Admittance linearity

Friedly (F4) expanded the theoretical analysis of Hart and McClure and included second-order perturbation terms. His analysis shows that the linear response of the combustion zone (i.e., the acoustic admittance) is not sign-ficantly altered by the incorporation of second-order perturbation terms. However, the second-order perturbation terms predict changes in the propellant burning rate (i.e., transition from the linear to nonlinear behavior) consistent with experimental observation. The analysis including second-order terms also shows that second-harmonic frequency oscillations of the combustion chamber can become important. 

Linear sweep voltammetry, capacitance-voltage and automated admittance measurements have been applied to characterize the n-GaAs/room temperature molten salt interphase. Semiconductor crystal orientation is shown to be an important factor in the manner in which chemical interactions with the electrolyte can influence the surface potentials. For example, the flat-band shift for (100) orientation was (2.3RT/F)V per pCl" unit compared to 2(2.3RT/F)V per pCl" for (111) orientation. The manner in which these interactions may be used to optimize cell performance is discussed. The equivalent parallel conductance method has been used to identify the circuit elements for the non-illum-inated semi conductor/electrolyte interphase. The utility of this... 

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 interpretation of dielectric measurements assumes that the sample behavior can be represented by a linear, time-invariant admittance. The meaning of each of these terms is examined in turn. 

The admittance of a linear time-invariant sample can now be defined. In the sinusoidal steady state at angular frequency oo, both v(t) and i(t) are sinusoids having some phase difference cp (see Fig. 2). 

Each node mutual admittance 7 (i k) is the sum of the admittances between two given nodes i and k. The current Y kV in the mutual admittances between nodes i and k is negative if the voltages of nodes i and k have the same assumed polarity relative to the reference node. The current Y kV is positive if the voltages of nodes i and k have the opposite assumed polarity relative to the reference node. In a linear network, we have... 

FIGURE 9.1. The real part of the nondimensional nozzle admittance yyJMi as a function of the nondimensional frequency k for various values of the nozzle-entrance Mach number M, with y = 1.2 for longitudinal modes in a nozzle having a velocity linear with distance [20]. 

This polarizability involves a set of characteristic times jlkT, Id(, illoil, and (IcIIodI), between any two of which the correlation function may take a fairly simple form. However, this example indicates that for a linear system which falls to show normal mode behaviour the frequency-dependent admittance may be much more straightforward than the corresponding correlation function. 

An expression for the complex admittance, Y(jf), of an axon membrane is obtained by linearizing the Hodgkin-Huxley (HH) equations (I) and by applying a Laplace transformation (13, 14). The membrane admittance is then given by the general expression... 

Description of ion-channel kinetics via admittance analysis provides a framework within which linear kinetic models can be compared to macroscopic data (from a population of channels) in a membrane. Analysis of conduction via driving-point-function determinations also provides proper data (from a true linear analysis) for comparison with the relaxation times obtained from microscopic data from one or a small number of channels in a membrane patch isolated by a micropipette (4). In Markov modeling, the open- and closed-time distributions are fitted to sums of exponential functions (15). 

Data Analysis. Complex admittance determinations were fitted by an admittance function (13, 14, 16) based on the linearized HH equations (I). Admittance measurements were made under steady-state conditions (see Figures 2 and 4). Series resistance (Rs), the access resistance between the two voltage electrodes and up to the inner and outer surfaces of the axon membrane was not removed from measurements. Instead Rs was included and determined in the fit of the steady-state admittance model to the data. The measured complex admittance, therefore, is... 

The difference in potential between the reference and working electrodes, e(t), and the current response of the cell, i(t), are sampled simultaneously and converted to digital numbers. The Fourier transforms of these data arrays are then computed, yielding the frequency domain equivalents, the spectra, E(cj) and /(u ). If the cell response is linear then the cell admittance, y(u ), can be calculated using the following relation ... 

A,-ev is the reversible faradaic admittance magnitude and G(u ) is the kinetic correction which is related to the observed linear cot< spectrin as shown in Equation 2 (cot contains the important thermo-d amic (E, Ejq, n, T, F, R) and kinetic parameters (ks,o, and diffusion coefficients). All one does is rearrange Equation 1 to the form... 

However, if we excite the same series RC-circuit with a controlled current step and record the voltage across the RC circuit, the voltage will increase linearly with time ad infinitum. The time constant is infinite. Clearly, the time constant is dependent not only on the network itself, but on how it is excited. The time constant of a network is not a parameter uniquely defined by the network itself. Just as immittance must be divided between impedance and admittance dependent on voltage or current driven excitation, there are two time constants dependent on how the circuit is driven. The network may also be a three-or four-terminal network. The time constant is then defined with a step excitation signal at the first port, and the possibly exponential response is recorded at the second port. 

Correspondence between Components of an Equivalent Electric Circuit and Algebraic Models of (Integral) Admittances and Complex Admittances in the Linear Approximation... 

FIGURE 11.9 Cole-Cole (Nyquist) plots of the impedance (a) and of the admittance (b) of a parallel RC circuit in the linear case. The apex of the semicircle occurs when the angular frequency m equates the kinetic constant Kq (=1/Tc), which corresponds to the point intersecting the bisector in the admittance plot... 

In the restricted frame of small amplitude signals and homogeneous, isotropic, and linear properties, the complex admittance is deduced from the previous equation by Fourier transformation... 

In the restricted case of linear constitutive properties, by multiplying the previous transformed function (shown in Equation 11.51) by the capacitance, one is able to express the transformed effort, and then deduce the well-known expressions of the complex transfer fnnctions that are the admittance and the impedance ... 

The negative sign merely corresponds to an orientation convention of potential differences and currents as explained hereafter. It can be dropped when this feature is meaningless. This is the case when AC techniques are employed with sufficiently small signal amplitudes for approximating both Fourier-transformed impedances as well as linear operators. Consequently, the admittances become proportional through the square of the coupling factor. 

FIGURE 12.5 Plots of the electric admittance I=f(E) in response to a linear ramp of electrode potential E imposed to a thin layer of electroactive substance (left) and of the physical chemical capacitance n =f(fi) of the redox couple (right). 

It can be remarked that the shape of this expression corresponds to the already discussed expression of the relationship between two linear admittances belonging to different energy varieties (see Chapter 12, Section 12.5.3). 

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