of conduction. As noted in... 

This example shows three t5rpes of circuit diagrams for two relaxation time constants with the same R and C values, resulting in a Nyquist plot that usually shows two time constants (in the case of the Ladder model only one time constant was observed due to selected values for R, R, Cj, parameters) but with various characteristic fitting parameters. CMe always has to attempt to select a circuit that represents the geometrical distribution of impedance-related processes that physically occiu- inside of the system. A criterion of simplicity also has to be applied based on selection of only feasible processes with defined and well-resolved impedance characteristics to model the system. Computer-assisted fitting is necessary to resolve the overlapping arcs in the Z and M planes, but this method typically is unable to resolve arcs with time constants that are different by less than at least a factor of 100. 

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