Electrical equivalent model circuits

Figure 3 Electrical equivalent circuit model commonly used to represent an electrochemical interface undergoing corrosion. Rp is the polarization resistance, Cd] is the double layer capacitance, Rct is the charge transfer resistance in the absence of mass transport and reaction intermediates, RD is the diffusional resistance, and Rs is the solution resistance, (a) Rp = Rct when there are no mass transport limitations and electrochemical reactions involve no absorbed intermediates and nearly instantaneous charge transfer control prevails, (b) Rp = Rd + Rct in the case of mass transport limitations.

Figure 5 Nyquist, Bode magnitude and Bode phase angle plots for hypothetical corroding interfaces with Rp = 10, 100, or 1,000 ohms, Cd, = 100 tF, and Rs = 10 ohms using the electrical equivalent circuit model of Fig. 3a.

Fig. 10.11 The proposed electrical equivalent circuit model for defective polyester-coated galvanized mild steel at different relative humidity [49].

Fig. 2 Electrical equivalent circuit models for a TSM resonator (a) transmission line model (TLM) and (b) Butterworth-vanDyke lumped element model (LEM). Circuit elements are defined in the main text.

Many treatments of this subject have used an electrical equivalent circuit model to simulate the corroding metal/ electrolyte interface [1,38,39]. The simplest form of such a model is shown in Fig. 4. The three parameters discussed above... 

FIG. 4—Electrical equivalent circuit model simulating a simple corroding metal/electrolyte Interface. See also Fig. 5. R. is the solution resistance. R, is the polarization resistance. C Is the double layer capacitance. 

If the electrical equivalent circuit model is a series connection of model elements, the overall model equation is the vector addition of the single model element equations in time domain and frequency domain, respectively. Time domain correspondences for single model elements to certain excitation signals can be calculated analytically as described in the following sections. In general these transformations can be easily computed for models consisting of lumped elements. For models containing distributed elements, fractional calculus is required. 

An electrical equivalent circuit model, composed of a resistor and a capacitor which were connected in series, was adopted to represent the soymilk-electrode system. High frequency impedance measurements presented an increase in resistance and constant value in reactance for different coagulation times. This resistance is attributed to the bulk resistance of soymilk mixture and changed minimally when the coagulation process is finished [50]. 

A model for the ac response of real electrodes is the simple electric equivalent circuit consisting of a resistance R and capacitance Q conneeted in series (Fig. 12.12a). It follows from the rules for ac circuits that for this combination... 

A generalised model of electrical equivalent circuit for painted surfaces has been considered in many of the recent publications. Googan ( 2) used it to study vinyl coatings free of defects and coatings containing defects. Electrocoatings were also evaluated. Muslanl et al (27) in their investigation of mild steel... 

Modeling and optimization of chemical sensors can be assisted by creating equivalent electrical circuits in which an ordinary electrical element, such as a resistor, capacitor, diode, and so on, can represent an equivalent nonelectrical physical parameter. The analysis of the electrical circuit then greatly facilitates understanding of the complex behavior of the physical system that it represents. This is a particularly valuable approach in the analysis and interpretation of mass and electrochemical sensors, as shown in subsequent chapters. The basic rules of equivalent circuit analysis are summarized in Appendix D. Table 3.1 shows the equivalency of electrical and thermal parameters that can be used in such equivalent circuit modeling of chemical thermal sensors. 

Electrical equivalent circuit representing the model of Ershler-Randles. 

To understand the electrical behaviour of the LAPS-based measurement, the LAPS set-up can be represented by an electrical equivalent circuit (see Fig. 5.2). Vbias represents the voltage source to apply the dc voltage to the LAPS structure. Re is a simple presentation of the reference electrode and the electrolyte resistance followed by a interface capacitance Cinterface (this complex capacitance can be further simulated by different proposed models as they are described, e.g., in Refs. [2,21,22]). In series to the interface capacitance, the insulator capacitance Cj will summarise the capacitances of all insulating layers of the LAPS device. The electrical current due to the photogeneration of electron-hole pairs can be modelled as current source Ip in parallel to the... 

FIGURE 11.9 Basic capacitor electrical equivalent circuit comprising a capacitance, a series inductance, a series resistance, and a parallel resistance. This simple model can fit a DLC behavior in first approximation for a given frequency. 

One can show [42] that, when the surface mechanical impedance is not large, the distributed model in the vicinity of resonance (where we make measurements) can be reduced to the simpler lumped-element model of Fig. 13.8(b). This modified Butterworth-van Dyke (BVD) electrical equivalent circuit comprises parallel static and motional arms. The static... 

