Electrical Circuits Containing Resistances

First, electrical circuits containing resistances only are presented, followed by circuits containing R, C, and L elements in transient and ac conditions. To understand the concept of impedance, the notions of Laplace and Fourier transforms are presented and must be understood thoroughly. In this chapter, impedance plots are also presented, along with several examples for various circuits. Next, methods for determining impedances, including fast Fourier transform-based techniques, are discussed. 

If the real part of the impedance Z is plotted on the x-axis and the imaginary part is plotted on the y-axis of a chart, a "Nyquist plot" is obtained. For an electric circuit containing resistance and a capacitor in parallel, the impedance Z in Nyquist plot is shown in Figure 8.6. Note that in this plot, the... 

A dilTerent approach is used in UTE C20-453 [120], where specimens are burned in an enclo.sed cabinet and the change in resistance of a copper wire or other electrical circuits contained in the cabinet measured. Tests of this type appear to be more realistic, since they directly assess the corrosive effects of fire gases on actual components, but the acid gas type of test described above is simpler. 

Electrical conductivity cell. Impedance is a measure of the total opposition to the flow of a sinusoidal alternating current in a circuit containing resistance, inductance, and capacitance. Inductance and capacitance together are called the reactive part of the circuit. The changes in impedance that occur in a microbial culture can be measured by placing two metal electrodes into the culture medium and introducing an alternating potential into the circuit. 

Until now, only circuits containing resistances and capacitances have been discussed. Inductive effects in electrical circuits appear when alternative electrical current flow creates a magnetic field interacting with the flowing current of course, in a strait wire the inductance is very small, but in looped wires or a coil it becomes larger. The inductive effects always lead to positive imaginary impedances, as will be shown in what follows. Let us first consider the circuit in Fig. 2.40, which contains inductance L in series with resistance Rq and a nested coimection of two (RQ circuits, i.e., LRo(Ci(Ri(R2C2))). The complex plane and Bode plots for this circuit without inductance were presented in Fig. 2.39. 

In Chap. 2 we saw the responses of electrical circuits containing the elements R, C, and L. Because these are linear elements, their impedance is independent of the ac amplitude used. However, in electrochemical systems, we do not have such elements we have solution-electrode interfaces, redox species, adsorption, etc. In this and the following chapters, we will learn how to express the electrochemical interfaces and reactions in terms of equations that, in particular cases, can be represented by the electrical equivalent circuits. Of comse, such circuits are only the electrical representations of physicochemical phenomena, and electrical elements such as resistance, capacitance, or inductance do not exist physically in cells. However, such a presentation is useful and helps in our understanding of the physicochemical phenomena taking place in electrochemical cells. Before presenting the case of electrochemical reactions, the case of an ideally polarizable electrode will be presented. 

For time t > % this current is zero because the carrier would have reached the substrate. The current I(t) can be detected via the voltage it induces in the external circuit. Shortly, the equivalent electrical circuit of the XTOF experiment under the small-signal condition contains the coupling capacitance Cl (the sum of the amplifier and the parasitic capacitances) and f L ( l is the load resistance). The total current is the sum of the conduction current due to the drift of photogenerated charge and the displacement current and is equal to zero (for further details, see Ref [15]),... 

Any electrochemical cell can be represented in terms of an equivalent electrical circuit that comprises a combination of resistances and capacitances (inductances only for very high frequencies). This circuit should contain at the very least components to represent ... 

The second meaning of the word circuit is related to electrochemical impedance spectroscopy. A key point in this spectroscopy is the fact that any -> electrochemical cell can be represented by an equivalent electrical circuit that consists of electronic (resistances, capacitances, and inductances) and mathematical components. The equivalent circuit is a model that more or less correctly reflects the reality of the cell examined. At minimum, the equivalent circuit should contain a capacitor of - capacity Ca representing the -> double layer, the - impedance of the faradaic process Zf, and the uncompensated - resistance Ru (see -> IRU potential drop). The electronic components in the equivalent circuit can be arranged in series (series circuit) and parallel (parallel circuit). An equivalent circuit representing an electrochemical - half-cell or an -> electrode and an uncomplicated electrode process (-> Randles circuit) is shown below. Ic and If in the figure are the -> capacitive current and the -+ faradaic current, respectively. 

