Current distribution frequency dependent

Combining these two properties, it is easily seen that STSA transforms the original broad band noise into a signal composed of short-lived tones with randomly distributed frequencies. Moreover, with a standard suppression rule (one that depends only on the relative signal level as measured in the current short-time frame) this phenomenon can only be eliminated by a crude overestimation of the noise level. Using the result of Eq. 4.20 in the case where Q= 1, it is easily shown that the overestimation needed to make the probability of appearance of musical noise negligible (below 0.1%) is about 9 dB [Cappe, 1991],... 

It must be emphasized that the mathematical simplicity of equations (13.1) and (13.2) is the consequence of a specific time-constant distribution. As shown in this chapter, time-constant distributions can result from nonuniform mass transfer, geometry-induced nonuniform current and potential distributions, electrode porosity, and distributed properties of oxides. At first glance, the associated impedance responses may appear to have a CPE behavior, but the frequency dependence of the phase angle shows that the time-constant distribution differs from that presented in equation (13.7). 

The results obtained during the Couette flow of aqueous solutions of polyethylene oxide and other water-soluble polymers appear especially promising since they showed an appreciable increase in the current noise level with shear rate. The current noise level depended also on the viscosity (molecular weight) of the solution. A slight increase of thermal noise was recorded also. The pseudoplasticity exponent n in the Ostwald-de Waele power law formula and the exponent a in the l/f -frequency distribution of the current noise were interrelated. This relation appeared to be generally valid. 

In all real systems, some deviation from ideal behavior can he observed. If a potential is applied to a macroscopic system, the total current is the sum of a large number of microscopic current filaments, which originate and end at the electrodes. If the electrode surfaces are rough or one or more of the dielectric materials in the system are inhomogeneous, then all these microscopic current filaments would be different. In a response to a small-amplitude excitation signal, this would lead to frequency-dependent effects that can often be modeled with simple distributed circuit elements. One of these elements, which have found widespread use in the modeling of impedance spectra, is the so-called constant phase element (CPE). A CPE is defined as... 

The real importance of current distribution problems in impedance measurements, however, lies in the fact that the distribution is frequency-dependent. This arises because of the influence of interfacial polarization combined with the geometrical aspects of the arrangement. 

In general, interfacial impedance is partly capacitative as well as resistive in nature. At high frequencies, the capacitance short-circuits the interface, and the primary distribution is observed for the ac part of the current. As the frequency is lowered, the interface impedance increases, causing a changeover to the secondary distribution. Of necessity, this effect leads to a frequency dependence of the equivalent circuit parameters which describe the system. Of course, if the primary distribution is uniform, there will be no frequency dispersion arising from this source. 

The question of the frequency dependence of the current distribution and its effect on the measured impedance of a solid state electrochanical system has been hardly considered, although it is important in discussing the impedance of, for example, porous gas electrodes on anion conductors, of rough electrodes (discussed below), and also perhaps of polycrystalline materials. In aqueous electrochemical situations the effects has been considered with respect to the rotating disk electrode, where there may be severe current distribution problems. 

As pointed out by de Levie, however, the most important weakness in the model is the assumption that the current distribution is normal to the macroscopic surface, that is a neglect of the true current distribution. For a rough surface, the lines of electric force do not converge evenly on the surface. The double layer will therefore be charged unevenly, and the admittance will be time and frequency dependence. 

Bias-induced reverse piezoelectric response Broadband dielectric spectroscopy (BDS) Dielectric permittivity spectrum Dielectric resonance spectroscopy Elastic modulus Ferroelectrets Electrical breakdown Acoustic method Characterization Dynamic coefficient Interferometric method Pressure and frequency dependence of piezoelectric coefficient Profilometer Quasistatic piezoelectric coefficient Stress-strain curves Thermal stability of piezoelectricity Ferroelectric hysteresis Impedance spectroscopy Laser-induced pressure pulse Layer-structure model of ferroelectret Low-field dielectric spectroscopy Nonlinear dielectric spectroscopy Piezoelectrically generated pressure step technique (PPS) Pyroelectric current spectrum Pyroelectric microscopy Pyroelectricity Quasistatic method Scale transform method Scanning pyroelectric microscopy (SPEM) Thermal step teehnique Thermal wave technique Thermal-pulse method Weibull distribution... 

The current density pattern on the surface of an electrode depends on the electrode shape and position [9, 11, 12, 14, 17], It affects the corrosion behavior of the electrodes considerably. If electrode polarization is ignored, it was shown in [12] that on a disk electrode, with the surface in the same level as the surface of the surrounding insulator, the current density increases from the center of the disk while approaching the edge, with theoretically an infinite value at the edge. This assumption (no electrode polarization) can be made if the potential on the electrolyte side of the double layer is equal to that of the electrode. The current density under this condition is called primary current distribution. This state prevails at high frequency when the double layer capacitance behaves as a short circuit [14]. 

This same explanation can be given by applying the transformation matrices in Equations 1.176, Equations 1.177 and 1.178 in the case of a completely transposed line. When a line is untransposed, the transformation matrices are no longer useful — except Equation 1.179, which can be used as an approximation of a transformation matrix of an untransposed horizontal line. In the case of an untransposed line, the transformation matrix is frequency-dependent as explained in Section 1.5.1 thus, the modal voltage and current distributions vary as the frequency changes. Also, the current distribution differs from the voltage distribution. 

It is well known that current is distributed near a conductor s surface when its frequency is high. Under such a condition, the resistance (impedance) of the conductor becomes higher than that at a low frequency because the resistance is proportional to the cross section of the conductor. This is called frequency dependence of the conductor impedance. As a result, the propagation constant and the characteristic impedance are also frequency dependent. 

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