The Equivalent Circuit Representation

Usually one considers only the part of the circuit inside the dashed line, since the experiment is set up in such a way that only one of the electrodes is studied at a time. 

The combination of the double-layer capacitance Qi and the Faradaic resistance jRp (also referred to as the charge-transfer resistance, i ct) represents the interface. How do we know that Cai and Rp must be put in a parallel rather than in a series combination Simply because we can observe a steady direct current flowing when 

The equivalent circuit just described also makes it dear why conductivity measurements are routinely done by applying a small AC signal. If the appropriate frequency is chosen, the capacitive impedance assodated with Cji can be made negligible compared to the Faradaic resistance Rp, which is thus effectively shorted, leaving the solution resistance R as the only measured quantity. 

The difference between polarizable and nonpolarizable interfaces can be easily understood in terms of this equivalent circuit. A high value of Rp is associated with a polarizable interface, whereas a low value of Rp represents a non-polarizable interface. 

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Apart from these practicalities, there is an important new concept contained in the equivalent circuit representation, which is the load impedance, Zi. The load impedance in this context is the ratio of the stress, a, and the speed, ii, at the crystal surface. The load impedance is normaUzed to area (unlike the mechanical impedance). 

In the simple-spring model, the crystal is in contact with an immobile object. The model can be extended to cover situations where the object takes part in the oscillation to some extent. A typical object of this kind would be a small (< 10 im) sphere [40]. Figure 2c depicts the physical situation and the equivalent circuit representation. Note that the motion occurs into the lateral direction even though the spring is drawn vertically. In the following, we assume a spring constant independent of frequency, labeled its. From Fig. 2c, we infer the load to be ... 

The input impedance of an antenna is defined as the ratio between the voltage and the current at its terminals, when the antenna is in transmit mode. The equivalent circuit representation is displayed in Fig. 26.2. 

In another study of DMFC anodes, shovm in Figure 16.10, the complex-plane impedance plots were studied as a function of the current density applied. The diameters of the semicircles were found to decrease with increasing current density, as expected, but the new feature observed is an inductive branch of the curves. This can be modeled, of course, by adding an inductive element to the equivalent circuit representation, in series with the Faradaic resistance, but the physical origin of this added circuit element is still open for debate. There is a tendency to associate it with sluggish adsorption of CO, formed as an intermediate in the oxidation of methanol. However, unlike the adsorption pseudocapacitance, which is well understood (cf Section 11.2), there is no theory for the dependence of the pseudoinductance on potential, coverage or any other measured parameter. 

Figure 6.7 Complex impedance of a polycrystalline ceramic sample (a) representation of the equivalent circuit of a component (b) the impedance spectrum of the equivalent circuit in (a) (c) the impedance spectrum of a typical ceramic sample. Each semicircular arc represents one component with an equivalent circuit as in (a) that at the highest frequency corresponds to the repose of the bulk, that at middle frequencies to the grain boundary response, and that at lowest frequencies to the electrodes.

Figure 8.12 (a) Nyquist plot obtained for the all-solid-state cell, ITOAVO3/PEO-H3PO4/ ITO(H) at 8°C, with the electrolyte being unplasticized. The WO3 layer was 0.3 pm in thickness (as gauged during vacuum evaporation with a thin-film monitor), while the electrolyte thickness was 0.24 mm (achieved by using 0.3 mm spacers of inert plastic placed between the two ITO electrodes), (b) Schematic representation of the equivalent circuit for this cell. 

While a good equivalent-circuit representation of the transport processes in a fuel cell can lead to an increased understanding, it is not as good as taking a 1-D sandwich model and taking it into the frequency domain. These models typically analyze the cathode side of the fuel cell. °2.3i3 3i4 pj g j ost comprehensive is probably that of Springer et al. °2 The use of impedance models allows for the calculation of parameters, like gas-phase tortuosity, which cannot be determined easily by other means, and can also allow for the separation of diffusion and migra-... 

Figure 17. Equivalent circuit representation of the injection/recombination process. (Reproduced with permission from Ref. [81].)...

In corrosion systems, a salt film may cover an electrode that is itself covered by a porous oxide layer. If two different layers are superimposed, the geometrical analysis shows that the equivalent circuit corresponds to that described in Section 9.3.1 with an additional series Rti a. circuit to take into account the effect of the second porous layer. The circuit shown in Figure 9.5 is approximate because it assumes that the botmdary between the inner and outer layers can be considered to be an equipotential plane. This plane will, however, be influenced by the presence of pores. The circuit shown in Figure 9.5 will provide a good representation for systems with an outer layer that is much thicker than the inner layer and with an inner layer that has relatively few pores. 

Figure 3.40 Equivalent circuit representation of an electrode incorporating an adsorbed redox active species (see text for notation). The components responsible for the ER (electroreflectance) response are shown within the dotted frame, where VF is the interfacial admittance, and the subscript F refer to the current and potential associated with the faradaic or interfacial modulation.

FIGURE 10.8 Representation of the equivalent circuit used by Sinha and Munichandraiah (2008) to describe the EIS of LiMn2O4 electrode in 1 M LiAsF6/(ethylene carbonate + dimethyl carbonate) depicted in Figure 10.7. 

