Equivalent circuit series resistance

FIGURE 12.12 Equivalent circuits with resistance and capacitance in series (a) and in parallel b). 

A model for the ac response of real electrodes is the simple electric equivalent circuit consisting of a resistance R and capacitance Q conneeted in series (Fig. 12.12a). It follows from the rules for ac circuits that for this combination... 

FIG. 7 Simplified equivalent circuit for charge-transfer processes at externally biased ITIES. The parallel arrangement of double layer capacitance (Cdi), impedance of base electrolyte transfer (Zj,) and electron-transfer impedance (Zf) is coupled in series with the uncompensated resistance (R ) between the reference electrodes. (Reprinted from Ref. 74 with permission from Elsevier Science.)... 

The impedance data have been usually interpreted in terms of the Randles-type equivalent circuit, which consists of the parallel combination of the capacitance Zq of the ITIES and the faradaic impedances of the charge transfer reactions, with the solution resistance in series [15], cf. Fig. 6. While this is a convenient model in many cases, its limitations have to be always considered. First, it is necessary to justify the validity of the basic model assumption that the charging and faradaic currents are additive. Second, the conditions have to be analyzed, under which the measured impedance of the electrochemical cell can represent the impedance of the ITIES. 

Dielectric losses arise from the direct capacitive coupling of the coil and the sample. Areas of high dielectric loss are associated with the presence of axial electric fields, which exist half way along the length of the solenoid, for example. Dielectric losses can be modeled by the circuit given in Figure 2.5.3. The other major noise source arises from the coil itself, in the form of an equivalent series resistance, Rcoii. Exact calculations of noise in solenoidal coils at high frequencies and small diameters are complex, and involve considerations of the proximity and skin depth effects [23],... 

Variations of resistance with frequency can also be caused by electrode polarization. A conductance cell can be represented in a simplified way as resistance and capacitance in series, the latter being the double layer capacitance at the electrodes. Only if this capacitance is sufficiently large will the measured resistance be independent of frequency. To accomplish this, electrodes are often covered with platinum black 2>. This is generally unsuitable in nonaqueous solvent studies because of possible catalysis of chemical reactions and because of adsorption problems encountered with dilute solutions required for useful data. The equivalent circuit for a conductance cell is also complicated by impedances due to faradaic processes and the geometric capacity of the cell 2>3( . 

Fig. 10.1 Equivalent circuits used to represent the semiconductor-electrolyte interface, (a) A more complete approach taking into account the series resistance (Ry), the depletion layer (Csc, Rsc), an oxide surface film...

To evaluate the magnitude of capacitive currents in an electrochemical experiment, one can consider the equivalent circuit of an electrochemical cell. As illustrated in Figure 24, in a simple description this is composed by a capacitor of capacitance C, representing the electrode/solution double layer, placed in series with a resistance R, representing the solution resistance. 

Figure 5.10 is EIS of marmatite electrode in O.lmol/L KNO3 solution with different pH modifiers at open circuit potential. This EIS is very complicated. Simple equivalent circuit can be treated as the series of electrochemical reaction resistance R with the capacitance impedance Q == (nFr )/(icR ) resulting fi-om adsorbing action, and then parallel with the capacitance Ca of double electric... 

In practice, poor charge mobility, energetic disorder, carrier trapping, and physical aberrations comphcate device characterization. The effects of these nonidealities are often modeled according to an equivalent circuit shown in Fig. 12. Incorporating all specific series resistive elements as R, and all specific parallel resistances as R, one obtains the expression... 

The actual SPICE model of this circuit is shown in Fig. 10.2. Note that there are two resistors in series with each of the capacitors, Rl and R2. These resistors model the approximate equivalent series resistance (ESR) of the tantalum capacitors in the circuit at the switching frequency. 

In drawing an appropriate equivalent circuit, it is clear that the resistance of the solution should be placed first in the intended diagram, but how should the capacitative impedance be coupled with that of the interfacial resistance One simple test decides this issue. We know that electrochemical interfaces pass both dc and ac. It was seen in Eq. (7.103) that for a series arrangement of a capacitor and a resistor, the net resistance is infinite for = 0, i.e., for dc. Our circuit must therefore have its capacitance and resistance in parallel for under these circumstances, for = 0, a direct current can indeed pass the impedance has become entirely resistive.51... 

The equivalent circuit of an electrochemical cell is shown in Fig. 5.6. It can be represented by a capacitive divider consisting of Cw and CAux connected in series. Figure out how the voltage V and charge Q are distributed across this divider when the resistances are (a) finite (b) infinite. 

