Capacitance equivalent electrical circuit with

We found an equivalent electrical circuit that fits best the LixC6 electrode behavior at high frequency. The circuit consists of a resistor R in parallel with a constant phase element (CPE). The latter is defined with a pseudo-capacitance Q and a parameter a with 0< a <1 [6], The impedance of... 

Rational optimization of performance should be the main goal in development of any chemical sensor. In order to do that, we must have some quantitative tools of determination of key performance parameters. As we have seen already, for electrochemical sensors those parameters are the charge-transfer resistance and the double-layer capacitance. Particularly the former plays a critical role. Here we outline two approaches the Tafel plots, which are simple, inexpensive, but with limited applicability, and the Electrochemical Impedance Spectroscopy (EIS), based on the equivalent electrical circuit model, which is more universal, more accurate, and has a greater didactic value. 

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. 

Since the unloaded QCM is an electromechanical transducer, it can be described by the Butterworth-Van Dyke (BVD) equivalent electrical circuit represented in Fig. 12.3 (box) which is formed by a series RLC circuit in parallel with a static capacitance C0. The electrical equivalence to the mechanical model (mass, elastic response and friction losses of the quartz crystal) are represented by the inductance L, the capacitance C and the resistance, R connected in series. The static capacitance in parallel with the series motional RLC arm represents the electrical capacitance of the parallel plate capacitor formed by both metal electrodes that sandwich the thin quartz crystal plus the stray capacitance due to the connectors. However, it is not related with the piezoelectric effect but it influences the QCM resonant frequency. 

The a.c. impedance technique [33,34] is used to study the response of the specimen electrode to perturbations in potential. Electrochemical processes occur at finite rates and may thus be out of phase with the oscillating voltage. The frequency response of the electrode may then be represented by an equivalent electrical circuit consisting of capacitances, resistances, and inductors arranged in series and parallel. A simplified circuit is shown in Fig. 16 together with a Nyquist plot which expresses the impedance of the system as a vector quantity. The pattern of such plots indicates the type and magnitude of the components in the equivalent electrical network [35]. 

Figure 9.6 Equivalent electrical circuit for a electrode coated by two superimposed porous layers with the capacitance C(2 indicated by dashed lines to denote its experimental inaccessibility.

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. 

In EIS, the impedance of the system is measured by applying a small ac signal and by the frequency scanned (typically between 10-100 kHz and 1 Hz or less). Stable impedance spectra can be obtained with electrically charged redox markers in solution. The data can be fitted with an equivalent electrical circuit, where the most important components are the charge transfer resistance Ret and the double layer/biolayer capacitance Cdi. 

The EIS data was fitted with an equivalent electrical circuit in order to characterize the electrochemical properties of the composite. The experimental results and values obtained from equivalent circuit modeling have shown that both the charge transfer resistance and the double-layer capacitance decreased with the increase of the incorporated PANA into the composite structure. The impedance spectra for nanofibers may be described by the equivalent circuit of (R(Q(R)). ... 

Murran reported a position sensor based on capacitance sensing methods [5]. The equivalent electrical circuit model of a pair of electrodes (ground electrode and control electrode) with an electrolyte droplet is shown in Fig. 7a. The equivalent capacitance is given by... 

The electrical impedance of a pyroelectric detector is almost that of a pure capacitance. Hence an output signal only appears when the input radiation is changing. For maximum output the rate of change of the input radiation should be comparable with the electrical (RC) time constant of the element. Figure 3.10 is the equivalent electrical circuit of a pyroelectric detector (Putley [3.11, 51J). Assume that the element receives radiation over an area A normal to the polar axis of the material and that this produces a modulated temperature rise (3.4). 

GRAPH 10.30 Three dipoles with common flow inductive CPE, resistor, and capacitive CPE and their representation as an equivalent electrical circuit in series (a) and as a single unit of Eormal Graph (b). Each CPE is built by complementing its own paths with its own evolution mode before being added to the resistance. As the linear circuit is not powered, the common flow is zero. 

Similarly to the previous case the element Rs of the equivalent electric circuit for this corrosion system represents the resistance of the 0.5M solution of NaCl electrolyte used as the corrosive environment. Elements in parallel in the equivalent electric circuit characterize the protective properties of AI2O3 layer deposited on the bulk Al. The element Cb specifies the AI2Q3 layer capacitance, which depends on the thickness of this layer and on the dielectric properties of the material. The resistor Rb in such a system represents the resistance of the protective layer, and depends on properties of the material forming the layer, and varying with the thickness of the layer and its material composition. The... 

Illustration of the principle of the impedance analyzer (a) and the equivalent electric circuits (b) of a leaky dielectric film (R is the resistance, and C is the capacitance of the material studied). The equivalent electric circuit of a real sample containing the electrodes with resistance r, polymer alignment layers with resistance Rp and capacitance Cp (c) the equivalent circuit of the empty cell with capacitance (d). 

The experimental conductance and capacitance Cm as a function of at a frequency of 50 cps and for a silver chloride crystal 0.01 cm thick are shown in Figs. 9 and 10. Above 1000 cps the conductance and capacitance are dependent on frequency, but are constant with within experimental accuracy, the value at 1592 cps being 3.1 Xl0 ohm" and 5.5 X 10" F, respectively. The two space charge regions and the bulk region in the silver chloride crystal can be represented by an equivalent electrical circuit shown in Fig. 11, where Rg and Cs represent the surface space charge region and R and Cg the bulk crystal. We assume that the electrode-solution interface, the solution, and the crystal-solution interface do not contribute to the measured Rm and Cj. ... 

Fig. 9 Top The simulation cell consists of a BMI-PFe ionic liquid electrolyte surrounded by two CDC electrodes held at constant potentials (blue C atoms red, the three sites of BMI cations and green, PFs" anions). Bottom Equivalent electric circuit. Rtuik is the resistance of the electrolyte in the bulk region, Ri is the resistance of the electrolyte adsorbed in an electrode slice, Ci and C2 are the capacitances of the two slices. Reprinted with permission from ref, 135. Copyright 2014 American Chemical Society. 

An actual plot of impedance in the complex plane for the polyethylene oxide NaSCN complex (n = 22, x = 2.0) is shown in Figure 5. This impedance plot represents an equivalent electrical circuit having a capacitor and a resistor in parallel with no resistor in series because the semicircle starts right at the origin. Here, the diameter of the semicircle gives the resistance of the circuit, the capacitance value is given by the formula ... 

There is an equivalence between the differential equations describing a mechanical system which oscillates with damped simple harmonic motion and driven by a sinusoidal force, and the series L, C, R arm of the circuit driven by a sinusoidal e.m.f. The inductance Li is equivalent to the mass (inertia) of the mechanical system, the capacitance C to the mechanical stiffness and the resistance Ri accounts for the energy losses Cc is the electrical capacitance of the specimen. Fig. 6.3(b) is the equivalent series circuit representing the impedance of the parallel circuit. 

Faradic impedance (//) is directly related to the rates of charge transfer reactions at and near the electrode/electrode interface. As shown in Figure 3.1, the Faradaic impedance acts in parallel with the double-layer capacitance Cd, and this combination is in series with the electrolyte resistance Rei The parameters Rei and Cd in the equivalent circuit are similar to the idea of electrical elements. However, X/ is different from those normal electrical elements because Faradaic impedance is not purely resistive. It contains a capacitive contribution, and changes with frequency. Faradaic impedance includes both the finite rate of electron transfer and the transport rate of the electroactive reagent to the electrode surface. It is helpful to subdivide Zj into Rs and Cs, and then seek their frequency dependencies in order to obtain useful information on the electrochemical reaction. 

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