Frequency-dependent complex impedance

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. 

The quartz disk is used as the bottom plate of a cell culture vessel and is moimted in a temperature controlled crystal holder (37 °C). The surface electrodes on either side of the quartz are connected to an impedance analyzer (Solatron Instruments, SI-1260) operating in continuous wave mode. The frequency-dependent complex impedance Z(J) returned by the impedance analyzer is expressed as magnitude of impedance Z (f) and phase shift between voltage and current (f). The raw data is analyzed by the well-known Butterworth-Van Dyke (BVD) equivalent circuit with the liunped impedance elements Co, Rq, iq, Cq and Zl. Rq, Lq and Cq represent the piezoelectric properties of the unperturbed resonator itself, whereas Co summarizes its dielectric properties and all parasitic contributions arising from contacts and wiring. The load material in contact with the resonator surface is represented by the complex impedance Zl. As long as the resonator is not loaded too... 

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. 

Since sound speed is a frequency dependent complex quantity, it therefore follows that the characteristic impedance of the media will also be frequency dependent and complex. If the frequency dependence of sound speed is not known, it can be estimated from the attenuation coefficient as follows. For the rubber composites of interest here, usually a A is essentially independent of frequency. Using Kramers-Kronig relationships (5) it can then be shown that ... 

In the 1920s, impedance was applied to biological systems, including the resistance and capacitance of cells of vegetables and the dielectric response of blood suspensions. ° Impedance was also applied to muscle fibers, skin tissues, and other biological membranes. " The capacitance of the cell membranes was found to be a function of frequency, and Fricke observed a relationship between the frequency exponent of the impedance and the observed constant phase angle. In 1941, brothers Cole and Cole showed that the frequency-dependent complex... 

The energy dissipative and inertia-free resistance R impedes not only the current flow -energy-conserving inert elements, capacitance C, and inductance L are also included in the complex impedance by their frequency-dependent reactive impedances Xc(m) and Xl(co). 

Frequency dependent electrode impedance is also important in bioelectric stimulation applications, where relatively larger currents and complex waveforms are often used to stimulate excitable tissues. 

Electrochemical impedance is usually measured by applying an AC potential to an electrochemical cell and then measuring the current through the cell (Barsoukov and Macdonald, 2005, Ivers-Tiff et al., 2003, Orazem and Tribollet, 2008, Springer et al., 1996). The response to this potential is an AC current signal. According to ASTM G-15, the definition of electrochemical impedance is the frequency-dependent, complex valued proportionality factor, AE/Ai, between the applied potential (or current) and the response cmrent (or) potential in an electrochemical cell. This factor becomes the impedance when the perturbation and response are related linearly (the factor value is independent of the perturbation magnitude) and the response is caused only by the perturbation. 

In the equivalent circular base approach the impedance function of a foundation is obtained from the elastic solution of a rigid massless circular base resting on the surface of the soil for each degree of freedom independently. The impedance function for each degree of freedom is a frequency dependent complex expression, where its real part represents the elastic stiffness (spring constant) of the soil-foundation system and its imaginary part represents the damping in the soil-foundation system. The impedance function is expressed as ... 

F/gwre 5 JO, (a) Complex impedance spectra (Nyquist plots) of the CH4,02) Pd YSZ system at different Pd catalyst potentials. Open circuit potential U R =-0.13 V. Dependence on catalyst potential of the individual capacitances, C4i (b) and of the corresponding frequencies, fmii, at maximum absolute negative part of impedance (c).54 Reprinted with permission from Elsevier Science. 

Frequency-dependent measurements of the materials dielectric impedance as characterized by its equivalent capacitance, C, and conductance, G, are used to calculate the complex permitivity, e = d — id, where co = 2nf, f is the measurement frequency, and C0 is the equivalent air replacement capacitance of the sensor. 

The variables are frequency-dependent and represent Y (admittance), Z (impedance), V (voltage), and I, or i (current). The relationship between angular (on) and linear (F) frequency is o) 2nF. Both the admittance and the impedance are complex numbers, consisting of real and imaginary parts. Thus for admittance... 

The above-described situation is but an exception rather than the rule. Generally, the diamond electrode capacitance is frequency-dependent. In Fig. 12 we show a typical complex-plane plot of impedance for a single-crystal diamond electrode [69], At lower frequencies, the plot turns curved (Fig. 12a), due to a finite faradaic resistance Rp in the electrode s equivalent circuit (Fig. 10). And at an anodic or cathodic polarization, where Rf falls down, the curvature is still enhanced. At higher frequencies (1 to 100 kHz), the plot is a non-vertical line not crossing the origin (Fig. 12b). Complex-plane plots of this shape were often obtained with diamond electrodes [70-73],... 

However, real electrochemical systems exhibit much more complex behaviours. They are not simply resistive. The electrochemical double layer adds a capacitive term. Other electrode processes, such as diffusion, are time and/or frequency dependent. Therefore, for an actual electrochemical system, impedance is used instead of resistance. The impedance of an electrochemical system (defined as Ziot)) is the AC response of the system being studied to the application of an AC signal (e.g., sinusoidal wave) imposed upon the system. The form of the current-voltage relationship of the impedance in an electrochemical system can also be expressed as... 

The complex-plane impedance diagram is given in Figure 4.126. At both high and medium frequencies the complex-plane impedance is characterized by a well pronounced semicircle, while at the low-frequency range a tail appears. This tail s shape is strongly dependent on the value of the CPE exponent, as can be seen in AppendixD (Model Dll). 

Impedance data are often represented in complex-impedance-plane or Nyquist format, as shown in Figure 16.1. The data are presented as a locus of points, where each data point corresponds to a different measurement frequency. One disadvantage of the complex-impedance-plane format is that the frequency dependence is obscured. This disadvantage can be mitigated somewhat by labeling some characteristic frequencies. In fact, characteristic frequencies should always be labeled to allow a better understanding of the time constants of the underlying phenomena. 

Figure 6. Complex impedance curves depending on (A) the applied frequency at a constant potential of -0.40 V vs. SCE and (B) the applied potential at a constant frequency of 1000 Hz for Ti( )2/Ti ordinary and long nanotube array.

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