Impedance function

The simplest electrical elements that can be part of an electrical equivalent circuit are resistance R, capacity C and induction L, with the following impedance functions ... 

It is often experimentally observed that in the Nyquist plot, semi-circles are obtained with a centre point below the x-axis. Analysis of the situation tells us that the double-layer capacity is not a suitable description of the system occurring and should be replaced by an element with an impedance function given in Equation 2.51 ... 

Finiteness of the impedance function The impedance must be finite valued at / —> 0 and / —> and the impedance must be continuous and finite... 

This method of estimating Rc is useful when it can be applied, since the determination is not based on any presumed model of the corrosion damage process or any of the assumptions that come with assignment of an equivalent circuit model. This method is particularly helpful when there is more than one time constant in the spectrum, or the impedance spectrum is particularly complicated. Caution is warranted however. This method of estimation can be in serious error for samples with large capacitance-dominated low-frequency impedances. As a general rule, for this estimation method to be reasonably accurate, the impedance function must exhibit a clear DC limit, or a diffusional response that can be modeled by a constant phase element in equivalent circuit analysis (75). 

The traditional way is to measure the impedance curve, Z(co), point-after-point, i.e., by measuring the response to each individual sinusoidal perturbation with a frequency, to. Recently, nonconventional approaches to measure the impedance function, Z(a>), have been developed based on the simultaneous imposition of a set of various sinusoidal harmonics, or noise, or a small-amplitude potential step etc, with subsequent Fourier- and Laplace transform data analysis. The self-consistency of the measured spectra is tested with the use of the Kramers-Kronig transformations [iii, iv] whose violation testifies in favor of a non-steady state character of the studied system (e.g., in corrosion). An alternative development is in the area of impedance spectroscopy for nonstationary systems in which the properties of the system change with time. 

The detailed derivation leads to an impedance function, having an equivalent circuit where serially to the adsorption capacitance there are a resistance and a -> Warburg impedance [1]. Important points are as follows ... 

The (ice)-1/2 frequency dependence appears also in the impedance function of unsupported systems, that is, when the transport of the electroactive species proceeds via coupled diffusion and migration. 

To illustrate the technique, let s consider the case in which the system is stimulated by an applied voltage at a discrete frequency and the response current is measured at the same frequency. At zero frequency, or d.c., impedance is equivalent to resistance as defined by Ohm s law R = V/I. When the impressed voltage is oscillated at a particular frequency, the system responds by passing an oscillating current. If the amplitude of the input voltage is sufficiently small (typically < 10 mV), the system is linear, and the frequency of the response wave (current) matches the frequency of the perturbation (voltage). However, the response current wave may differ from the perturbation in amplitude and phase (Fig. 1). The ratio of the amplitudes of the perturbation to the response waveforms and the phase shift between the signals define the impedance function. 

The real component (Z ) of the impedance function corresponds to the in-phase... 

Figure 11.10 plots the imaginary -Z" vs. the real part Z" of the complex impedance (Argand plot) exemplary for undoped CoTi03/La at 400°C under synthetic air. All plots showed semicircles and could be described with the impedance function of a parallel RC circuit equivalent. 

An automated data fitting software was developed allowing to describe the impedance spectra as the impedance function of a circuit equivalent, e.g., consisting of a resistor and a capacitor in parallel. Also other elements, such as inductances or constant phase elements (CPE), could be implemented. The admittance of the parallel RC circuit is just the sum of the admittances of the two elements which give the impedance44 ... 

With the phase

real part Z and the imaginary part Z" is given by ... 

In this case R is the extrapolated dc-resistance of the material, C the capacitance, and o) the angular frequency. The use of these fit functions reduces the complex impedance function describing the electrical properties of each material to one value for the resistance and the capacitance, respectively. Although the capacitance can be assigned to the geometric capacitance of the IDC, the resistance values under the respective measuring conditions are taken for the determination of the material sensitivity. 

Macdonald et al. and Dougherty and Smedley used a polynomial approximation of the impedance function, followed by analytical integration of the polynomials. However, extrapolation of polynomials over a large frequency range may be unreliable. Haili extrapolated Z" as proportional to (0 as (0 0 and as inversely proportional to ro as co ... 

