Transfer function impedance

These relations offer the possibility to examine measured transfer functions (impedance spectra) on errors caused by time instability or time drift. However, KK-checking techniques have fundamental problems in their application to practical measurements. Therefore many attempts have been made to overcome these limitations by means of different interpolation procedures. An attempt is the Z-HIT approximation, applied by Schiller et al. [2001], Agarwal et al. [1995], and Ehm et al. [2000], an approximation formula for the calculation of the impedance modulus course from the phase angle by integration. 

AC electrogravimetry consists in the simultaneous measurements of the electrochemical impedance and the mass/potential, or electro-gravimetric, transfer function. Impedance measurements are already well documented in the literature hence, in this text, only the electrogravimetric transfer function measurement will be described. It uses a fast QCM whose general concepts are first given below. 

Although chromate is the best aqueous corrosion inhibitor available, its use has been severely curtailed due to toxicity and environmental concerns ( ). One of the more successful non-chromate treatments involves the use of phosphate/phosphonate combinations. This treatment employs high levels of orthophosphate to promote passivation of the metal surfaces. Therefore, it is important to control calcium phosphate crystallization so that high levels of orthophosphate may be maintained in the system without fouling or impeding heat-transfer functions. 

In the case of potential steps, it is more direct to think in terms of characteristic time constants than in terms of impedances. Two time constants can be defined and used to describe the cell transfer function ts = RsCdl. The units... 

The experimental set-up is shown in Fig. 7-1 an electrochemical interface with low level noise and a transfer function analyzer (TFA) were used for measurements of the EHD impedance. A matched two-channels 24 db/octave low pass filter (F) was used to remove HF noise and the ripple due to electric network supply, this analog filtering allows the TFA to operate with an increased sensitivity. These instruments were controlled by a computer, which recorded the data. 

In the EHD impedance method, modulation of the flow velocity causes a modulation of the velocity gradient at the interface which, in turn, causes a modulation in the concentration boundary layer thickness. As demonstrated previously in Section 10.3.3 and Fig. 10.3 the experiment shows a relaxation time determined solely by the time for diffusion across the concentration boundary layer. Although there is a characteristic penetration depth, 8hm, of the velocity oscillation above the surface, and at sufficiently high modulation frequencies this is smaller than the concentration boundary layer thickness, any information associated with the variation of hm with w is generally lost, unless the solution is very viscous. The reason is simply that, at sufficiently high modulation frequencies, the amplitude of the transfer function between flow modulation and current density is small. So, in contrast to the AC impedance experiment, the depth into the solution probed by the EHD experiment is not a function... 

Electrogravimetric transfer functions Am/A (o>) and AmlAq associated with the electrochemical impedance, A /A/(a>) were determined for the anodic dissolution of Fe in sulfuric acid. This gave the unique possibility of discriminating between the change in mass due to metal dissolution and the relaxation of the film thickness on the basis of their individual frequency responses [35, 36]. 

Another passive method is the transference function method (TFM) introduced by Muramatsu [6]. The method consists of an oscillator that drives a crystal through a known measuring impedance and a radiofrequency voltmeter which measures the transference modulus of the system. Muramatsu [6] neglected the effect of the parasitic capacitance and his expression for the quartz impedance resulted in a nonlinear relationship between the measured resistance R with the ac voltage divider and the value of R measured by an impedance analyser. Calvo and Etchenique [74] improved the method and introduced an analytical expression to fit the entire transfer function around resonance in order to obtain the same values of R, L and C as measured by a frequency response analyser. 

Note that the quantities in bold are vectors represented in the complex plane. In Fig. 12.5 a transfer function spectrum obtained in 0.25 s is shown for 100 frequencies around 10 MHz. It can be used for real time evaluation of the quartz electro-acoustical impedance when the viscoelastic properties change. 

V0/V/ (fti), to the analytical expression with recovery of the complete quartz impedance near resonance (admittance, conductance and impedance). Although the voltage divider method does not measure the transfer function phase and hence it is not possible to demonstrate the validity of BVD circuit, it has the advantage of speed. Also passive methods like TFM can be applied under high viscous damping so that the shear wave phase never crosses zero and the EQCM no longer resonates. 

The electrochemical impedance is a fundamental characteristic of the electrochemical system it describes. A knowledge of the frequency dependence of impedance for a corroding system enables a determination of an appropriate equivalent electrical circuit describing that system. Such a circuit is typically constructed from resistors and capacitors. Table 1 shows the transfer functions... 