The complications and sources of error associated with the polarization resistance method are more readily explained and understood after introducing electrical equivalent circuit parameters to represent and simulate the corroding electrochemical interface (1,16-20). The impedance method is a straightforward approach for analyzing such a circuit. The electrochemical impedance method is conducted in the frequency domain. However, insight is provided into complications with time domain methods given the duality of frequency and time domain phenomena. The simplest form of such a model is shown in Fig. 3a. The three parameters (Rp, Rs, and C d,) that approximate a corroding electrochemical inter-... 

EIS data analysis is commonly carried out by fitting it to an equivalent electric circuit model. An equivalent circuit model is a combination of resistances, capacitances, and/or inductances, as well as a few specialized electrochemical elements (such as Warburg diffusion elements and constant phase elements), which produces the same response as the electrochemical system does when the same excitation signal is imposed. Equivalent circuit models can be partially or completely empirical. In the model, each circuit component comes from a physical process in the electrochemical cell and has a characteristic impedance behaviour. The shape of the model s impedance spectrum is controlled by the style of electrical elements in the model and the interconnections between them (series or parallel combinations). The size of each feature in the spectrum is controlled by the circuit elements parameters. 

In the previous section we considered the conditions under which mechanical resonances would occur in a TSM resonator. In considering only the mechanical properties of the crystal, however, we neglected consideration of how these resonances would actually be excited or detected. The device uses a piezoelectric substrate material in which the electric field generated between electrodes couples to mechanical displacement. This allows electrical excitation and detection of mechanical resonances. In constructing a practical sensor, changes in resonant frequency of the device are measured electrically. The electrical characteristics of the resonator can be described in terms of an equivalent-circuit model that describes the impedance (ratio of applied voltage to current) or admittance (reciprocal of impedance) over a range of frequencies near resonance. 

Figure 3.5 Equivalent-circuit models to describe the near-resonant electrical characteristics of the resonator (a) distributed model (b) lumped-element model. (Reprinted with permission. See Refs. [7 14J. (a) 1994 American Institute of Physics and (b) 1993 American Chemical Society.)...

TSM resonator electrical characteristics are typically described in terms of electrical admittance, defined as the ratio of current flow to applied voltage (the reciprocal of impedance). The total TSM resonator admittance can be determined from inspection of the equivalent circuit model ... 

The equivalent circuits (Figure 3.5) can be used to describe the electrical response of the perturbed device. The lumped-element model. Figure 3.Sb, is most convenient to use. When the resonator has a surface perturbation, the motional impedance increases, as represented by the equivalent-circuit model of Figure 3.7. This model contains the elements C , Li, C, and Ri corresponding to the unperturbed resonator. In addition, the surface perturbation causes an increase in the motional impedance Z(n as described by the complex electrical element Ze in Figure 3.7a. This element is given by [12]... 

The electrical characteristics of the TSM resonator with a generalized surface perturbation can be described by the equivalent-circuit model of Figure 3.7b [14]... 

Yang and Thompson [21] have noted that when a TSM resonator is operated in a liquid, fringing electric fields can enter the liquid, making Co sensitive to the dielectric properties of the liquid. This sensitivity, which can be considered to arise from changes in the parasitic capacitance Cp, is especially pronounced when both electrodes are immersed. Tiean et al. [22] have noted that under these circumstances, a parallel conductance must be added to the equivalent-circuit model to account for conduction through the liquid between electrodes. 

The electrical characteristics of the TSM resonator with a generalized surface perturbation can be described by the equivalent-circuit model of Figure 3.7b. Measurements can be made on a dry TSM resonator to determine C o,L, Ci, and R. Fixing these parameters and fitting the equivalent-circuit model to data measured on an immersed device determines R2 and L2. Equations 3.21 can then be used to determine the components of from L2 and R2. 

The equivalent circuit model of Figure 3.7 can be used to describe the near-resonant electrical characteristics of the quartz resonator coated by a viscoelastic film. The surface film causes an increase in the motional impedance, denoted by the complex element Zg. From Equation 3.19, this element is proportional to the ratio of the surface mechanical impedance Zj contributed by the film to the characteristic shear wave impedance Zq of the quartz. 

In this section we will describe how a proper accounting for film dynamics, based on a model of the thin-film/acoustic-wave interactions, can be used to quantitatively evaluate the shear modulus values as a function of temperature. As described in Section 3.1, an equivalent-circuit model can be used to relate the measured TSM electrical characteristics to the elastic properties, density, and thickness of a polymer film coating the device. Consequently, measurements made with polymer-coated TSM devices can be used to extract the shear elastic properties of the film. 

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