Perturbation amplitude

For a linear dissipative system, an impedance may be defined, and the proportionality constant between power and the square of the perturbation amplitude is simply related to the impedance. In the electrical case... 

Figure 1.14. Scheme of an undulation perturbation of the interface of a thread with radius Tc- a(z, t) is the perturbation amplitude. The thread breaks with a characteristic wavelength into drops of diameter cIr. ... 

The summation over final states /> has been carried out by the completeness relation, or equivalently, by matrix multiplication. Thus, the zeroth moment is just an equilibrium average of the square of the perturbation amplitude. The first moment can likewise be expressed as an equilibrium property. 

Figure 14.5 Behavior of the perturbation-amplitude growth coefficients 1 /rv and l/rs...

We will consider the cold-gas-convex surface of the flame front as a curved cell of the flame which had been formed after the plane flame lost its stability. The steady state of the convex flame is a result of the nonlinear hydrodynamic interaction with the gas flow field (see Zeldovich, 1966, 1979). In the linear approximation the flame perturbation amplitude grows in time in accordance with Landau theory, but this growth is restricted by nonlinear effects. 

Guo et al. [7], as shown by the Nyquist plots in Figure 6.10. In their impedance measurements, different amounts of Nafion ionomer in the catalyst layer, ranging from 0.33 to 1.13 mg/cm2 (dry weight) were examined. The active area of their fuel cells was 1.0 cm2. The fuel cells were operated in H2/air gas feeding mode with a flow rate of 220 cm3/min (at standard temperature and pressure) for both sides. The cell temperature as well as the humidification temperature for both electrodes were controlled at 70°C. The cell s AC impedance was measured using a Gamry PC4/750-DHC2 potentiostat. The perturbation amplitude was set at 5 mV in potentiostatic mode, and the frequency was scanned from 0.01 Hz to 100 kHz with 10 points per decade. 

Figure 7.5 Lissajous representation of the signals presented in Figure 7.4 with frequency as a parameter. The signals were normalized by the perturbation amplitude such that the values for current and potential ranged between 1.

Example 8.1 Guideline for Linearity We wish to establish a guideline for the perturbation amplitude needed to maintain linearity under potentiostatic regulation. An electrochemical system that follows a Tafel law is polarized at a potential V. If a large potential sinusoidal modulation is superimposed, write the current response in the form of a Taylor series and calculate the complete expression of the dc current. By considering only the first three terms of the Taylor series, write the expression cf the current under the form of the first three harmonics. 

Remember 8.4 The optimal perturbation amplitude depends on the polarization curve for the system under study. 

Example 8.2 Influence of Ohmic Resistance on Linearity As an extension of Example 8.1, establish a guideline for the perturbation amplitude needed to maintain linearity under potentiostatic regulation for a system with a nonnegligible Ohmic resistance. 

Figure 8.4 Impedance results obtained for the time-domain results presented in Figure 8.3 by use of the Fourier analysis presented in Section 7.3.3 with potential perturbation amplitude AV as a parameter a) Nyquist representation b) real part of the impedance as a function of frequency and c) imaginary part of the impedance as a function of frequency.

Figure 8.7 Lissajous representation for current and applied potential time-domain signals corresponding to a system presented in Figure 7.8 but with a potential perturbation amplitude AV = 100 mV.

The frequency dependence of the surface overpotential is shown more clearly in Figure 8.9, where the magnitude of the surface overpotential is presented as a fvmction of frequency with the magnitude of the applied potential perturbation as a parameter. The approach to unity at low frequencies for small perturbations shows that the surface overpotential is properly scaled by Rt/ Rt + Re)- For larger perturbation amplitudes, the nonlinear response reduces the effective charge-transfer... 

The optimal perturbation amplitude may be best determined experimentally. Distortions in Lissajous plots at low frequency (see Figure 8.3) may be attributed to a nonlinear response. If the shape is distorted from an ellipse, one should reduce the amplitude. A second approach is to compare the impedcince response for several amplitudes as demonstrated in Figure 8.4. If the magnitude of the impedance at low frequencies depends on amplitude of perturbation, the perturbation amplitude is too large. 

Following the discussion presented in Example 8.2, estimate the effect an Ohmic resistance has on the maximum potential perturbation amplitude that can be applied to am electrochemical system while satisf5ong the guidelines presented in Example 8.1. 

Should the same potential perturbation amplitude be applied for all parts of the polarization curve Give examples to demonstrate your answer. 

Researchers have reported that, for impedance measurements on human skin under fixed-amplitude galvanostatic modulation, significant changes in skin properties were observed that could be attributed to the impedance measurement. The magnitude of the skin impedance varied from about 10 Ocm at high frequency to 100 fcOcm at low frequency. The perturbation amplitude was 0.1 mA on an exposed skin sample of 1 cm area. Explain the reasons for their observation and suggest an improved experimental protocol. 

All impedance measurements should begin with measurement of a steady-state polarization curve. The steady-state polarization curve is used to guide selection of an appropriate perturbation amplitude and can provide initial hypotheses for model development. The impedance measirrements can then be made at selected points on the polarization curve to explore the potential dependence of reaction rate constants. Impedance measurements can also be performed at different values of state variables such as temperature, rotation speed, and reactant concentration. Impedance scans measured at different points of time can be used to explore temporal changes in system parameters. Some examples include growth of oxide or corrosion-product films, poisoning of catal5dic surfaces, and changes in reactant or product concentration. 

Figure 2.43 Finite amplitude, oscillatory perturbation amplitudes 0.01, 0.001 and 0.0001...

Searby and Rochwerger [9] developed a model describing the effect of an acoustic field on the stability of a laminar, premixed flame, treated as a thin interface between two fluids of different densities and under the influence of a periodic gravitational field. Their model is an extension of the work by Markstein [8] and is consistent with the more recent flame theory of Clavin and Garcia-Ybarra [16]. Bychkov [17] later solved the problem analytically, presenting the following linear equation for the perturbation amplitude, /, of a flame under the influence of an acoustic field [17] ... 

In this study we consider the same problem in the framework of MCPT and propose a modification which restores the invariance to the choice of Fermi-vacuum. This involves calculating the perturbed quantities by all possible choices and constructing a weighted average. The number of parameters in the theory agrees with that of a Jeziorski-Monkhorst-type MRCC parametrization. The redundancy of a Jeziorski-Monkhorst parametrization however does not show up in the present approach due to the fact that perturbational amplitudes corresponding to different... 

Grant and Middleman [22] reported that (1.36) correlated experimental data for capillary breakup of low viscosity liquid jets when a value of C = 13 was selected. It is instructive to use this value of C to evaluate the initial perturbation amplitude Co- Taking for the estimate the unperturbed cross-sectional radius a = 1 mm, one can find Co = 10 exp( 13) = 2.26 x 10 m = 2.26 nm. How plausible such estimates are for liquid jets whose profile is visibly perturbed at the nozzle exit, remains an open question. 

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