To understand the electrical behaviour of the LAPS-based measurement, the LAPS set-up can be represented by an electrical equivalent circuit (see Fig. 5.2). Vbias represents the voltage source to apply the dc voltage to the LAPS structure. Re is a simple presentation of the reference electrode and the electrolyte resistance followed by a interface capacitance Cinterface (this complex capacitance can be further simulated by different proposed models as they are described, e.g., in Refs. [2,21,22]). In series to the interface capacitance, the insulator capacitance Cj will summarise the capacitances of all insulating layers of the LAPS device. The electrical current due to the photogeneration of electron-hole pairs can be modelled as current source Ip in parallel to the... 

T. N. Rudakov, J. H. Flexman, V. T. Mikhaltsevitch, P. A. Hayes, W. P. Chisholm, C. N. Aitken, and J. H. Feldman, Nuclear Quadrupole Resonance Scanner for Detecting Explosives in Mail, Has Low Equivalent Series Resistance Switch to Switch Capacitance of Tunable circuit Enclosed by Electromagnetic Shield, US Patent Application No. US2006012366-A1 (2006). 

A basic electric equivalent circuit to describe an EDLC is presented in Figures 1.22a and b, which shows the Nyquist (Figure 1.22b) plot of an ideal capacitor C, in series with a resistance... 

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. 

FIGURE 11.9 Basic capacitor electrical equivalent circuit comprising a capacitance, a series inductance, a series resistance, and a parallel resistance. This simple model can fit a DLC behavior in first approximation for a given frequency. 

The capacitance and the series resistance have values which are not constant over the frequency spectrum. The performances may be determined with an impedance spectrum analyzer [70], To take into account the voltage, the temperature, and the frequency dependencies, a simple equivalent electrical circuit has been developed (Figure 11.10). It is a combination of de Levie frequency model and Zubieta voltage model with the addition of a function to consider the temperature dependency. 

FIGURE 11.10 DLC equivalent circuit for capacitance and series resistance dependencies as a function of frequency, voltage, and temperature. 

Consider the situation under potentiostatic conditions. Here, the potential control takes care that the sum of the potential drop across the double layer, DL, and through the electrolyte up to the position of the RE (and possibly additional external series resistances) is constant, i.e. that U = DL + I Rn or / = (U - DL)/Rn. Rn is the sum of the uncompensated cell resistance and possible external resistances and I the total current through the cell. Hence, a perturbation of a state on the NDR branch towards larger values of Dl causes, on the one hand, a decrease of the faradaic current If, and, on the other hand, a decrease of the current through the electrolyte, I. The charge balance through the cell, which can be readily obtained from the general equivalent circuit of an electrochemical cell (Fig. 8), tells us whether the fluctuation is enhanced or decays ... 

Fig. 8. General equivalent circuit of an electrochemical cell. C double layer capacitance qSDL potential drop across the double layer Zp faradaic impedance Rij series resistance (comprising the uncompensated ohmic cell resistance and all external resistances). V is a potentiostatically fixed voltage drop. (It differs from the potentiostatically applied voltage by the constant potential drop across the RE see footnote 3).

For practical solar cells, the ideal equivalent circuit will be modified to include the series resistance from Ohmic loss in the two electrodes and the shunt resistance from leakage currents (Fig. 5.36a). The diode current for a realistic setup is then given by... 

The experimental impedance is always obtained as if it were the result of a resistance and capacitance in series. We have already seen in (11.20) and (11.21) the relation between an RC series combination and the Rct + zw combination. It can be shown for the full Randles equivalent circuit for this simple charge transfer reaction, see Fig. 11.4, on separating the in-phase and out-of-phase components of the impedance, that... 

Fig. 3. Equivalent circuit of amorphous silicon solar cell showing voltage source, photoconductive resistance, and fixed series resistance.

To estimate the effects of electrode polarization, the equivalent circuit of Fig. 16 can be used. It shows a blocking layer capacitance Cb (actually the series combination of two identical capacitors — one at each electrode interface) together with a parallel R — C circuit representing the bulk material. The separate thicknesses of the blocking layer 2tb and the total specimen length, L, must be used to construct the capacitances and resistance. The blocking layer capacitance Cb has the value... 

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. 

Transmission line — This term is related to a more general concept of electric -> equivalent circuits used frequently for interpretation of experimental data for complex impedance spectra (-> electrochemical impedance spectroscopy). While the complex -> impedance, Z, at a fixed frequency can always by obtained as a series or parallel combinations of two basic elements, a resistance and a capacitance, it is a much more compli-... 

EIS data analysis is commonly carried out by fitting it to an equivalent electric circuit model. An equivalent circuit model is a combination of resistances, capacitances, and/or inductances, as well as a few specialized electrochemical elements (such as Warburg diffusion elements and constant phase elements), which produces the same response as the electrochemical system does when the same excitation signal is imposed. Equivalent circuit models can be partially or completely empirical. In the model, each circuit component comes from a physical process in the electrochemical cell and has a characteristic impedance behaviour. The shape of the model s impedance spectrum is controlled by the style of electrical elements in the model and the interconnections between them (series or parallel combinations). The size of each feature in the spectrum is controlled by the circuit elements parameters. 

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