Phase angle error

Ratio and phase angle errors also occur due to the need for a portion of the primary current to magnetize the core and the requirement for a finite voltage to drive the current through the burden. These errors must be small. Current transformers are classified (in descending order of accuracy) into types AT, AM, BM, C and D. Ratio errors in class AT must be within the limits of ( 0.1 per cent for AT and ( 5 per cent for D, while the phase error limit on class A1 is ( 5 minutes to ( 2 minutes in types CM and C. 

The amplitudes and the phases of the diffraction data from the protein crystals are used to calculate an electron-density map of the repeating unit of the crystal. This map then has to be interpreted as a polypeptide chain with a particular amino acid sequence. The interpretation of the electron-density map is complicated by several limitations of the data. First of all, the map itself contains errors, mainly due to errors in the phase angles. In addition, the quality of the map depends on the resolution of the diffraction data, which in turn depends on how well-ordered the crystals are. This directly influences the image that can be produced. The resolution is measured in A... 

Fig. 1.22 RARE sequence. Here the formation of the first spin echo is conventional. The CPMG form of spin echo is used to avoid the accumulation of flip angle errors over the echo train. However, before the second echo can be acquired, the phase-encoding has to be rewound to undo the dephasing of the spins. Therefore, a phase encoding step of equal...

This pulse transfer function has a zero at z = a and a pole at z = +1. It cannot produce any phase-angle advance since the pole lies to the right of the zero (a is less than 1). The pole at -I-1 is equivalent to integration (pole at s = 0 in continuous systems) which drives the system to zero steadystate error for step disturbances. 

Figure 6.3 Isomorphous replacement phase determination (Marker construction), (a) Single isomorphous replacement. The circle with radius Fpp represents the heavy-atom derivative, while that with radius Fp represents the native protein. Note that the circles intersect at two points causing an ambiguity in the phase angle apg and apt, represent the two possible values, (b) Double isomorphous replacement. The same construction as that in single isomorphous replacement except that an additional circle with radius Fpn2 (vector not shown for simplicity) has been added to represent a second heavy-atom derivative. Note that all three circles (in the absence of errors) intersect at one point thus eliminating the ambiguity in the protein phase angle ap. Fm and Ppy represent the heavy-atom vectors for their respective derivatives.

FIG. 5.9 Phase angles in an acentric X-N analysis phase angle as calculated with spherical-atom form factors and neutron positional and thermal parameters tpx is the unknown phase of the X-ray structure factors which must be estimated for the calculation of the vector AF. Use of FX — FN introduces a large phase error. Source Coppens (1974). 

In practice, the quarter-wave plates will possess some imperfection in retardation. Furthermore, the phase angle of the plates may not be zero relative to the mechanical rotation device. Both sources of error can be taken into account. For example, an imperfect quarter-wave plate with retardation S = tc/2 + (3 and a phase offset of <)> would produce the following Stokes vector for the (P/RQ) 5G,... 

Fig. 8. Dependence of BR-24 m.p. spectra on pulse errors. Spectrum (b) was obtained for a phase error of 1° of the -Hy pulses spectrum (c) was obtained for a flip-angle error of -1-1% of the -Hr pulses. Spectrum (a) is a reproduction of the r = 3-/as BR-24 spectrum in Fig. 5. Note the common shift with respect to spectrum (a) of all lines in spectrum (b) and the larger linewidths in spectrum (c).

Difference maps also allow adjustments to be made to displacement parameters. If the displacement parameter of an atom in the model is too large, it will be spread over more space than necessary, and its peak height will be lower than it should be. As a result, there will be a positive peak at the atomic center in the difference map if the displacement factor is too small, a negative valley will appear in that position [Figure 9.8(b)]. The final difference Fourier map is generally not completely flat because it contains indications of both errors in the data ( Ft, 1) and inadequacies in the model ( Fc ), including, of course, the relative phase angle, a hkl)c-... 

Preliminary three-dimensional atomic coordinates of atoms in crystal structures are usually derived from electron-density maps by fitting atoms to individual peaks in the map. The chemically reasonable arrangement of atoms so obtained is, however, not very precise. The observed structure amplitudes and their relative phase angles, needed to calculate the electron-density map, each contain errors and these may cause a misinterpretation of the computed electron-density map. Even with the best electron-density maps, the precisions of the atomic coordinates of a preliminary structure are likely to be no better than several hundredths of an A. In order to understand the chemistry one needs to know the atomic positions more precisely so that better values of bond lengths and bond angles will be available. The process of obtaining atomic parameters that are more precise than those obtained from an initial model, referred to as refinement of the crystal structure, is an essential part of any crystal structure analysis. 

In the first, reflections with low accuracy in individual intensities (those that are completely or nearly completely overlapped) are simply discarded. This works best when direct methods are used for the structure solution because substantial errors even in some of the normalized structure amplitudes may affect phase angles of many other reflections. 

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