Perturbation waveforms

If, however, we start the system with a given non-uniform distribution, corresponding to n = 2 say, and a value for ji such that the initial point lies beneath the neutral stability curve, then the spatial amplitudes will not decay. Rather the positive real parts to the eigenvalues will ensure that the perturbation waveform grows. The system may move to a state which is varying both in time and position—a standing-wave solution. 

The theory and practical implementation of the Fourier Transform Faradaic Admittance Measurement technique was introduced and developed by Smith and coworkers.This elec-troanalytical technique now rests on a sound theoretical basis and has been proven in many experimental situations. In this method a multiple-frequency perturbation waveform is used and the cell admittance is calculated at each of these frequencies using the measured current and potential. Measurements over a substantial range of frequencies, at least over two orders of magnitude, are required before any conclusive interpretations can be made concerning the mechanism and kinetics of an electrochemical system. The main advantage of this Fourier transform approach is that the cell admittance can simultaneously by measured at a large number of frequencies. 

Fig. 3.9 Complex plane plots of numerictilly simulated impedances, in Ci, for different perturbation waveforms with 1 % noise added (a) rectangular pulse, (b) exponentially decaying perturbation, (c) quasi-random noise, (d) sum of sine waves with constant amplitudes and zero phases (From Ref. [105] with permission of editorial board)...

The above mentioned difficulties can be successfully overcome, again, by the simultaneous application of dual dynamic perturbing waveforms to both the disk and ring electrodes. An example of this technique is described below. 

Fig. 10.10 Voltammetry with periodical renewal of the diffusion layer (a) perturbation waveform (b) sequence of chronoamperometric signals recorded at each potential step (c) i vs. E response...

Differential Pulse Voltammetry. As illustrated in Figure 38, in the Differential Pulse Voltammetry (DPV) the perturbation of the potential with time consists in superimposing small constant-amplitude potential pulses (10 < AEpUise <100 mV) upon a staircase waveform of steps of constant height but smaller than the previous pulses (1 < AEbase < 5 mV). 

In conclusion, in Pulse and Multipulse techniques the perturbation is given as an arbitrary sequence of constant potentials without any a priori restriction on the duration of each individual potential of the sequence or on the particular waveform employed, and the current is sampled at a pre-fixed time. 

In Multipulse techniques, the potential waveform consists of a sequence of potential pulses Ei, E2, , Ep, and the initial conditions of the system are only regained after the application of the last potential step [1-6]. When the potential waveform is a staircase of constant pulse amplitude AE, the perturbation includes as a limiting... 

Scheme 7.3 Square wave voltammetry, (a) Potentialtime waveform, (b) current-potential response. The black dots in (a) indicate the time at which the current is measured. / denotes the frequency of SW perturbation...

Another more sophisticated approach is to make a Fourier Transform analysis of the response in the way proposed by Bond et al. [84, 85]. In this case, the perturbation is a continuous function of time (a ramped square wave waveform) which combines a dc potential ramp with a square wave of potential that can be described as a combination of sinusoidal functions. Under these conditions, the faradaic contribution to the response generates even harmonics only (i.e., the non-faradaic current goes exclusively through odd harmonics). Thus, the analysis of the even harmonics will provide excellent faradaic-to-non-faradaic current ratios. 

In a CPA, because a large secondary dispersion is added to the stretcher, the spectrum is mapped to the temporal waveform in the amplifier. For that reason, the phase change added to the pulse shaper cannot cause a large change in the pulse shape in the CPA. Self-phase modulation depends on the temporal amplitude transition. Therefore, the phase perturbation can be regarded as almost linear in the amplifier. Thus, the inverse of the spectrum phase difference between the acquired and the target phase is added to the pulse shaper. 

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. 

Consider, for example, a test sample of material with a well-defined geometry as shown in Fig. 2. Reversible electrodes are attached to opposite planar faces of the test article, and a sinusoidal electrical potential (V ) is applied via a waveform generator. The current response is monitored with a frequency response analyzer (FRA), which converts the signal to the frequency domain. The amplitude (A) of the input wave is adjusted to the range in which the system responds linearly, about 10 mV. Thus, the perturbation can be described by the following equation ... 

Figure 1 Sinusoidal perturbation and respionse waveforms illustrating change in amplitude (A and B) and phase shift ((]>).

There are many other ways by which the capacitance of the interphase can be determined, with the use of single or repetitive waveforms, with either the current or the potential being used to perturb the system. The choice depends on available instrumentation, the nature of the system being studied and often the personal preference and experience of the researcher. 

In LEIS, the full electrochemical impedance spectrum of the sample/electrolyte interface can be obtained at the submillimeter level. The system works by stepping a probe tip across the sample surface (the smallest step size is 0.5 pm) while the sample (connected as the working electrode) is perturbed by an ac voltage waveform (usually about the open-circuit potential with an amplitude typically of 20 mV). The probe tip consists of two separated platinum electrodes, separated by a known distance. Measurement of the potential difference between the two electrodes allows the calculation of the potential gradients above the sample surface, which then give the current density. Comparison of the in-phase and out-of-phase current flow produces the impedance data, as with the regular EIS. The data can be plotted as Bode or Nyquist charts for specific points on the surface, or impedance maps of the sample surface can be obtained. 

Many applications of this strategy are based on extensions of the concepts of impedance developed earlier in this chapter (41-43). However, the excitation waveform is usually an impulse in potential (rather than a periodic perturbation), and a transient current is measured. One records both E t) and i t) as observed functions. Then both are subjected to transformations, and comparisons are made in the frequency domain between E s) and i s). Ratios of the form i s)IE s) are transient impedances, which can be interpreted in terms of equivalent circuits in exactly the fashion we have come to understand. The advantages of this approach are (a) that the analysis of data is often simpler in the frequency domain, (b) that the multiplex advantage applies, and (c) the waveform E(f) does not have to be ideal or even precisely predictable. The last point is especially useful in high-frequency regions, where potentiostat response is far from perfect. Laplace domain analyses have been carried out for frequency components above 10 MHz. 

Methods for measuring the impedance can be divided into controlled current and controlled potential [2, 4, 81]. Under controlled potential conditions, the potential of the electrode is sinusoidal at a given frequency with the amplitude being chosen to be sufficiently small to assure that the response of the system can be considered linear. The ratio of the response to the perturbation is the transfer function, or impedance, Z, when considering the response of an AC current to an AC voltage imposition and is defined asE = IZ, where E and I are the waveform amplitudes for the potential and the current respectively. Impedance may also be envisaged as the resistance to the flow of an alternating current. 

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