Network Circuit Analysis

To construct an equivalent circuit of a complicated electrode process (e.g., a porous electrode) and calculate its impedance, more knowledge about the network circuit may be necessary. In this section, we will spend some time discussing network circuit analysis. 

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The method of approach will be akin to that used in the analysis of electrical networks where current to and from nodes and along individual circuit branches can be calculated. In fact, a principal purpose of this chapter will be to introduce the concepts of electrical network circuit analysis to the study of diffusion mechanisms. The physical picture to which this discussion will relate throughout the chapter is that for self-diffusion in a crystallogra-phically perfect, homogeneous material. Thus, no attention will be given to grain boundary or surface diffusion. 

The fundamental laws for circuit analysis are Ohm s law and Kirchhoff s laws. Ohm s law, described above, can be used to find the current, voltage, and power associated with a resistor. However, in some cases Ohm s law by itself cannot analyze the circuit. Analytical solutions for most electric networks need to combine Ohm s law and Kirchhoff s laws, the latter being also known as Kirchhoff s current law (KCL) and Kirchhoff s voltage law (KVL). 

The aim of network analysis is the investigation of the amplitude and phase response of a two- or four-port network. Impedance analysis determines the complex impedance or admittance of a device. This method is appropriate for quartz resonators in order to obtain more complete information than is conceivable by merely considering the shift of the resonance frequency. The method especially allows the determination of the equivalent circuit elements (BVD) presented in Fig. 8. Actually many commercial instriunents directly provide this information. Determination of the physical parameters, or their effective values, for accurate modeling of the sensor behavior based on Eq. 5 requires mathematical procedures which fit the calculated curves (e.g., with Eq. 2) to the experimentally measured values. It is recommended to include an external capacitance parallel to Co to accoimt for uncompensated para-... 

There are several advantages in adopting this new viewpoint of a reaction network for its analysis, visualization and reduction. The main advantage is that within this representation a reaction network becomes analogous to a general linear electrical circuit network, so that the procedures of electric circuit analysis are applicable. 

Table 39.2 lists the pressure-volume relationships for various geometries the fluidic capacitance is found simply by differentiating with respect to pressure. For small deformations, volume varies linearly with applied pressure, such that the capacitance is not a function of the pressure it merely defines the proportionality between increases in pressure and increases in stored mass. For such cases, the fluid circuit analysis is linear, because flow rate and pressure drops are related via linear expressions. For large deformations (i.e., the membrane limit), the fluidic capacitance is a function of the pressure this implies that the fluidic circuit behavior will be nonlinear. Obviously, once the fluidic resistance, capacitance and inductance have been identified via geometry (and material properties), complicated networks can be analyzed using commercially available circuit analysis software such as SPICE [42]. 

Sneak circuit analysis is usually inductive and can be very difficult to perform without the software to aid in producing network trees and other graphics. Much of this software is proprietary, and a large portion of the sneak circuit analysis work done to date has been accomplished by large aerospace and weapons contractors (notably Boeing and General Dynamics). 

Electrical network finite-difference models for study of various phenomena occurring in solid-state devices, circuits, and systems have been widely reported in Hterature (Ellison 1987 Fukuoka and Ishizuka, 1984 Riemer, 1990). One of the advantages of such a technique is a simple physical interpretation of the phenomena in question in terms of electrical signals and parameters existing in the network/circuit model (see Fig. 11.45). For all but very simple cases, the equivalent circuits are sufficiently complex that computer solution is required. It is important to note, however, that once the equivalent circuit is established the analysis can readily be accompHshed by existing network analysis programs, such as SPICE (SPICE2G User s Manual). 

In another type of measurement, the parallel between mechanical and electrical networks can be exploited by using variable capacitors and resistors to balance the impedance of the transducer circuit. These electrical measurements readily lend themselves to computer interfacing for data acquisition and analysis. 

In order to assess the effect of Cp on the conductance measurements, the analysis of the parallel RC network of Figure 8.9a may be carried out in a manner analogous to the series network discussed previously. The situation differs from the series circuit in that in this case, the voltage is the same across R and C, and the currents ic and iR in Cp and R are different. The instantaneous currents are given by... 

Immittance — In alternating current (AC) measurements, the term immittance denotes the electric -> impedance and/or the electric admittance of any network of passive and active elements such as the resistors, capacitors, inductors, constant phase elements, transistors, etc. In electrochemical impedance spectroscopy, which utilizes equivalent electrical circuits to simulate the frequency dependence of a given elec-trodic process or electrical double-layer charging, the immittance analysis is applied. 

The immittance analysis can be performed using different kinds of plots, including complex plane plots of X vs. R for impedance and B vs. G for admittance. These plots can also be denoted as Z" vs. Z and Y" vs. Y, or Im(Z) vs. Rc(Z), and Im( Y) vs. Re( Y). Another type of general analysis of immittance is based on network analysis utilizing logarithmic Bode plots of impedance or admittance modulus vs. frequency (e.g., log Y vs. logo)) and phase shift vs. frequency ( vs. log co). Other dependencies taking into account specific equivalent circuit behavior, for instance, due to diffusion of reactants in solution, film formation, or electrode porosity are considered in - electrochemical impedance spectroscopy. Refs. [i] Macdonald JR (1987) Impedance spectroscopy. Wiley, New York [ii] Jurczakowski R, Hitz C, Lasia A (2004) J Electroanal Chem 572 355... 

In order to understand electrochemical impedance spectroscopy (EIS), we first need to learn and understand the principles of electronics. In this chapter, we will introduce the basic electric circuit theories, including the behaviours of circuit elements in direct current (DC) and alternating current (AC) circuits, complex algebra, electrical impedance, as well as network analysis. These electric circuit theories lay a solid foundation for understanding and practising EIS measurements and data analysis. 

If bulk recombination is important in the depletion layer, then we cannot separate hole and electron flows in the above manner and the Zr, / scp network collapses to a frequency-independent resistor I D, as shown in Fig. 100. In this figure IFis a Warburg impedance for the hole current. This is too complex, as it stands, for analysis and a simpler case can be derived if Css is dominant and the frequency range is such that W can also be neglected. Under these circumstances, I D, Raan and 7 ssp further collapse to a simple resistor Rr, leading to the equivalent circuit shown in Fig. 101, which has been applied to p-GaAs under illumination and n-GaAs under hole injection. 

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