Impedance Operational

Application to small amplitude methods the operational impedance the Fast Fourier Transform... 

Evidently, it makes sense to define quotients A (s)/Ajj(s) or A i(s)/ AE(s) as so-called operational impedances or operational admittances... 

Equation (108) is universal for any small amplitude perturbation method and may, in principle, be used as the starting point for the derivation of a response vs. time relation for a given perturbation function. It is easily verified [53] that substitution of s = ico into any expression of an operational impedance or admittance delivers the complex impedance or admittance as they are defined in Sect. 2.3.1. 

On the other hand, the relative simplicity of expressions for the operational impedance (sometimes even as compared with the complex impedance) has led to the method of Laplace Plane Analysis of results of relaxation techniques [76—78, 81, 82]. The primary problem of the method is how to obtain Laplace transformed data of both the perturbation P(f) and the response R(t). To that end, several possiblities may be distinguished. 

The operative impedance Z(p) for oxygen sensors at the absence of polarization can be expressed as follows [45] ... 

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 ... 

This method can also be applied for diagnostics of the solid electrolyte sensors. In this case, i (t) = const or i (j ) = l/p. Operative impedance at the absence of polarization can be determined from the following equation ... 

In order to derive the familiar reactances e.g. Xj the sub-transient reactance, it is first necessary to obtain the operational impedances . (In control theory terminology these would be called transfer functions .)... 

The equation for the operational impedance that relates the cl-axis flux linkages to the stator current id and the rotor excitation v/ is,... 

At this stage operational impedances and time constants have been derived for synchronous machines, and for induction machines, if appropriate substitutions are made as shown in Reference 23. 

Table 2.1 Operational impedance of linear electrical elements...

The operational impedance is the ratio of the Laplace transform of the potential to the Laplace transform of the current (Eq. 2.88). It is usually used for an arbitrary perturbation signal. For the periodic signal it is equivalent to the definition using Fourier transformation. What follows are examples of the applicatirMi of the Laplace technique to the determinatirHi of the current-potential relations and the impedances. 

The operator impedance of the circuit is simply a connection of the three elements in series ... 

As was mentioned in Sects. 2.8.1 and 2.8.2, application of the Laplace transform to the transient potential and current permits determination of the operational impedance. Such a method was initially introduced by Pilla [90-93] and applied to studies using mercury electrodes. Using a fast potentiostat a small potential step was applied, and both voltage and current transients were measured. Of course, because of the nonideal potentiostat response, the potential increase was not a rectangular step but occiured more slowly. Examples of the measured potential and current transients are shown in Fig. 3.6. Such data acquisition was extended to longer times and then extrapolated as the integration had to be continued to inlinity. 

This method was later critically reviewed and modified by Barsoukov et al. [94]. They showed that the weakness of the direct Laplace transform lies in its large sensitivity to noise. Instead of direct numerical integration, Eq. (2.21), they proposed instead to fit first the time-domain data to a carrier function, which could then be directly transformed. The operator impedance of stable systems is... 

By invoking the equivalence of Eqs (56) and (57), it is apparent that the information required to calculate the operational impedance is contained in the input and output linear magnitude spectra and Sy. In practice these are cumbersome to compute. The power and cross-power spectra give the same basic information, are faster to compute, and can be applied to measurements to which linear magnitude spectra cannot (Roth [1970]). 

Generators typically used in m-g sets have substantially higher internal impedance than equivalent kVA-rated transformers. Because of this situation, m-g sets are sometimes supplied with an oversized generator that will be lightly loaded, coupled with a smaller motor that is adequate to drive the actual load. This approach reduces the initial cost of the system, decreases losses in the motor, and provides a lower operating impedance for the load. 

The term "operating impedance" is defined as the complex ratio of the voltage applied to a load to the current flowing in the load when it is operating under normal power and in its normal environment. In many cases, this impedance differs substantially from the "self-impedance" or "cold impedance" of the load. In antenna systems a radiator has a certain self-impedance when operating alone. When it is combined in an array to form a directional antenna, its operating impedance may differ substantially from its self-impedance because of coupled impedance from other radiators of the array. 

Many loads have an operating impedance which differs with applied power level. A dummy load, for example, may have an operating impedance which varies with applied power levels. Meaningful impedance measurements must, therefore, be made at normal power level. 

Quite often, in complex antenna systems, it is found that one or more of the elements has a negative operating impedance that is, the total of the coupled Impedance from all other elements exceeds the self-impedance of that element, and the element actually returns power to the transmitter. 

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