Circuit ladder network

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. 

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 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...

If we now substitute the complex p from Eq. (43) into Eq. (62) and a similar expression for Pp into Eq. (63), we readily find that the resulting impedances each lead to a simple ladder network whose hierarchical form is consonant with the sequential processes adsorption then reaction. But for the full ceU there are two identical interface impedances in series. The circuit for a half-ceU with total impedance Zr is shown in Figure 2.2.7a. The fiiU-cell impedance is just 2Zr . The normalized elements of Figure 2.2.7a are readily found to be given by... 

When terminated by a load Ziis), the entire circuit consisting of L-C ladder network filter and load (Fig. 4.35) must have the transfer function T2 (s)/ (s )> as specified in Eq. (4.63). The L-C ladder network... 

FIGURE 7.127 (a) Passive ladder network with grounded inductors, (b) active circuit simulation of (a). Copyright 2005 by Taylor Francis Group... 

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... 

Hence at low and very low frequencies, the impedance response of the ladder network is the same as that in Fig. 1.83 except that the impedance is divided by a factor N (related to the thickness of the film) and the locus of impedance points is shifted along the real axis in the first quadrant of the complex plane by a factor (iV/3)(f g + R + R ). A further point should be noted At high frequencies a second semicircle can also be observed, which is due to the intrinsic conduction properties of the solid polymer. At these frequencies interfacial behavior can be neglected. We can describe the network in terms of a randies equivalent circuit with parallel components C /N and NR and a series component NR. The latter quantities are defined 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. 

To illustrate the heuristic approach with a simple example and also to flavor why it is an attractive option when intelligently applied, consider the following problan. A very long ladder network consists of 1-Q resistors as shown in Fig. 9.2. The problem is to determine the input resistance at terminals TT . One line of attack is to truncate the network at AA and calculate the input resistance when (a) a short-circuit is created at AA and (b) when an open circuit is allowed at AA . 

Passive circuit realizations of filters have very low sensitivities to element values. Lossless filters designed to maximize power transfer between source and load have the lowest possible sensitivities in the passband (Schaumann, Ghausi, and Laker, 1990). These networks are realized as double-terminated LC ladders as shown in Fig. 7.107, and the corresponding active hlter realizations based on ladder simulations begin with the passive hlter schematic rather than with the transfer function. 

We now examine a method of ladder simulation that utilizes element substitution. This is a direct approachin which Antoniou sgeweraZized impedance converter (GIC), hsted as circuit number 15 in Table 7.3 (see Section 7.7), is used to realize grounded inductors in the passive network. Figure 7.126 shows the GIC... 

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... 

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