Impedance-Related Functions

The impedance has frequently been designated as the ac impedance or the complex impedance. Both these modifiers are redundant and should be omitted. Impedance without a modifier always means impedance applying in the frequency domain and usually measured with a monochromatic signal. Even when impedance values are derived by Fourier dansformation from the time domain, the impedance is still defined for a set of individual frequencies and is thus an alternating-current impedance in character. 

Impedance is by definition a complex quantity and is only real when 0=0 and thus Z(m) = Z(a ), that is, for purely resistive behavior. In this case the impedance is completely frequency-independent. When Z is found to be a variable function of frequency, the Kronig-Kramers (Hilbert integral transform) relations (Macdonald and Brachman [1956]), which holistically connect real and imaginary parts with each other, ensure that Z (and 9) cannot be zero over all frequencies but must vary with frequency as well. Thus it is only when Z( u) = Z, independent of frequency, so Z = R, m ordinary linear resistance, that Z(m) is purely real. 

There are several other measured or derived quantities related to impedance which often play important roles in IS. All of them may be generically called immit-tances. First is the admittance, Y = Z = Y + jY . In the complex domain where v, i, and Z are all taken complex, we can write v = Zi or alternatively i = Yv. It is also customary in IS to express Z and Y in terms of resistive and capacitance components as Z = I s(ft)) - yX/m) and Y = Gp(co) + jBp (o), where the reactance Xs = [ Cj(( )] and the susceptance Bp = (oCp( o). Here the subscripts s and p stand for series and parallel.  

The other two quantities are usually defined as the modulus function M = jcoCcZ = M + jM and the complex dielectric constant or dielectric permittivity e = AT = YI(j(t)Cc) = s -js . In these expressions C = SqAJI is the capacitance of the empty measuring ceU of electrode area Ac and electrode separation length /. The quantity fb is the dielectric permittivity of free space, 8.854 x 10 F/m. The dielectric constant e is often written elsewhere as e or to denote its complex character. Here we shall reserve the superscript asterisk to denote complex conjugation thus Z = Z - jZ . The interrelations between the four immittance functions are summarized in Table 1.1.1. 

The modulus function M = was apparently first introduced by Schrama [1957] and has been used appreciably by McCrum et al. [1976], Macedo et al. [1972], and Hodge et al. [1975, 1976]. The use of the complex dielectric constant 

---

For a linear system, the phase and amplitude of the impedance relate to each other. Consequently, if we know the frequency dependence of the phase we can calculate the amplitude of the impedance as a function of frequency. Similarly, we can deduce the frequency dependence of the phase from that of the amplitude of the impedance. The calculation can be achieved by the Kramers-Kronig (K-K) transforms. This is a useful check on the validity of a measured impedance spectrum. For information on K-K transforms, see Appendix C. 

Treatments of this problem are greatly simplified by uncoupling the long-term diffusion due to Edc from the rapid diffusional fluctuations due to Eac- We do that by recognizing that Edc sets up mean surface concentrations that look like bulk values to the ac perturbation because of the difference in time scale. In Section 10.3, we defined the faradaic impedance in terms of bulk concentrations thus the current response in ac voltammetry as a function of is readily obtained by substituting the surface concentrations imposed by directly into these impedance relations. Since this strategy is simple and intuitive, we will pursue it. More rigorous treatments are available in the literature for the interested reader (2, 3, 5). The results are the same by either approach. 

Both impedance and admittance are called immittances. Between other related functions, complex dielectric constant should also be mentioned [78]. It is used in the analysis of dielectric relaxation and is obtained from the measured admittance (impedance) of a cell with a given dielectric (liquid or solid) ... 

The determined eddy-eurrent parameter is the inductance of the eomplex impedance measured by impedance analyzer at j=100 kHz. Therefore the impulse response function from chapter 4.2.1. is used for calculation. The depth of the cracks is big in comparison to coil size. For presentation the measured and pre-calculated data are related to their maxima (in air). The path X is related to the winding diameter dy of the coil. 

Examination of the membranes with a variety of physicochemical techniques, from related electrochemical approaches (as electrochemical impedance spectroscopy (EIS), voltammetry and chronoamperometry) to more sophisticated characterization methods (spectroscopy and microscopy), actually serves the same end as the theory and leads to a deeper understanding of the chemistry behind the functioning of these sensors [5, 6],... 

Equation (108) is universal for any small amplitude perturbation method and may, in principle, be used as the starting point for the derivation of a response vs. time relation for a given perturbation function. It is easily verified [53] that substitution of s = ico into any expression of an operational impedance or admittance delivers the complex impedance or admittance as they are defined in Sect. 2.3.1. 

Due to its high potency and selectivity imetit (24) proved to be a valuable tool in pharmacological trials related to H3 receptors, but in spite of that the potential toxicity of its isothiourea moiety emerges to impede further pharmacological development. A similar situation arose with the H3 antagonist thioperamide, the clinical development of which was ceased due to hepatotoxicity which is most probably related to its thiourea functionality. 

It is useful to consider the solution of Maxwell s Equations (5.1) for plane electromagnetic waves in the absence of boundary conditions, which can be written as exp[i(/ 2 — u>t) assuming propagation in z-direction of cartesian coordinates. The quantity / is the complex propagation constant of the medium with dominant real part for dielectrics and dominant imaginary part for metals. The impedance of the medium, Z, defined as ratio of electric to magnetic field is related to / by Z = ojp,0/f3 with /x0 = 1.256 x 10 6 Vs/Am. As it can be derived from Maxwell s equations, the impedance is related to the conductivity/dielectric function by the following expression ... 

The impedance spectroscopy is most promising for electrochemical in situ characterization. Many papers have been devoted to the AB5 type MH electrode impedance analysis [15-17]. Prepared pellets with different additives were used for electrochemical measurements and comparing. Experimental data are typically represented by one to three semicircles with a tail at low frequencies. These could be described to the complex structure of the MH electrode, both a chemical structure and porosity [18, 19] and it is also related to the contact between a binder and alloy particles [20]. The author thinks that it is independent from the used electrolyte, the mass of the electrode powder and the preparing procedure of electrode. However, in our case the data accuracy at high frequencies is lower in comparison with the medium frequency region. In the case, the dependence on investigated parameters is small. In Figures 3-5, the electrochemical impedance data are shown as a function of applied potential (1 = -0.35V, 2 = -0.50V and 3 = -0.75V). 

---