Phase angle displacement

Since a phase angle displacement can occur across the trausformer due to its method of connection it is necessary to identify this displacement. The numbering system for this is based on the hands of a clock. Each five-minute position on a clock gives 30° phase displacement hence,... 

Step 11. At this point a computer program refines the atomic parameters of the atoms that were assigned labels. The atomic parameters consist of the three position parameters x,j, and for each atom. Also one or six atomic displacement parameters that describe how the atom is "smeared" (due to thermal motion or disorder) are refined for each atom. The atomic parameters are varied so that the calculated reflection intensities are made to be as nearly equal as possible to the observed intensities. During this process, estimated phase angles are obtained for all of the reflections whose intensities were measured. A new three-dimensional electron density map is calculated using these calculated phase angles and the observed intensities. There is less false detail in this map than in the first map. 

The previous equations indicate that the velocity and acceleration are also harmonic and can be represented by vectors, which are 90° and 180° ahead of the displacement vectors. Figure 5-4 shows the various harmonic motions of displacement, velocity, and acceleration. The angles between the vectors are called phase angles therefore, one can say that the velocity leads displacement... 

Sliop test facilities should include instrumentation with the capability of continuously monitoring and plotting rpm, peak-to-peak displacement, and phase angle (X-Y-Y ). Presentation of vibration displacement and phase marker by use of an oscilloscope makes visualization easier. 

In capacitive circuits the current leads the voltage producing it. Here current is maximum 90 deg earlier than the voltage, as shown hy 4 in sketch D. Again, if the circuit has resistance, the angle of the phase difference (displacement) is less than 90 deg. ... 

Consider two frequencies given by the expression Zj = flsin(ft)/) and Z2 = i sin(ft)/- -), which are shown in Figure 43.9 plotted against cot as the X-axis. The quantity, (p, in the equation for X2 is known as the phase angle or phase difference between the two vibrations. Because of (p, the two vibrations do not attain their maximum displacements at the same time. One is seconds behind the other. Note that these two motions have the same frequency, co. A phase angle has meaning only for two motions of the same frequency. 

With damped vibration, the damping constant, c, is not equal to zero and the solution of the equation gets quite complex assuming the function, X =Xo sin(ft)/ — ). In this equation, cj) is the phase angle, or the number of degrees that the external force, Fo sin(ft)/), is ahead of the displacement, Xo sin(ft)/ — cj>). Using vector concepts, the... 

During the test, the rotor s speed, vibration displacement, and corresponding phase angle, filtered to rotor speed (IX), shall be measured and recorded. 

Figure 13.5 Amplitude and phase angle of the third harmonic of the field induced strain and dielectric displacement of a 60/40 PZT thin film.

Fig. 10.18. Effects of surface roughness on EHD impedance (amplitude ratio, H(p)IH(p- 0), and phase lag, 9, against scaled frequency, p and comparison with the behaviour of a uniform disc—asymptotic line marked (a) —and an array of UMEs— asymptotic line marked (b). The frequency shift is deduced from the displacement between the two sections of the phase angle diagram where the data superimpose for different n . The modulation frequency, to2, at which the data deviate from that of a uniform electrode, is related to the amplitude of the surface roughness or the spacing between the elements of the UME array. Data from Reference [121], for Fe(CN)i reduction on smooth Pt at 120 rpm 4 240 rpm, and on a rough, Pt-coated silver electrode (roughness scale 5 (im, disc diameter 6 mm) at O 120 rpm + 240 rpm A 500 rpm and x 1000 rpm.

The four librations (torsional oscillations or rocking motions) arise because the crystal-field potential prevents the I2 molecule from rotating as it would in the gas phase. There are some special crystals, called plastic crystals, in which symmetric molecules that interact weakly can still undergo hindered rotation in the solid phase, but l2( ) is not one of these. The librational motions for each I2 occur about two axes (a, /3) perpendicular to the 1—1 bond direction. The librations of the two I2 molecules in the same unit cell are coupled—giving rise to SL, AL and SL, AL vibrations, where SL denotes symmetric libration (angle displacements in phase) and AL denotes antisymmetric libration (angle displacements out of phase). 

In this case, measurements of the amplitudes of displacement and force, as well as the phase angle, are not required. The frequency of the force is varied while its amplitude is kept constant (ref. 6 p. 34-48). The frequency at which the amplitude of the displacement is maximum is taken as the resonance frequency. Usually several relatively decreasing maxima are obtained. 

Calculated phase angle The phase angle o calculated from the atomic positional and displacement parameters for a model structure. 

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-... 

This process may be repeated as many times as needed until all atoms in the unit cell are located and the following Fourier map(s) do not improve the model. Equations 2.132 to 2.134 may be combined with a least squares refinement using the observed data, which results in a more accurate model of the crystal structure, including positional and displacement parameters of the individual atoms already included in the model. The success in the solution of the crystal structure is critically dependent on both the accuracy of the initial model (initial set of phase angles) and the accuracy of the experimental structure amplitudes. Needless to say, when the precision of the latter is low, then the initial model should be more detailed and precise. 

After all three independent atoms (peaks 1 through 3 in Table 6.6) have been included in computations assuming identical displacement parameters in an isotropic approximation (5 = 0.5 A ), the resulting Rp = 6.9% without refinement. This value is excellent because i) the powder diffraction pattern is relatively simple with minimum overlap, and ii) the powder particles used in the diffraction experiment were nearly ideal (spherical), thus preferred orientation effects were also minimized. The following electron density distribution Figure 6.13 and Table 6.7) was obtained using the newly determined set of phase angles. 

False peaks (e.g. peak No. 4 in Table 6.6, which is easily recognizable in Figure 6.12) appear on Fourier maps due to a variety of reasons i) the largest contribution comes from the truncation of the Fourier summation (Eq. 2.133) because only a limited amount of diffraction data is available (see Table 6.4) ii) the structure amplitudes are not exact, especially when powder diffraction data were used in combination with Le Bail s extraction, and iii) phase angles calculated using atomic parameters, which are not fully refined, are still imprecise because we used randomly assigned displacement parameters and assumed completely random distribution of Ni and Sn in two possible sites. 

Wetting is a fundamental process in which one fluid phase is displaced completely or partially by another fluid phase from the surface of a solid. One useful parameter to describe wetting is the contact angle 0 of a liquid drop on a solid substrate... 

The mechanical impedance of the resonator may in principle be calculated from the amplitudes of the force and the displacement signals and the phase angle. However, sufficiently precise measurement of the amplitude of the displacement is difficult around the resonance frequency as is used in MLR measurements. Instead, the phase angle is measured at several frequencies. The phase angle 6 is expressed as... 

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