RC circuit

According to this model, the SEI is made of ordered or disordered crystals that are thermodynamically stable with respect to lithium. The grain boundaries (parallel to the current lines) of these crystals make a significant contribution to the conduction of ions in the SEI [1, 2], It was suggested that the equivalent circuit for the SEI consists of three parallel RC circuits in series combination (Fig. 12). Later, Thevenin and Muller [29] suggested several modifications to the SEI model ... 

Alternately, for potential-step experiments (e.g., chronoamperometry, see Section 3-1), the charging current is die same as that obtained when a potential step is applied to a series RC circuit ... 

Fig. 12. Parallel RC circuit for the automatic correction of thermograms. Reprinted from (40) with permission.

The distribution of potential in TC is practically the same as that near the flat surface if the electrolyte concentration is about 1 mol/1 [2], So the discharge of TC may be considered as that of a double electric layer formed at the flat electrode surface/electrolyte solution interface, and hence, an equivalent circuit for the TC discharge may be presented as an RC circuit, where C is the double layer capacitance and R is the electrolyte resistance. 

Figure 18b.5b shows the equivalent circuit of the metal solution interface composed of C(i and the solution resistance Rs. When a voltage pulse, E, is applied across such a Rc circuit, the transient current flow... 

The insertion of an RC circuit in the feedback electronics right before the high-voltage amplifier for the z piezo has some other advantages. First, much of the high-frequency noise is efficiently filtered out. Second, it facilitates the realization of the electronics for spectroscopic study, which we will discuss in the following section. 

In this section we will look at the response of an RC circuit to a pulse input. We will work with the circuit below ... 

The current from the anode of the photomultiplier is fed to an RC circuit which effectively acts as an integrator, summing the current pulses and tending to average out the noise. RC times of 1 sec are usually satisfactory for most studies. However, for very weak signals, RC of longer times can be used. 

Exercise. In an RC circuit with non-Ohmic resistance R(V) the Fokker-Planck equation for the voltage V would be, according to the above phenomenological argument,... 

Consider an RC-circuit with linear resistance R. The charge Q on the condenser obeys macroscopically an equation with damping due to R, and must therefore be supplemented with a noise term,... 

The mechanical programmers were replaced by electronic programmers. These depended mainly on different solid state time-logic RC circuits to increase the power to the column oven to supply the desired heating rates. This allowed multilinear pro-... 

By the use of frequency filters (RG circuits) which cut off the high frequency noise from a low frequency signal. Care must be taken to avoid distortion of the signal. An RC circuit as shown below is a low pass filter of time constant t = RG. This gives a rough value of the cut-off frequency. 

At relatively low applied frequencies, a conductance cell may be represented as the double-layer capacitance Cs in series with the solution resistance R, as shown in Figure 8.8a. When a sinusoidal voltage es is applied to the series RC circuit, the instantaneous current i is the same in every part of the circuit and is given by... 

Figure 8.8 Series RC circuit (a) circuit (b) current-voltage relationships (c) frequency dependence of impedance Z and phase angle .

From the preceding expression and the expressions for ec and eR, we see that as the signal frequency is increased, Xc decreases, Z approaches R, and G approaches 1/Z. The potential across Cs also decreases gradually, and the phase angle 4> between i and es approaches zero, as is illustrated in the plot of versus co in Figure 8.8c. For an RC circuit, there is a frequency for which Xc = R. The reciprocal of this frequency is called the time constant x. When R = 1/coC, 1/co = RC. Thus x = RC, in seconds. 

As stated in Sect. 6.4.1, it has been assumed that the measured experimental currents and converted charges when a potential Ep is applied can be considered as the sum of a pure faradaic contribution, given by Eqs. (6.130) and (6.131), and a non-faradaic one, /pnf and Qpnl. In order to evaluate the impact of these non-faradaic contributions on the total response, analytical expressions have been obtained. If it is assumed that initially the monolayer is at an open circuit potential, rest, and then a sequence of potential pulses , E2, -,Ep is applied, the expression for the non-faradaic charge Qp.nf can be deduced from the analogy between the solution-monolayer interface and an RC circuit [53] (shown in Fig. 6.24), so the following differential equation must be solved ... 

The power generation of SC-SOFC is dependent on the resistance of the materials. The electrolyte itself, the chemical reactions, and the overpotential contribute to the impedance, which is measured with a load of half the short circuit current applied to the cell. Figure 3 shows the impedance spectra of a particular cell, fitted to an equivalent resistor/capacitor (RC) circuit. Usually, R1 is considered to be the electrolyte resistance with R2 and R3 as the overpotential of the electrodes. The inductance of the cables and the relaxation frequency of R2 and C2 tended to introduce error into the measurement of Rl. Therefore, R1 is usually measured together with R2 as R1 + R2 [31], Some cells may be significantly affected by the electrolyte resistance, which depends on thickness. 

First of all, membrane capacitance of nerves has a negative frequency dependence below 300 Hz, namely capacitance increases with frequency in this region. This is a behavior which cannot be explained by a simple RC circuit and indicates the presence... 

Fig. 2.55 (a) Series-parallel RC circuit which could, depending on component values, produce the frequency response shown in (b). 

Rs (Figure 1.22a). The double layer capacitance is represented by the capacitance C, and Rs is the series resistance of the EDLC, also named the equivalent series resistance (ESR). This series resistance shows the nonideal behavior of the system. This resistance is the sum of various ohmic contributions that can be found in the system, such as the electrolyte resistance (ionic contribution), the contact resistance (between the carbon particles, at the current collector/carbon film interface), and the intrinsic resistance of the components (current collectors and carbon). Since the resistivity of the current collectors is low when A1 foils or grids are used, it is generally admitted that the main important contribution to the ESR is the electrolyte resistance (in the bulk and in the porosity of the electrode) and to a smaller extent the current collector/active film contact impedance [25,26], The Nyquist plot related to this simple RC circuit presented in Figure 1.22b shows a vertical line parallel to the imaginary axis. 

Basically, the impedance behavior of a porous electrode cannot be described by using only one RC circuit, corresponding to a single time constant RC. In fact, a porous electrode can be described as a succession of series/parallel RC components, when starting from the outer interface in contact with the bulk electrolyte solution, toward the inner distribution of pore channels and pore surfaces [4], This series of RC components leads to different time constant RC that can be seen as the electrical response of the double layer charging in the depth of the electrode. Armed with this evidence, De Levie [27] proposed in 1963 a (simplified) schematic model of a porous electrode (Figure 1.24a) and its related equivalent circuit deduced from the model (Figure 1.24b). 

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