Ladder network

The Warburg impedance is sometimes modeled by RC ladder networks having impedance of (ice)-1/2 frequency dependences. However, such modeling is somewhat misleading, due to the different underlying physics. 

In view of the difficulties found in providing a satisfactory solution, it would be advisable to include the ladder network referred to in H.l and re-calculate the results. If this does not improve the situation significantly then two main improvements should be considered. Firstly use a neutral earth resistor at the source to restrict the earth return current to between 50 and 100 amps. This will directly reduce Em to values below Etouchso- Secondly reduce the fault clearing time L from 0.5 to... 

Cable theory Representation of a cylindrical fiber as two parallel rows of resistors (one each for the intracellular and extracellular spaces) connected in a ladder network by a parallel combination of resistors and capacitors (the cell membrane). 

KKTs are tools brought to network theory by the work of Kramers (1926) and Kronig (1929) on X-ray optics. Just as the reciprocity theorem, they are purely mathematical rules of general validity in any passive, linear, reciprocal network of a minimum phase shift type. By minimum-phase networks, we mean ladder networks that do not have poles in the right half plane of the Wessel diagram. A ladder network is of minimum phase type a bridge where signal can come from more than one ladder is not necessarily of the minimum-phase type. The transforms are only possible when the functions are finite-valued at all frequencies. With impedance Z = R- -jX the transforms are ... 

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...

The simplest method of taking into account the distribution of pores of different sizes is to use the transmission line ladder network (Fig. 9.4,9.14,9.17, or 9.19) and use different values for the parameters ri, r, and Ci or interfacial impedances z-, and calculate the total admittance by the addition of the admittances of the small pore elements. Such a method was used, for example, by Macdonald et al. [446,447] and Pyun et al. [448]. Although such a model can be used to simulate impedance spectra assuming changes in parameters with the position in the pore, it is difficult to obtain the pore parameters from the experimental spectra. 

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... 

Porous Electrode. Most battery electrodes comprise an open stmcture consisting of small particles compressed together, as shown in Figure 4.5.6. This structure does not have well-defined pores (such as cylindrical) but rather an irregular network of interconnected space between particles filled with electrolyte. The absence of well-defined pores complicates ab initio deduction of electrolyte impedance however, the frequency dependence of the impedance of porous materials is well described by the ladder-network approach originally proposed by de Levie [1963] for cylindrical pores. See also Section 2.1.6 for treatment of various geometries of porous electrodes. 

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 4.34 Passive LP filters (a) lossless L-C ladder network, odd order, (b) lossless L-C ladder network, even order. 

The highest degree term in the denominator of a mapped BP or BS transfer function is twice that of the original LP transfer function. This means that the s-domain transfer function must have twice as many poles, and the L-C ladder network must correspondingly have twice as many elements (Fig. 4.38). 

We first examine an operational simulation. Consider the doubly terminated ladder network shown in Fig. 7.122. This network represents a low-pass filter that has been frequency denormalized (frequency scaled up to the frequencies of interest). Interior node voltage V) and branch currents D and I2 have been identihed in the hgure. The hrst step in obtaining the operational simulation of this network is to write the network equations using Kirchhoff s and Ohm s laws. The equations are... 

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