Impedance equation

The impedance equations used here are cited in terms of o) (in radians per second)... 

Remember 11.2 The Warburg impedance, equation (11.52), applies for diffusion in an infinite stagnant domain. This expression applies as a high-frequency limit for diffusion in a finite domain. 

In case of the appearance of the blocking reaction layer on the surface of zirconia with polarization resistance Rf (see Figure 4.17, b), the operative impedance equation would read as follows ... 

B. In order to calculate the reaction faradaic impedance equations describing the current, j(ti, 0) [Eq. (127)] and the rate of formation of absorbed species (i], 0) [Eq. (129)] should be linearized, giving ... 

Here, the substitution p = jco has been made to obtain the impedance. Equation (22) leads to the Randles equivalent circuit for an electrode reaction and is equivalent to eqn. (70) given below. In all electrode reactions, the impedance due to the reaction must have a differential capacity in parallel with it, to give the impedance of the interface... 

Such decompositions can of course also be made of the Cole impedance equation as well as the conductivity and permittivity equations. Equations 9.30 and 9.31 illustrate how extremely compact the Cole and Cole—Cole equations are. 

The addition of salt to a polymer may lead to formation of dipoles between ions and chain molecules. These dipoles have a (longitudinal) component along the chains as it is sketched in Figure 25. The impedance measurements reflect relaxation of these longitudinal dipole components since they relax sufficiently slowly. Dipole relaxations occur as maxima in plots of Z as a function of frequency. It is analogous to complex viscosity sketched in Figure 15. Real and imaginaiy part of complex impedance (Equation 53) read ... 

The complex impedance (Equation 48) might be generalized as indicated in Equation (49). It reads ... 

The relationship between the real and imaginary impedance of eireuit that eon-sists of elements sueh as resistors and eapaeitors can be represented by a Nyquist plot. If the impedanee of a circuit exhibits a Nyquist plot that is similar to that exhibited by a polymer electrolyte, then this circuit model can be used to represent the electrolyte and is known as the equivalent circuit for the electrolyte. The impedance data is commonly analyzed by fitting them to the impedance equation of the equivalent circuit. Now, we will diseuss the selected eireuit elements together with their corresponding Nyquist plots. 

To calculate the impedance, equations for the total current and the changes in the surface coverage must be linearized ... 

In this case, the impedance equation may be rearranged into... 

A few examples of the complex plane plots that can be obtained in such a case are shown in Fig. 5.6. Many other possible plots are discussed in Ref. [225]. In a similar way, impedance equations for three [11, 57, 226] or more adsorbed species can be obtained [227]. Macdonald et al. [1, 227] studied A1 corrosion in KOH and considered three adsorbed species of aluminum and adsorbed hydrogen. They considered six possible mechanisms and found one that explained well changes of the impedance with the electrode potential. Examples of the experimental... 

This is the traditional semicircle complex impedance equation. 

The more frequently used parameter to describe the loss behavior of coils is the quality factor. Several definitions are available in the literature [57]. The quality factor, in general, is defined as the ratio between the imaginary part and Ihe real part of the impedance (Equation 9.39) ... 

The electrode impedance equation has positive signs for the anode impedance and negative for the cathode impedance. Equation (8.56) was the time... 

Differentiating Equation 5.116 over x and calculating the impedance (Equation 5.93), gives... 

Tests with a 1. 8-Ah MS-TiO cell demonstrated outstanding power 97% of the capacity measured at the 1C discharge rate was delivered at the 50C rate (Fig. 1) [11]. These results were eorrelated to obtain the impedance equations required for the vehicle simulation tests. 

At the minimal c.d., Bp, typical diffusion control is observed consistent with the local plateau of the current-voltage curve. A diffusion term is associated with a transport limitation across the passive layer, whereas the inductive loop at the low-ffequency end visualizes the contribution of a deca5ung film protection at increasing potentials. Application of a diffusion impedance equation to a finite-thickness layer (5 = 3 nm) given by... 

Using Eq. 1-2 for resistance R and Eq. 1-3 for capacitance C, which are frequency-dependent parameters when analyzed over a broad frequency range, a general impedance equation (Eq. 1-10) can be developed as a function of the frequency-dependent characteristic electrical properties of the analyzed material or system. This expression takes into account the frequency-dependent values of relative permittivity e((o) and resistivity p(co) or its inverse, electrical conductivity a(co), combined with the geometry factors of the electrode surface area (/4) and the sample s thickness between the bounding electrodes (d) ... 

For the liquid methane example in Table 4,1, the surface evaporation impedance equation, with X=512 kJ/kg and effective thermal conductivity (in region 2) keff=0.30 W/mK compared with the static value of 0.19 W/mK, becomes ... 

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