An electrical measuring instrument contains electrical circuits incorporating capacitance, inductance, and resistance. In the absence of resistance, a circuit tends to oscillate with a definite frequency /when disturbed. For optimum performance an amount of resistance is incorporated that is barely sufficient to damp the oscillations resulting from transient inputs the circuit is then said to be critically damped. For a critically damped circuit it can be shown that the root-mean-square (rms) fluctuations in voltage V and in current /are given by... 

It is possible to represent the entire electrochemical system including the instrumentation (potentiostat, etc.) as a single electrical circuit. The solution is usually spatially discretized into a network of resistance elements (see for example Coles et al., 1996). Double-layer charging can also be incorporated into these models by defining each element to contain a capacitor as well as a resistor. 

Sah and coworkers have developed a quasi-analytic calculation that can be expressed as a detailed equivalent electrical circuit. " This development is summarized by Jansen et al. and used to justify the application of the simplified equivalent circuit shown in Figure 12.8 to the analysis of the impedance response of semiconductors containing deep-level electronic states. This circuit was used to analyze the impedance data presented in Section 18.2 (see, e.g.. Figure 18.4). In Figure 12.8, C is the space-charge capacitance, Rn is a resistance that accounts for a small but finite leakage current, " and the parameters Ri...Rk and Ci... Q are attributed to the response of discrete deep-level energy states. 

For the purposes of this discussion, we need only be concerned with electrical circuits that contain capacitances, C, and resistances, R. The resistance is the dissipative element, formally analogous to the dashpot in the mechanical model case. It is defined by Ohm s law ... 

The AT-cut quartz resonator can be modeled mechanically as a body containing mass, compliance, and resistance. Figurel-a) shows the mechanical vibration motion depicting the vibration of the quartz resonator. An electrical network called an equivalent electrical circuit consisting of inductive, capacitive and resistive components can represent this mechanical model. Figure... 

An equivalent electric circuit in such a pore was modeled by a transmission RC circuit containing similar resistances of electrolyte within a pore with branching electric capacitances C of EDL of similar values within a pore (Fig. 27.1 lb). At high frequencies, the phase angle is 45° and the intercept at the Z axis is equal to solution resistance outside the pore. At the frequency of - 0, extrapolation of the dependence to the Z axis yields the sum of ionic resistance inside the pore structure and outer serial resistance. 

Any electrochemical cell can be represented in terms of an equivalent electrical circuit that comprises a combination of resistances, capacitances or inductances as well as mathematical components. At least the circuit should contain the doublelayer capacity, the impedance of the faradaic or non-faradaic process and the high-frequency resistance. The equivalent circuit has the character of a model, which more or less precisely reflects the reality. The equivalent circuit should not involve too many elements because then the standard errors of the corresponding parameters become too large (see Sect. II.5.7), and the model considered has to be assessed as not determined, i.e. it is not valid. 

To understand the impedance of electrochemical objects, it is necessary to understand the behavior of simple electrical circuits, first in steady state, then in transient conditions. Such circuits contain simple linear electrical elements resistance, capacitance, and inductance. Then the cmicept of electrical impedance will be introduced. It demands an understanding of the Laplace and Fourier transforms, which will also be presented. To understand impedance, it is necessary to thoroughly understand the complex plane and Bode plots, which will be presented for a few typical connections of the electrical elements. They can be computed using Excel, Maple, Mathematica, and specialized programs such as ZView. Several examples and exercises will be included. 

Electrical circuits may contain three passive elements resistors, capacitors, and inductors. The behavior of the capacitance and inductance is different from that of the resistance. A constant current caruiot flow through a capacitance, but an electrical charge can accumulate in it, and it is different at each voltage applied. The fundamental relation between charge and voltage is given as... 

Usually, quite compact layers are obtained. The simplest electrical equivalent model represents the solution resistance in series with the capacitance of a SAM, CsAM (Fig- 12.2a). More detailed analysis reveals that the layers are rarely purely capacitive and their capacitance is in parallel with their resistance, Rsam. leading to a circuit R iCsam Rsam)- Moreover, a diffuse double layer exists at the SAM/solution interface [485,486]. In such a case, the electrical equivalent circuit contains a diffuse-layer capacitance, C, in parallel with the resistance, (Fig. 12.2b). 

In order to calculate equivalent impedance for a wide fiaquency range, the equivalent electrical circuit for a ceramic material can be introduced as an impedance containing two capacitance C and Cp, an inductance L and a resistance R. 

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