In addition to the differential equations, the boundary conditions must also be taken into account. For this case, the ionic current is usually assumed to vanish at the end of the liquid channel, whereas the electronic current vanishes at the outer edge of the electrode, so that i (L) = 0 and /2(0) = 0. For such boundary conditions, the generalized solution of the model leads to the following impedance function [92,95,97,160,190] ... 

Notice that the impedance function obtained depends on the position also, whereas measured impedances do not. The two quantities relate to each other as... 

A transfer function, the admittance function, can be defined as y ((o) = iy((o)l6 , where iy(co)l represents the amplitude and c ) is the phase angle. The impedance function, Z(co), is the inverse of the admittance function, Z(a))=[y ((o)]" and since both the amplitude and the phase angle of the output may change with respect to the input values, the impedance is expressed as a complex number, Z = Z i -i- where Zjeji is the real part and Z is the imaginary part. 

The overall admittance (Equation 2.3) for a parallel resistor-capacitor (RC) circuit is given by the sum of the conductance (l/R) and capacitance contributions, where the resistance (R) represents the dissipative component of the dielectric response, while the capacitance (C) describes the storage component. The impedance function for that circuit is... 

As was pointed out by Cole and Cole, dielectric response corresponding to the function of Fq. (39) may be decomposed into the circuit shown in Figure 2.1.7, which contains a constant-phase element (CPF). The CPF is an empirical impedance function of the type... 

An important requirement for a valid impedance function is that the system be linear. Theoretically, this implies that the real and imaginary components transform correctly according to the Kramers-Kronig relationships (discussed later in this section). Practically, linearity is indicated by the impedance being independent of the magnitude of the perturbation, a condition that is easily (although seldom) tested experimentally. 

The problem of acquiring impedances at sufficiently low frequencies is amply demonstrated by the data (Syrett and Macdonald [1979]) for 90 10 Cn Ni alloy corroding in flowing seawater (Figure 4.4.1). Thus, for an exposure time of 22 h, the impedance function can be defined over the entire bandwidth, and an accurate value for Rp may be obtained by probing the interface at frequencies above 0.01 Hz. On the other hand, at much longer exposure times, frequencies as low as 0.0005 Hz are not sufficient to completely define the interfacial impedance, and considerable extrapolation is required to acquire a value for Rp. 

We will exemplify this approach for the finite-length-diffusion case. The impedance function for this case is described in Section 2.1.3, Eq. (135) for a reflective boundary and Eq. (136) for a transmissive one. The first case corresponds to diffii-... 

Here functions R(v) and C(v) can be obtained by piecewise-linear interpolation of the dependence of R and C parameters obtained by fitting the experimental spectra at different voltages (such as in Figure 4.5.4) to the impedance function in Eq. (10). Any other suitable smooth interpolation can be used. The impedance function has to be expressed in terms of electric parameters, as described in Section 4.5.1.3. For use in a discretized equivalent circuit, the values obtained from the fit have to be divided or multiplied by the number of chains, depending on the series or parallel position of the electric element. So, for series resistors it has to be divided, and for parallel, multiplied. It should be considered that the low-frequency limit of Re Z), used as a fitting parameter in the equation, is not always a simple sum of the discrete elements that constitute a transmission line. In particular, in Eq. (10) the Ra is 1/3 of the specific resistance multiplied by the transmission line length, as can be seen from Eq. (8). Therefore resistance of single chain shown in Eig 4.55 will be Ra 3/N. 

Another similar method suitable for power sources testing is based on current interrupf . If the load is applied for a long time, so that it can be assumed that the Systran under test has stabilized and no longer changes in the time interval similar to the necessary measuiranent duration, the load is disconnected and sampling starts in the instant of disconnection. Acquired data values v[i], /[i] are again used in a fit to a selected carrier function v(t) to obtain its parameters which are then substituted into the equation of this function s Laplace transform to obtain V(5 ). The impedance function is then obtained as shown below. 

The DRT method uses the fact that every impedance function that obeys the Kramers-Kronig relations can be represented as a differential sum of infinitesimally small RC elements (Figure 16.7). This sum goes from 0 to oo [16]. 

FIGURE 1.23 Piston impedance functions (a) resistance function, (b) reactance function. 

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