Frequency Dependence of Impedance

For a low frequency f fc ifc- critical frequency, which will be defined later) 

Similarly to the conductor internal impedance explained earlier, the earth-return impedance in Equation 1.15 is frequency dependent as the penetration depth is frequency dependent. Equation 1.15 is approximated considering ln(l + X) for a small x by 

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Fig. 5.9 Frequency dependence of impedance of fresh electrodes of different Ru content in 5 M NaCl at room temperature at OCP. 

Figure 8.8 Series RC circuit (a) circuit (b) current-voltage relationships (c) frequency dependence of impedance Z and phase angle .

Similar conclusions on the character of conductance in the polycrystalline diamond films were derived in [33], The resistive intercrystallite boundaries can involve nonlinear resistance in polycrystalline diamond films moderately doped with boron [34]. Later, more sophisticated analysis [35-37] of the frequency dependence of impedance of polycrystalline diamond films resulted in a conclusion that at higher temperatures, in addition to the aforementioned electric conductance caused by the motion of free holes in the valence band, a second component of conductance manifests itself. The second component is due to the hopping of charge carriers between local traps possibly associated with the intercrystallite boundaries. 

Two features manifest themselves in the impedance behavior of diamond electrodes, namely, the effects of semiconductor nature of diamond (with moderately doped films) and a specific type of frequency dependence of impedance, which is characteristic of a great majority of films. 

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... 

Dutoit, E.C., VanMeirhaeghe, R.L., Cardon, F., Gomes, W.P. Investigation on frequency-dependence of impedance of nearly ideally polarizable semiconductor electrodes CdSe, CdS and Ti02. Ber. Bunsenges. Phys. Chem. 79, 1206-1213 (1975)... 

FIGURE 5.1.9 Frequency dependence of impedance of the several 100 tiF capacitors. MLC, multilayer ceramic capacitor TAG, Ta electrolytic capacitors ALC, A1 electrolytic capacitors SPTAC, Ta electrolytic capacitors with polymer electrode OSALC, A1 electrolytic capacitors with polymer electrode. 

Neufeld, P. and Queenan, E. D., Frequency Dependence of Polarisation Resistance Measured with Square Wave Alternating Potential , Br. Corros. J., 5, 72-75, March (1970) Fontana, M. G., Corrosion Engineering, 3rd edn., McGraw-Hill, pp 194-8 (1986) Dawson, J. L., Callow, L. M., Hlady, K. and Richardson, J. A., Corrosion Rate Determination By Electrochemical Impedance Measurement , Conf. On-Line Surveillance and Monitoring of Process Plant, London, Society of Chemical Industry (1977)... 

Figure 5 presents the capacitance-frequency dependence from impedance spectroscopy measurements for CS48 and CS15 in acidic and organic medium. In the low frequency region (from ImHz to lOOmHz) nearly a complete penetration of the ions into the pores is allowed and the quite stable values indicate the domination of the capacitive behavior at the electro 1 ytc/carbon interface. All the curves show a typical drop of... 

Frequency dependent complex impedance measurements made over many decades of frequency provide a sensitive and convenient means for monitoring the cure process in thermosets and thermoplastics [1-4]. They are of particular importance for quality control monitoring of cure in complex resin systems because the measurement of dielectric relaxation is one of only a few instrumental techniques available for studying molecular properties in both the liquid and solid states. Furthermore, It is one of the few experimental techniques available for studying the poljfmerization process of going from a monomeric liquid of varying viscosity to a crosslinked. Insoluble, high temperature solid. 

In situ frequency dependent electromagnetic-impedence measurements provide a sensitive, convenient, automated technique to monitor the changes in macroscopic cure processing properties and the advancement of the reaction in situ in the fabrication tool. This chapter discusses the instrumentation, theory, and several applications of the techniques, including isothermal cure, complex time—temperature cure, resin film infusion, thick laminates, and smart, automated control of the cure process. 

The first two quantities calculated from S(t, y) depended on measurements of times the second two depend on measurements of amplitudes. From the amplitude of the reflection from the top surface of the cell the impedance, and hence the density, can be found using eqns (8.58) and (8.59). The impedance is plotted in Fig. 9.4(c). Finally, knowing the thickness and the impedance of the cell, the attenuation can be deduced from the amplitude of the echo from the interface between the cell and the substrate the weaker this echo, the greater the attenuation. The attenuation calculated from (8.60), neglecting frequency dependence, is plotted in Fig. 9.4(d). It is also possible to calculate the frequency dependence of the attenuation using (8.70) (Daft et al. 1989). 

What Is Impedance Spectroscopy The words impedance spectroscopy imply the dependence of impedance on a wavelength and therefore on frequency. The frequency here is not that of an incident light beam, but of an alternative current applied to a cell, and then the question is What is impedance ... 

Fig. 3. The frequency dependence of Pt L electrode impedance active component. L 1-0.5M K2S04 2- 0.1M K3Fe(CN)6 / 0.1M K4Fe(CN)6 3- 0.5M H2S04.

Fig. 5. The frequency dependence of Pt 0.5M H2S04 electrode impedance reactive component at relative length of extended conductor electrode parties equal to 1- 1.0 2- 0.8 3-0.64 4- 0.52 5- 0.40.

Frequency dependence of the three components of impedance (resistive, capacitative, and inductive), showing the resonance frequency co0 = (LQ y2. 

It should be noted that the presence of diffusion controlled corrosion processes does not invalidate the EIS method but does require extra precaution. In the case of a finite diffusional impedance added in series with the usual charge transfer parallel resistance shown in Fig. 3b, the frequency-dependent diffusional impedance can be described as (21)... 

Finally, the basic equivalence of the two measuring techniques should be appreciated. Although there are many ways to approach such a comparison, the following simplified explanation will, we hope, give a more intuitive feeling for the relationship between EIS and PR measurements. As stated above, both techniques rely on the frequency dependence of the impedance of the double-layer capacitance in order to determine the polarization resistance. EIS uses low frequencies to force the capacitor to act like an open circuit. PR measurements use a slow scan rate to do the same thing. To make comparisons, the idea of equivalent scan rate is useful. Suppose that a particular electrochemical system requires EIS measurements to be made down to 1 mHz in order to force 99% of the current through Rp. What would the equivalent scan rate be for PR measurements A frequency of 1 mHz corresponds to a period of 1000 s. If the sine wave is... 

In general, the frequency dependence of the dif-fusional impedance and the geometry of diffusion are correlated. The (ice)-1/2 frequency dependence corresponds to the semi-infinite planar diffusion such a frequency dependence is valid only if the characteristic length, (D/ce)1/2 of diffusion is much shorter than any size of the electrode or the thickness of the electrolyte layer from which diffusion proceeds. Otherwise spherical, or bounded diffusion with different frequency dependence is observed. 

Immittance — In alternating current (AC) measurements, the term immittance denotes the electric -> impedance and/or the electric admittance of any network of passive and active elements such as the resistors, capacitors, inductors, constant phase elements, transistors, etc. In electrochemical impedance spectroscopy, which utilizes equivalent electrical circuits to simulate the frequency dependence of a given elec-trodic process or electrical double-layer charging, the immittance analysis is applied. 

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