Gabrielli C, Keddam M, Takenouti LI (1990) New trends in the investigation of electrochemical systems by impedance techniques multi-transfer function analysis. Electrochim Acta 35 1553-7... 

Impedance transfer functions may be determined through use of an input signal containing more than a single frequency. Such signals may be a tailored multi-sine... 

The methods described in this chapter and this book apply to electrochemical impedance spectroscopy. Impedance spectroscopy should be viewed as being a specialized case of a transfer-function analysis. The principles apply to a wide variety of frequency-domain measurements, including non-electrochemical measurements. The application to generalized transfer-function methods is described briefly with an introduction to other sections of the text where these methods are described in greater detail. Local impedance spectroscopy, a relatively new and powerful electrochemical approach, is described in detail. 

While the emphasis of this book is on electrochemical impedance spectroscopy, the methods described in Section 7.3 for converting time-domain signals to frequency-domain transfer functions clearly are general and can be applied to any t3q>e of input and output. Some generalized transfer-function approaches are described in Chapters 14 and 15. 

Remember 7.3 The techniques described here for measuring impedance spectra are, in fact, very general and can be used to measure any transfer function. 

Four state variables may be defined, for example, for the rotating disk described in Qiapter 11. These may include the rotation speed, the temperature, the current, and the potential. At a fixed temperature, three variables remain from which a transfer function may be calculated. As shown in Table 7.1, the generalized transfer functions include impedance, admittance (see Chapter 16), and two types of electrohydrodynamic impedance (see Chapter 15). 

Local impedance measurements represent another form of generalized transfer-function analysis. In these experiments, a small probe is placed near tiie electrode surface. The probe uses either two small electrodes or a vibrating wire to allow measurement of potential at two positions. Under the assumption that the electrolyte conductivity between the two points of potential measurement is uniform, the current density at the probe can be estimated from the measured potential difference AVprobe by... 

Remember 10.2 Transfer functions such as impedance provide therelationship between tioo oscillating variables, e.g., i and V. When the expression for i is given in terms of two or more oscillating variables, additional relationships must be found to relate the additional variables to i or to V. 

Theoretical developments show that it is possible to deduce hydrodynamic information from the limiting current measiuement, either in quasi-steady state where /(f) cx py t) or, at higher frequency, in terms of spectral analysis. In the latter case, it is possible to obtain the velocity spectra from the mass-transfer spectra, where the transfer function between the mass-transfer rate and the velocity perturbation is known. However, in most cases, charge transfer is not infinitely fast, and the analysis also requires knowledge of the convective-diffusion impedance, i.e., the transfer function between a concentration modulation at the interface and the resulting flux of meiss under steady-state convection. 

The proliferation of different and conflicting sets of notation for different impedance techniques makes necessary a unified approach for describing the transfer function resulting from all impedance measurements. A unified notation is presented in the following section for cases where the electrical properties (current or potential) are the measured output quantity and where the input forcing function is nonelectrical. A subsequent section addresses cases for which the input forcing fimction is electrical. 

Therefore, the elementary transfer functions that give the response of current and potential to the perturbation of the m input quantity are related by the electrochemical impedance. For a given experiment, only one input quantity is sinusoidally modulated aroimd a mean value, and all others are maintained constant by different regulations. 

Remember 14.3 Equation (14.22) provides the relationship between the usual electrochemical impedance response and the transfer function for cases where current or potential is the output. Equation (14.26) provides the corresponding relationship for cases where current or potential is the input. Equation (14.22) and equation (14.26) are distinguished by a minus sign resulting from the implicit function derivation. 

The examples of transfer functions presented here involve exclusively electrical quantities, but are distinct from the usual impedance measurements described in other parts of this book. 

By introducing the Faradaic impedance of the disk electrode Zdisk/ the transfer function can be written as... 

The transfer function V can be determined from the Faradaic impedance Zdisk and the transfer function M (v). It is an important kinetic parameter that allows evaluation of the frequency dependence of the amoimt of charge stored at the electrode surface. 

Remember 14.4 Transfer functions serve to isolate the influence of specific independent variables that contribute to the electrochemical impedance response of a system. 

The usual derivation based on the linearization procedure allows calculation of the electrochemical impedance and the M/U transfer function. Two types of behavior occur, depending on whether the average mass of the electrode changes continuously with time. 

---