As shown in the previous section, the mechanical properties of a quartz crystal close to resonance frequency can be expressed by means of a motional impedance. To complete the equivalent circuit of a quartz crystal, the capacitance, Co, must be added in parallel to the motional impedance. It results in the Butterworth-Van Dyke (BVD) equivalent circuit of a quartz crystal, as shown again in Fig. 8 for an unloaded quartz crystal [32]. In this notation common in electronic Hterature, Is is the dynamic inductance and is imder-stood here as a representation of the oscillating mass of the quartz crystal. Cs is the dynamic capacitance and reflects the elasticity of the oscillating body. Rs is the dynamic resistance and returns friction of the quartz slice as well as all kinds of acoustic damping. 

Fig. 1 Equivalent circuit representation of the quartz crystal including a load. Piezoelectric stiffening (described by the element 42 in Fig. 13, Chap. 2 in this volume) was neglected. The sample is represented by the load Zl...

From an electronic standpoint, an electrochemical cell can be regarded as a network of impedances like those shown in the equivalent circuit of Figure 15.4.1a, where Z and Z k represent the interfacial impedances at the counter and working electrodes, and the solution resistance is divided into two fractions, R and R, depending on the position of the reference electrode s contact with the current path (see Section 1.3.4). This representation can be distilled further into that of Figure 15.4. IZ . 

Figure 4.12 Bode plot representation of the impedance of the equivalent circuit in Figure 4.10a.

Figure 5.9 Representation of the impedance spectrum of the equivalent circuit in Figure 5.8 for...

Figure 5.10 Representation of the impedance spectrum of the equivalent circuit in Figure 5.8 for when Warburg impedance is much larger than the charge transfer resistance = 1000 Mil, IZ I = 1 Mil s , Cj, = 100 nF, = 10 il. (a) Nyquist plot and (b) Bode plot.

To discuss the results, the sensor is represented as a lossy capacitor, with both the capacitance C and the resistance R depending on frequency (Fig. 4 the frequency dependence of the equivalent-circuit elements is a consequence of the distributed nature of the processes in the sensor, which cannot be modeled appropriately by only two lumped elements with frequency-independent element values.). That simplifies the recognition of even small changes in the impedance, as changes at low frequencies become easily visible in the representation of the resistance R(f) and changes at higher frequencies become even more visible in the representation of the capacitance C(f). 

Figure 20 shows the experimental results (solid circles) for hexanoyl chitosan (M of 2 X 10 g mol )-based electrolyte system. The Nyquist plot shows a depressed semicircle with its centre below the horizontal axis by an angle 20°. The equivalent circuit may take the form of a resistor connected in parallel with a CPE.The value of R obtained by graphical means was 226 kfl. Open squares semicircle was calculated withi = 226 kO and = 5.3 x 10 s after Equations (19) and (20). The intercept of the open squares semicircle on the Z axis gives = 220 kQ. Thus the equivalent circuit that has been used to simulate the Nyquist plot gives a hue representation of the material. 

The impedance plot shown in Figure 2.1a vs. or the Nyquist plot) corresponds to an electrochemical cell (electrode/NaCl solution/electrode) and the equivalent circuit consists of a resistance (R) in parallel with a capacitor (C), which is represented as RQ, while Figure 2.1b shows the variation of the phase angle 4) = arc tan(Zi g/Z,e ) with frequency (4) vs./), but other typical impedance representations correspond to the variation of Z, and -Zj g with frequency (Bode plots), as indicated in Figure 2.1c and 2.1d. This latter representation allows the determination of the interval of frequency associated with a given relaxation process, between KT and 10 Hz, with a maximum frequency around 2 x 10 Hz, for the NaCl solution... 

Fig. 6 Equivalent circuit representation of a typical impedance cell used in microbiology. The symbols R, and Q represent the interfacial resistances and capacitances, respectively, while Rm is the resistance of the medium separating the two electrodes. 

FIGURE 18.13 (a) Equivalent circuit representation of an internal short, (b) Simulation of power generation as a function of variation in short and cell resistance during internal short. The maximum power generated in the short area is when the cell resistance equals the internal short resistance. (For color version of this figure, the reader is referred to the online version of this book.)... 

The nonideal characteristics of a transformer include core and winding losses, presence of leakage fluxes, and finite permeability of the core. Hence, the actual model should include the physical representations of these nonideal characteristics. This is shown in Fig. 10.105(b), where the shunt magnetization and core loss components have been ignored for the sake of convenience. Such approximations are common in transformer analysis and only cause trivial inaccuracies in computation. Figure 10.106 shows the equivalent circuit of Fig. 10.105(b) with the secondary-side impedance referred to the primary side. 

Macroscopic Modeling of Porous Electrodes, Fig. 1 Simple equivalent-circuit representation of a porous electrode. The total current density, i, flows through the separator or membrane to the electrolyte phase (2) and then into the solid or electronic phase... 

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. 

FIGURE 1.84. Equivalent circuit representation of the elements X and Z in the Fletcher model. 

Figure 3-1 is an equivalent circuit representation of an operational amplifier. In this figure, the input voltages are represented by and V-. The input difference voltage IV is the difference between these two voltages that is, IV = - v. The power supply connections arc... 

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