Phase errors

The accuracy of a VT depends upon its leakage reactance and the winding resistance. It determines the voltage and the phase errors of a transformer and varies with the VA on the secondary side. With the use of better core material (for permeability) (Section 1.9) and better heat dissipation, one can limit the excitation cunent and reduce the error. A better core lamination can reduce the core size and improve heat dissipation. 

Only the r.m.s. values and not the phasor quantities are considered to define the voltage error. The phase error is defined. separately. Together they form the eoinposite error. Refer to Table l.i,.S for measuring and Table l.s.6 for protection VTs. 

The phasor difference between /( and /, i.e. results in a composite error /. The phase displacement between I2 and /, by an angle 8 is known as the phase error. The current error will be important in the accurate operation of an overcurrent relay and the phase error in the operation of a phase sensitive relay. The composite error will be significant in the operation of a differential relay. 

This defines the maximum permissible current error at the rated current for a particular accuracy class. The standard accuracy classes for the measuring CTs may be one of 0.1, 0.2, 0.5, I, 3 and 5. The limits of error in magnitude of the secondary current and the phase error, as discussed in Section 15.6.1 and shown in Figure 15.18,... 

Voltage transformers are classified into types AL, A, B, C and D, in descending order of accuracy. The ratio errors for small voltage changes (within ( 10 per cent of the rating) vary between 0.25 per cent and 5 per cent and the phase errors between ( 10 minutes and ( 60 minutes. 

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. 

Figure 1 outlines the basic AO system. Wavefronts incoming from the telescope are shown to be corrugated implying that they have phase errors. Part of the light is extracted to a wavefront phase sensor (usually referred to as a wavefront sensor, WFS). The wavefront phase is estimated and a wavefront corrector is used to cancel the phase errors by introducing compensating optical paths. The most common wavefront compensator is a deformable mirror. The idea of adaptive optics was first published by Babcock (1953) and shortly after by Linnik (1957). 

The random phase error of a wavefront which has passed through turbulence may be expressed as a weighted sum of orthogonal polynomials. The usual set of polynomials for this expansion is the Zernike polynomials, which... 

The accuracy with which a wavefront sensor measures phase errors will be limited by noise in the measurement. The main sources of noise are photon noise, readout noise (see Ch. 11) and background noise. The general form of the phase measurement error (in square radians) on an aperture of size d due to photon noise is... 

Adaptive optics requires a reference source to measure the phase error distribution over the whole telescope pupil, in order to properly control DMs. The sampling of phase measurements depends on the coherence length tq of the wavefront and of its coherence time tq. Both vary with the wavelength A as A / (see Ch. 1). Of course the residual error in the correction of the incoming wavefront depends on the signal to noise ratio of the phase measurements, and in particular of the photon noise, i.e. of the flux from the reference. This residual error in the phase results in the Strehl ratio following S = exp —a ). 

Let do be the diameter of the telescope for which the cone effect causes a phase error a = 1 rad. Thus for a telescope L> the phase error is =... 

Closure Phases (III.9-10) Closure phases are obtained by triple products of the complex visibilities from the baselines of any subset of three apertures of a multi-element interferometer. Element-dependent phase errors cancel in these products, leaving baseline dependent errors which can be minimised by careful designs. Although there are many closure relations in a multi-element array, there are always fewer independent closure phases than baselines. Closure phases are essential for imaging if no referenced phases are available. 

Composite pulse A composite sandwich of pulses that replaces a single pulse employed to compensate for B] field inhomogeneities, phase errors, or offset effects. 

Execution Phase Errors (except where stated control is returned to ISIM monitor cannot be continued with a GO)... 

The core and valence monopole populations used for the MaxEnt calculation were the ones of the reference density (electrons in the asymmetric unit iw = 12.44 and nvalence = 35.56). The phases and amplitudes for this spherical-atom structure, union of the core fragment and the NUP, are already very close to those of the full multipolar model density to estimate the initial phase error, we computed the phase statistics recently described in a multipolar charge density study on 0.5 A noise-free data [56],... 

The XY problem gives rise to a constant phase error across the spectrum, the delay problem gives a linear phase error. To correct for this, we have two phase adjustment parameters at our disposal zero and first order. 

The quality of the measured phases ean be ehaiacterised by the averaged phase error (phase residual ( )Res) of s mimetry-related refleetions ... 

After analysis nodes 42(25), 50(18), 53(9), 98(56), 125(40), 189(26) are kept (the mean absolute phase errors are in parentheses). For reference the correct map using experimental phases is shown in Figure 6, and Figure 7 shows the best centroid map (for node 53). For reference the correct map using experimental phases is shown in Figure 6. The map correlation coefficient is 0.94. 

Figure 7. The centroid map for node 53 with a mean phase error of 90 and a map correlation coefficient of 0.94.

Here Xprotein is real space coordinate within the protein region, H(p(x)) is the expected, non-Gaussian histogram of the electron density and H°i (p(x)) is the observed histogram of protein density which may or may not have phase errors. 

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). 

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