Real and Imaginary Components

The difficulty with using the Ohmic-resistance-corrected Bode plots presented in the previous section is that an accurate estimate is needed for the electrolyte resistance and that, at high frequencies, the difference Zr — Re,est is determined by stochastic noise. As discussed in Section 16.1.4, these difficulties can be obviated by plotting the real and imaginary components of the impedance. 

As discussed in Section 21.2, the variances of stochastic errors aire equal for real and imaginary parts of the impedance. Thus, another advantage of presenting real and imaginary parts of the impedance as functions of frequency is that comparison between data and levels of stochastic noise can be easily represented. 

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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 ... 

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,... 

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... 

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... 

Evans, C. L., Potma, E. O., and Xie, X. S. 2004. Coherent anti-Stokes Raman scattering spectral interferometry Determination of the real and imaginary components of nonhnear snsceptihility )f for vibrational microscopy. Opt. Lett. 29 2923-25. 

Figure 7.7 Typical responses for the real and imaginary components of the dielectric constant to frequency.

By using the properties of even and odd functions, it is easy to show that P(x) is the unknown function if a(eo) is an even function. On the other hand, if imaginary components. By superposition, any function can be broken into odd and even components and treated separately. The results may then be combined. 

Rheological properties of filled polymers can be characterised by the same parameters as any fluid medium, including shear viscosity and its interdependence with applied shear stress and shear rate elongational viscosity under conditions of uniaxial extension and real and imaginary components of a complex dynamic modulus which depend on applied frequency [1]. The presence of fillers in viscoelastic polymers is generally considered to reduce melt elasticity and hence influence dependent phenomena such as die swell [2]. 

The frequency ft)T at which the values of the real and imaginary components are equal has special meaning. Thus from (5.28), when Zr6 = Z m,... 

In the impedance circuit analysis of an electrode/solution interface, the impedance and phase angle (or real and imaginary components) are measured at several frequencies. The imaginary component is plotted against the real component and the semicircle is then interpolated to those points. The solution resistance and the sum of solution and charge-transfer resistances, respectively, are then found by extrapolating the semicircle to = 0. 

Looking at the phenomenon of optical absorption by the medium from the viewpoint of classical wave mechanics, we see that the attenuation of electromagnetic radiation can be attributed to the interaction of the oscillating electric vector with the medium. Any phenomenon involving periodic oscillations can be decomposed to real and imaginary components. Thus, the ordinary refractive index n is the real part of the index of refraction n, which can be written as... 

In this section, the nonlinear problem of forming a least squares solution for the sine-wave amplitudes, phases, and frequencies is transformed into a linear problem. This is accomplished by assuming the sine-wave frequencies are known apriori, and by solving for the real and imaginary components of the quadrature representation of the sine waves, rather than solving for the sine-wave amplitudes and phases. The... 

One of the first attempts to estimate the control functions relies on a running discrete-Fourier transform (DFT (Discrete Fourier Transform)) [Moorer, 1977], Assuming the presence of one periodic note in a measurement x(n), the DFT length is set equal to the waveform s pitch period N. The real and imaginary components are then given by... 

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