Nonideal circuit elements

In all real systems, some deviation from ideal behavior can be observed. If a potential is applied to a macroscopic system, the total current is the sum of a large number of microscopic current filaments, which originate and end at the electrodes. If the electrode surfaces are rough or if one or more of the dielectric materials in the system is inhomogeneous, many of these microscopic current filaments would be different. In a response to a small-amplitude excitation signal this would lead to frequency-dependent effects, which can often be modeled with simple distributed circuit elements. For example, many capacitors in EIS experiments, most prominently the double-layer capacitor often do not behave ideally due to the distribution of currents and electroactive species. Instead, these capacitors often act like a constant phase element (CPE), an element that has found widespread use in impedance data modeling. 

The term constant phase element stems from the fact that the phase angle of the portion of a circuit represented by such an element is AC frequency-independent. The impedance of a CPE has the form of dependency on an effective CPE coefficient Q [ohm s ] [3, p. 211] as  

CPE with a depressed semicircle has been used to describe both the doublelayer capacitance and the low-frequency pseudocapacitance (Section 5-3). The double layer capacitor in electrochemical experiments often shows a CPE-like distributed behavior instead of that of a pure capacitor. Several theories have been proposed to accoimt for the nonideal behavior of the double layer, but none has been universally accepted. As a first approximation, one can treat a as an empirical constant and not worry about its physical basis. For double-layer analysis, the parameter Q, expressed in [s ohm cm ], is 

FIGURE 3-1 The Nyquist plot of a single CPE (O) and a parallel resistor and CPE circuits. 

FIGURE 3-2 The Nyquist plots of A. single CPE (ideally polarizable electrode) B. parallel resistor and CPE circuits 

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Equivalent circuit (formal or mathematical) modeling presents the system in hypothetical electrical circuits consisting of well-defined ideal and sometimes nonideal electrical elements. Measurement modeling explains the experimental impedances in terms of mathematical functions in order to obtain a good fit between the calculated and experimental impedances. In the latter case the parameters obtained do not necessarily have clear physicochemical significance. Such a model describes the system s response to various possible electrical input signals. 

Under this electrochemical configuration, it is commonly accepted that the system can be expressed by the Randles-type equivalent circuit (Fig. 6, inset) [23]. For reactions on the bare Au electrode, mathematical simsulations based on the equivalent circuit satisfactorily reproduced the experimental data. The parameters used for the simulation are as follows solution resistance, = 40 kS2 cm double-layer capacitance, C = 28 /xF cm equivalent resistance of Warburg element, W — R = 1.1 x 10 cm equivalent capacitance of Warburg element, IF—7 =l.lxl0 F cm (

charge-transfer resistance, R = 80 kf2 cm. Note that these equivalent parameters are normalized to the electrode geometrical area. On the other hand, results of the mathematical simulation were unsatisfactory due to the nonideal impedance behavior of the DNA adlayer. This should... 

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... 

In many cases, the use of ideal equivalent circuits is convenient but not always appropriate. Nonideal behavior might arise from interactions of species, resulting in frequency-dependent capacitances [C((D)]. Under these conditions, the physical process is more accurately described by a range of relaxation time constants instead of a unique value. Such distributed relaxation events are usually manifested as semicircles depressed below the real axis in the complex plane, and the angle of depression is related to the degree of nonideality. Various distribution functions and constant phase elements have been employed to describe such events. These nonidealities are especially evident in biological systems. 

This replacement was necessary to adapt the equivalent circuit to the nonideal behaviour of the aluminium oxide film. The exponent n of the CPE element can be regarded as a measure of the inhomogeneity of the film structure [17]. For an ideal capacitor the exponent n is one. For the calculation of the CPE values, the fitting program in Ref [18] was used. 

A possible but rough electrical DC/AC equivalent circuit for the electrode processes is shown in Figure 7.20. The electrodic part consists of three principal current paths in parallel. The elements are Cole-like as discussed in Section 9.2, and some of the used component symbols indicate that their values are nonideal, frequency-dependent. 

FIGURE 11.10 Electrical circuit of the parallel mounting of a capacitive common phase element and a resistor representing the generalized Cole-Cole model of nonideal dielectric relaxation. 

Figure 6.1.5.1 Equivalent circuit of the electrochemical cell and potentiostat measuring circuit incorporating nonideal elements.

Equivalent circuit (EC) analysis is relatively simple for a circuit containing ideal elements R, C, and L. It may also be carried out for circuits containing distributed elements that can be described by a closed-form equation, such as CPE, semi-infinite, finite length, or spherical diffusion. Many "ideal" resistances and capacitances chosen to represent a real physicochemical system are really nonideal as any resistor has a capacitive component and vice versa. However, for the broad frequency range utilized by UBEIS it is usually adequate to incorporate "ideal" resistors, capacitors, and inductances [29, p. 87]. The type of electrical components in the model and their interconnections... 

In a broad sense a parallel combination of charge transfer resistance and CPE elements, in series with finite diffusion element typically represent the circuit. When potential modulation is introduced, charge-transfer-related impedances decrease with increases in electrochemical potential and capacitance for the metal-polymer interface. The capacitance is usually nonideal due to film or electrode porosity [13] and typically is represented by the CPE element. If the film is formed as a reflective boundary, the angle is sometimes different from -90° because of inhomogeneity of the film and distributed values for diffusion coefficients. If two films are formed on the electrode, two RI CPE semicircles are often observed. 

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