Laplace transform periodic potentials

First of all, the mathematical background will be developed for the case of a simple electrode reaction O + n e = R. In this treatment, contrasts like potential versus current perturbation, large amplitude versus small amplitude, and single step versus periodical perturbation are emphasized. While discussing these principles, the most common methods derived from them will be briefly mentioned. On the other hand, it will be shown that, by virtue of the method of Laplace transformation, these methods have much in common and contain, in principle, the same information if the detected cell response is of the same order. 

One can envision three types of perturbation an infinitesimally narrow light pulse (a Dirac or S-functional), a rectangular pulse (characteristic of chopped or interrupted irradiation), or periodic (usually sinusoidal) excitation. All three types of excitation and the corresponding responses have been treated on a common platform using the Laplace transform approach and transfer functions [170]. These perturbations refer to the temporal behavior adopted for the excitation light. However, classical AC impedance spectroscopy methods employing periodic potential excitation can be combined with steady state irradiation (the so-called PEIS experiment). In the extreme case, both the light intensity and potential can be modulated (at different frequencies) and the (nonlinear) response can be measured at sum and difference frequencies. The response parameters measured in all these cases are many but include... 

The operational impedance is the ratio of the Laplace transform of the potential to the Laplace transform of the current (Eq. 2.88). It is usually used for an arbitrary perturbation signal. For the periodic signal it is equivalent to the definition using Fourier transformation. What follows are examples of the applicatirMi of the Laplace technique to the determinatirHi of the current-potential relations and the impedances. 

In the Laplace transform above, we assumed a real transform with s = a. As in the impedance technique, we usually apply a periodic cosine perturbation, and in such a case it is simpler to use the FT with s = jco. In general, a periodic potential perturbation, AE, applied to a circuit may be written as a complex analog of the simple periodic perturbation, see Eq. (2.56) ... 

As it is experimentally not easy to provide electric voltages to the capacitor which are exactly constant in time, it is useful to apply periodically alternating electromotive forces and to extrapolate experimental data to zero frequencies, i. e. constant potential differences. Combining Laplace transformation of Eqs. (6.18-6.20) with Equation (6.2) and the generalized Ohm Law (6.8), we get... 

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. 

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