Pure capacitor

A nonpolarizable interface behaves as a capacitor C and a resistor R in parallel a polarizable interface responds as a pure capacitor. The higher the resistance R, the closer the behavior of the former to the latter. For R —> °o, a nonpolarizable interface becomes polarizable. The condition / — < corresponds to Am —> 0. This condition is met when the amount of M+ in the null solution is negligibly small. 

It is evident that the current still leads the voltage but that the "phase angle, a, will vary from close to 90° at low frequencies to close to 0 at high frequencies. Also, at low frequency Z — 1 /tuC and at high frequency Zf — R. In other words, at low frequencies, the circuit behaves like a pure capacitor but at high frequencies it behaves like a pure resistor. Moreover, by fitting the observed current data as a function of frequency to calculated values of Zj and a, an accurate estimate of both R and C can be made. 

From our definition of a pure capacitor (i.e. one having no resistive component), we can say that the real impedance Z is zero. We see straightaway from equation (8.8) that the impedance is a function of frequency. The impedance of a capacitor is infinite when a DC voltage is applied (just put ru = 0 into equation (8.8)), while the imaginary impedance Z" decreases as the frequency co is increased. 

The electrode solution double-layer sometimes behaves as a capacitor. By assuming that it behaves as a pure capacitor with a capacitance of 10 F, what is its impedance at a frequency of 10 Hz ... 

Curves A and B (Fig. 5.1) describe the behavior of two interfaces that are fundamentally different. As we have mentioned already, Curve A below the breakdown voltage shows the typical response of a pure capacitor. Thought Experiments I and II also show that this capacitor is located at the interface between the electrode and the ionic sample. Any interface involving mobile charges always separates these charges. In other words, a capacitor forms spontaneously at such interfaces. Because... 

Another possibility is based on the fact that constant-phase elements represent processes that are not purely conservative (charging and discharging of a capacitor is a conservative process), first because of the fact that n is not equal to 1. From such an element, one can isolate the conservative contribution, which can then be further treated as a conservative process and simulated by a pure capacitor. The starting point for this separation is the existence of an angle frequency, to, for which48 ... 

Finally, it can be seen from Fig. 9.9a that the real impedance does not remain constant at low frequencies for the textile electrode, and this effect is more pronounced at higher electrolyte concentrations. Probably, Zr is influenced by other effects only occurring in the low-frequency range. This effect is frequently observed and described in the literature and is caused by non-uniformity of surfaces at the micro-scale, which in fact is the case for the textile electrodes. It is also not possible to explain this effect by a pure resistor or a pure capacitor in the electrical equivalent circuit. For this purpose, constant-phase elements are implemented as described in the theoretical discussion of electrochemical impedance spectroscopy (presented in Chapter 2, section 2.4). 

Note that this capacitance is linked to the electron transfer reaction and therefore has a faradaic origin and is not related to the double-layer charging process (this last capacitance corresponds to a pure capacitor see Sect. 6.4.1.5). In this sense, it has been called pseudo-capacitance [56]. The normalized current i/rCv is a ratio of capacitances since, from Eq. (6.161), y ev = (Icv/v)/(Qf F/RT)) = Ccv/Cf [48, 57],... 

The mathematical expressions which describe the impedance of some passive circuits are shown below, where a passive circuit is one that does not generate current or potential [129], In this regard, the impedance response of simple passive circuit elements, such as a pure resistor with resistance R, a pure capacitor with capacitance C, and a pure inductor with inductance L, are given, respectively ... 

One assumes that the double layer admittance is that of a pure capacitor. Thus its value should be independent of frequency. The graphical variation of Y with Y" is shown schematically in Fig. 11.9. More details may be found in Ref. 9. 

In a sinusoidal AC circuit, the current through a pure capacitor leads the voltage drop across this capacitor by 90°. The 90° phase relationship between Ic ( ) (the current through the capacitor) and Vc ( ) (the voltage across the capacitor) can be written as... 

Figure 3.1 shows a typical equivalent circuit of an electrochemical cell. Rel represents the electrolyte resistance between the working electrode surface and the point of reference electrode Cd is a pure capacitor of the capacity associated with the double layer of the electrode/electrolyte interface and Zf refers to the Faradaic impedance, which corresponds to the impedance of the charge transfer at the electrode/electrolyte interface. The connection of X, and Cd in Figure 3.1 is in parallel. The impedance X, can be subdivided in two equivalent ways, as seen in Figure 3.1 b ... 

For a circuit containing both capacitors and resistors, the ratio between the applied voltage signal and the resulting current signal is the impedance Z((o), which is a function of frequency. The impedance of a pure resistor is simply its resistance R, while the impedance of a pure capacitor is given by... 

Although the values of tlie two resistors are easily discerned, there is no region in which the circuit behaves as a pure capacitor. The slope never reaches a value of - 1, and (p never even comes close to - 90°, which one would have for a pure capacitor. 

For the very simplified situation that the sphere behaves electrically as a pure capacitor, and the solution as a pure resistance, the relaxation can be described by a Maxwell-Wagner mechanism, with T = e e/K, see (1.6.6.321. Although some success has been claimed by Watillon s group J to apply this mechanism for a model, consisting of shells with different values of e and K, generally a more detailed double layer picture is needed. In fact, this Implies stealing from the transport equations of secs. 4.6a and b. generedizing these to the case of a.c. fields. 

Electrochemical impedance spectroscopy (EIS) analysis of such electrodes is shown in Figure 7.2. At high frequencies, the imaginary part of the impedance tends to zero, whereas at low frequencies it increases sharply, thus approaching the variation of impedance with the frequency expected for a pure capacitor (see Chapter 1). In the intermediate frequencies, a semicircle can be observed, the amplitude of the loop varying with the nature of the activated material. This semicircle can be... 

When wc compare ilic last two equations, wc find that the voltage across a pure capacitor that results from a sinusoidal input signal is sinusoidal but lags the current by (see bigurc 2-9).. As we show later, this lag In smaller than 90 in a real circuit that also ci>ntains lesistanec. 

Capacitance If a sinusoidal voltage is applied across a pure capacitor, the impedance can be calculated according to the relationship... 

If the Nyquist plot is a straight line lying along the Z axis, then the sample can be represented by a pure capacitor (Figure 7).The Z = 0 and Z = -1/mC, thus, the total impedance, Z = -j/coC and decreases with increasing o). 

A3 PURE RESISTOR AND PURE CAPACITOR CONNECTED IN SERIES... 

FIGU RE 8 Nyquist plot for a pure resistor connected in series with a pure capacitor. 

The impedance of the equivalent circuit comprising a pure resistor and a pure capacitor in parallel connection is given by ... 

From Equation (12), knowing and R, the capacitance, C can be evaluated. The C value varies with different conduction processes in a material (Table 1). The various conduction processes in a material is one of the reasons wlty its electrical characteristics cannot be represented by pure resistors and pure capacitors. Thus, a new circuit element known as constant phase element (CPE) is introduced (Bottelberghs and Broers, 1976). 

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