Correlation function graphical analysis

Analysis of the 1D Correlation Function. Several publications describe the search for a simple graphical analysis [22,159,162-164] of the ID correlation function by means of a geometrical construction. It is the drawback of all such methods that polydispersity and heterogeneity are not considered. The methods are derived from the general generation principle of correlation functions (Fig. 8.20), resulting in equations (cf. Eqs. (8.23), (8.70) and (8.64)) for the first off-origin maximum, the depth of the first minimum or the initial slope iid (0) of ideal correlation functions. For the simplified case of a lamellar system we obtain... 

The First-Zero Method of Correlation Function Analysis. For the purpose of a practical graphical evaluation of the linear crystallinity, Eq. (8.67) can be applied to a renormalized correlation function y (x/Lapp). The method which has been proposed by Goderis et al. [162] is based on the implicit assumption that the first zero, Jto, of the real correlation function is shifted by the same factor as is the position of its first maximum, Lapp. 

As an alternative to nonlinear regression, a number of graphical correlations can be used quickly to find approximate values of ti and T2 in second-order models. The accuracy of models obtained in this way is often sufficient for controller design. In the next section, we present several shortcut methods for estimating transfer function parameters based on graphical analysis. 

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