Imaginary component

Two difierent components contribute to impedance the resistive or real component due to resistors and the reactive or imaginary component from AC circuitry elements, such as capacitors, inductors, etc. Unlike the resistive component, the reactive impedance affects not only the magnitude of the AC wave but also its time-... 

Nyquist Plot a graph of the frequency response of an electrode in which the imaginary component of the impedance is plotted as a function of the real component for a range of frequencies. 

Grossing and Zeilinger have studied this periodicity invariance quantitatively, as a function of 5 [gross88c]. Generalizing the parameter 6 (which they call a mixing parameter) to have possibly non-equal real and imaginary components, we write S... 

The RPA built-in data treatment extracts from the recorded torque signal 16 discrete values in order to calculate through a discrete FT (U.S. Patent 4,794,788) the real and imaginary components... 

Figure 3.6 The first set of Fourier transformations across <2 yields signals in V2, with absorption and dispersion compronents corresponding to real and imaginary parts. The second FT across /, yields signals in V, with absorption (i.e., real) and dispersion (i.e., imaginary) components quadrants (a), (b), (c), and (d) represent four different combinations of real and imaginary components and four different line shapes. These line shaptes normally are visible in phase-sensitive 2D plots.

The frequency-domain spectrum is computed by Fourier transformation of the FIDs. Real and imaginary components v(co) and ifi ct>) of the NMR spectrum are obtained as a result. Magnitude-mode or powermode spectra P o)) can be computed from the real and imaginary parts of the spectrum through application of the following equation ... 

The imaginary component, "(f), is the dilational viscosity modulus. This arises when the demulsifier in the monolayer is sufficiently soluble in the bulk liquid, so that the tension gradient created by an area compression/expansion can be short circuited by a transfer of demulsifiers to and from the surface. It is 90° out of phase with the area change. 

Equivalent Circuit Analysis. IS measurements yield values of V and Z the real and imaginary components of the impedance, as a function of f, the AC frequency. The data are usually displayed as Nvauist plots (Z, vs. Z ) or Bode plots (impedance modulus,... 

The fit values of / and g have substantial imaginary components, so are significantly out of phase with h (we set h = 1) and, hence, explain the large values of the circular-difference effect. Also there are substantial real components of / and g, which means a significant portion of / and g are in phase with h and, hence, explains the large values of the linear-difference effect. 

A.7 Show that for a 2 X 2 symmetrical matrix, the eigenvalues must be real (do not contain imaginary components). Develop a 2 X 2 nonsymmetrical matrix which has complex eigenvalues. 

This expression describes the variation of the complex modulus with frequency for a Maxwell model. It is normal to separate the real and imaginary components of this expression. This is achieved by multiplying through by (1 — icut) to give... 

Far from the Laue condition the absorption shows the normal photoelectric absorption, as would be measured (with allowance for density) in a liquid or gas of the same atomic species. Close to the Laue condition, the absorption is quantified by the imaginary parts of the susceptibilities, leading to imaginary components of the wavevectors. These imaginary components are always normal to the crystal surface and hence the planes of constant attenuation are parallel to the surface. The attenuation coefficient (n) normal to the surface is given by (n)=-4 lm(K o) (4.28)... 

Allen et a/. (1991) performed these computations for 1-octadecene droplets, and they measured the evaporation rate of the droplets as a function of laser power. To determine the absolute irradiance /, of the laser beam, they also measured the force on the particle exerted by the laser beam using the techniques discussed above. The photon pressure force is given by Eq. (87), which involves the complex refractive index. The real component of the refractive index n was determined from optical resonance measurements, and the imaginary component was obtained iteratively. That is, they assumed a... 

Table III shows that the experimental and predicted evaporation rates are in good agreement at all beam intensities. There is some inconsistency at the highest power levels. It was difficult to maintain the droplet in the center of the laser beam at the highest power level, and the measured evaporation rate is somewhat low as a result of that problem. Additional computations demonstrate that the predicted evaporation rate is quite sensitive to the choice of the imaginary component of N, so the results suggest that this evaporation method is suitable for the determination of the complex refractive index of weakly absorbing liquids. For strong absorbers, the linearizations of the Clausius-Clapeyron equation and of the radiation energy loss term in the interfacial boundary condition may not be valid. In this event, a numerical solution of the governing equations is required. The structure of the source function, however, makes this a rather tedious task.

The ratio of the amplitudes of the applied and the response signal Vmlim) and the phase shift between these signals (0) can be used to determine the impedance, which can be represented as a complex number. The real component of impedance is known as the resistance, R, and the imaginary component, given by X, is known as the reactance [19]. 

In protein crystallography we assume that all electron density is real, and does not have an imaginary component. In reciprocal space this observation is known as Friedel s law, which states that a structure factor F(h) and its Friedel mate F(—h) have equal amplitudes, but opposite phases. The correspondence of these two assumptions follows straight from Fourier theory and, in consequence, explicitly constraining all electron density to be real is entirely equivalent to introducing Nadditional equalities of... 

The separation between the real and imaginary components is achieved by use of the expression... 

Since the scattering amplitude has both real and imaginary components, we may write... 

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