Capacitive coupling model

A further assumption is that the time-averaged current is zero, because of the capacitive coupling of the power supply to the electrodes. With this model unequal-size electrode systems can be modeled, by using an area ratio a = Ag/Ap. The ratio between Vdc and Vrf has been calculated for 1 < a < 10, and is in excellent agreement [174, 175] with experimental data [176, 177]. 

Dielectric losses arise from the direct capacitive coupling of the coil and the sample. Areas of high dielectric loss are associated with the presence of axial electric fields, which exist half way along the length of the solenoid, for example. Dielectric losses can be modeled by the circuit given in Figure 2.5.3. The other major noise source arises from the coil itself, in the form of an equivalent series resistance, Rcoii. Exact calculations of noise in solenoidal coils at high frequencies and small diameters are complex, and involve considerations of the proximity and skin depth effects [23],... 

The above set of equations applies as shown to capacitively coupled systems. An identical set applies to the modeling of semiconductor devices [149]. Also, the similarity with the equations governing electrochemical systems (Section 8) is striking. The main difference is that continuity equations for electrons are not necessary, since there are no free electrons in the solution. 

Electric coupling of the body to environmental sources is usually due to proximity capacitance and, to a lesser extent, inductive (magnetic) fields. Capacitive coupling between two objects results from the electric field between them. The space or air gap separating the objects acts as a dielectric. The coupling can be simply modeled as a parallel-plate capacitor, which can be determined by... 

Lumped inductors in series connection are usually described by the n model as shown in Figure 9.59. The shunt capacitors depict the parasitic capacitances of coil windings to ground. Capacitive coupling between windings cause the self-resonance frequency /, as shown in Equation 9.38. Because of increasing effective inductance values in the vicinity of a parallel resonance, the inductors can only be used far below the self-resonance frequency (Figure 9.60). 

Modeling of High-Speed Interconnections. Modeling the electrical behavior of an interconnection involves two steps. First, the transmission line characteristics, such as the characteristic impedance, propagation constant, capacitance, resistance, dielectric conductance, and coupling parameters, must be calculated from the physical dimensions and material properties of the interconnection. In addition, structures, such as wire bonds, vias, and pins, must be represented by lumped resistance (R), inductance (L), and capacitance (C) elements. 

In this chapter, we will review the fundamental models that we developed to predict cathode carbon-support corrosion induced by local H2 starvation and start-stop in a PEM fuel cell, and show how we used them to understand experiments and provide guidelines for developing strategies to mitigate carbon corrosion. We will discuss the kinetic model,12 coupled kinetic and transport model,14 and pseudo-capacitance model15 sequentially in the three sections that follow. Given the measured electrode kinetics for the electrochemical reactions appearing in Fig. 1, we will describe a model, compare the model results with available experimental data, and then present... 

The pseudo-capacitive effect can be incorporated in the coupled kinetic and transport model through Eqs. (19) and (20). Here we choose to illustrate the effect through the kinetic model for simplicity. With considering the pseudo-capacitive current density, the kinetic model becomes... 

Here / is the current density with the subscript representing a specific electrode reaction, capacitive current density at an electrode, or current density for the power source or the load. The surface overpotential (defined as the difference between the solid and electrolyte phase potentials) drives the electrochemical reactions and determines the capacitive current. Therefore, the three Eqs. (34), (35), and (3) can be solved for the three unknowns the electrolyte phase potential in the H2/air cell (e,Power), electrolyte phase potential in the air/air cell (e,Load), and cathode solid phase potential (s,cath), with anode solid phase potential (Sjan) being set to be zero as a reference. The carbon corrosion current is then determined using the calculated phase potential difference across the cathode/membrane interface in the air/air cell. The model couples carbon corrosion with the oxygen evolution reaction, other normal electrode reactions (HOR and ORR), and the capacitive current in the fuel cell during start-stop. 

Fig. 9.22 Coupled lumped model with electrode capacitance.

The Coupled Lumped Simulink model presented in Section 9.5.1 was extended to analyze for the double layer capacitance behavior using the model given in Section 9.3.3. The Simulink block for calculating the capacitance behavior is shown in Figure 9.21. Results for a load decrease and increase are shown in Figure 9.22. Figure 9.23 shows the same data for the load decrease case but over an expanded timebase to more clearly show the transient behavior due to the electrochemical capacitance. Similar results are found for the load increase case. 

A first-order system coupled with dead time (transportation lag) is a good model for many process systems. The dead time (L or td) is the time that has to elapse before the output first starts to respond to a change in the input. The effect of a change in steam rate on the water temperature at the end of the pipe will depend not only on the resistance and capacitance effects in the tank but will also be influenced by the length of time necessary for the water to be transported through the pipe. The effect of dead... 

Here, w = m, n, and S. V represents the membrane potential, n is the opening probability of the potassium channels, and S accounts for the presence of a slow dynamics in the system. Ic and Ik are the calcium and potassium currents, gca = 3.6 and gx = 10.0 are the associated conductances, and Vca = 25 mV and Vk = -75 mV are the respective Nernst (or reversal) potentials. The ratio r/r s defines the relation between the fast (V and n) and the slow (S) time scales. The time constant for the membrane potential is determined by the capacitance and typical conductance of the cell membrane. With r = 0.02 s and ts = 35 s, the ratio ks = r/r s is quite small, and the cell model is numerically stiff. The calcium current Ica is assumed to adjust immediately to variations in V. For fixed values of the membrane potential, the gating variables n and S relax exponentially towards the voltage-dependent steady-state values noo (V) and S00 (V). Together with the ratio ks of the fast to the slow time constant, Vs is used as the main bifurcation parameter. This parameter determines the membrane potential at which the steady-state value for the gating variable S attains one-half of its maximum value. The other parameters are assumed to take the following values gs = 4.0, Vm = -20 mV, Vn = -16 mV, 9m = 12 mV, 9n = 5.6 mV, 9s = 10 mV, and a = 0.85. These values are all adjusted to fit experimentally observed relationships. In accordance with the formulation used by Sherman et al. [53], there is no capacitance in Eq. (6), and all the conductances are dimensionless. To eliminate any dependence on the cell size, all conductances are scaled with the typical conductance. Hence, we may consider the model to represent a cluster of closely coupled / -cells that share the combined capacity and conductance of the entire membrane area. 

Fig. 58. Calculated plot of the double layer potential as a function of position and time for a low value of the double layer capacitance. Model geometry ring-shaped WE and CE, symmetric RE close to the CE (weak negative global coupling). (Reproduced form A. Birzu, B. J. Green, N. I. Jaeger, J. L. Hudson, J. Electroanal. Chem. 504 (2001) 126, with permission of Elsevier Science.)...

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