Phase-angle

The structure amplitude, expressed earlier as a sum of exponents, can be also represented in a different format. Thus, by applying Euler s formula 

The sums of cosines and sines in Eq. 2.103 signify the real (A) and imaginary (B) components of a complex number, respectively, which the structure amplitude indeed is. Hence, considering the notations introduced in Eq. 2.87, Eq. 2.103 can be rewritten as  

As a result, the structure amplitude can be represented by its magnitude, lF(h), and phase angle, a(h), which varies between 0 and 27t. When the crystal structure is centrosymmetric (i.e. when it contains the center of inversion), each atom with coordinates (x, y, z) has a sjmimetrically equivalent atom with coordinates (-x, -y, -z). Thus, considering that cos(-y) = cos(y) and sin(-y) = -sin(y) and assuming that all atoms scatter normally, Eq. 2.103 can be simplified to 

When the anomalous scattering is present, the structure amplitude even for a centrosymmetric crystal becomes a complex number. This is shown in Eqs. 2.107 and 2.108. The first (general expression) is easily derived by combining Eqs. 2,101 and 2.103 and rearranging it to group both the real and imaginary components. 

The introduction of the center of inversion results in the cancellation of all sine terms in Eq. 2,107 and 

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Introducing the complex notation enables the impedance relationships to be presented as Argand diagrams in both Cartesian and polar co-ordinates (r,rp). The fomier leads to the Nyquist impedance spectrum, where the real impedance is plotted against the imaginary and the latter to the Bode spectrum, where both the modulus of impedance, r, and the phase angle are plotted as a fiinction of the frequency. In AC impedance tire cell is essentially replaced by a suitable model system in which the properties of the interface and the electrolyte are represented by appropriate electrical analogues and the impedance of the cell is then measured over a wide... 

Here tire are tire amplitudes of tire hannonics of tire density, is tire wavenumber and tire are arbitrary phase angles, which are necessary for a complete tlieoretical description of tliis stmcture (see section C2.2.3.4). The z direction is, by convention, nonnal to tire layers. 

The point z can also be located by establishing polar coordinates in the complex plane where r is the radius vector and 0 is the phase angle. Draw suitable polar coordinates for the Argand plane. What is r for the point 7 = 3 + 4i7 What is 0 in degrees and radians ... 

In this representative dihedral potential, V is the dihedral force constant, n is the periodicity of the Fourier term, (jtg is the phase angle, and (]) is the dihedral angle. 

The period of the interaction is 360/n. The phase angle shifts the curve to the left or right. For n=l and (j)o=0, the curve represents the situation where the energy is a minimum for the tra 5-conformation with a barrier of to the highest energy cw-conformation. A... 

The dihedral angle or torsional energy interaction in MM-t is of the general form of equation (12) on page 175 but explicitly includes n=l, 2, and 3 with a phase angle (j) =0 ... 

In the previous section we defined several characteristic properties of electromagnetic radiation, including its energy, velocity, amplitude, frequency, phase angle, polarization, and direction of propagation. Spectroscopy is possible only if the photon s interaction with the sample leads to a change in one or more of these characteristic properties. 

In the second broad class of spectroscopy, the electromagnetic radiation undergoes a change in amplitude, phase angle, polarization, or direction of propagation as a result of its refraction, reflection, scattering, diffraction, or dispersion by the sample. Several representative spectroscopic techniques are listed in Table 10.2. 

We commented above that the elastic and viscous effects are out of phase with each other by some angle 5 in a viscoelastic material. Since both vary periodically with the same frequency, stress and strain oscillate with t, as shown in Fig. 3.14a. The phase angle 5 measures the lag between the two waves. Another representation of this situation is shown in Fig. 3.14b, where stress and strain are represented by arrows of different lengths separated by an angle 5. Projections of either one onto the other can be expressed in terms of the sine and cosine of the phase angle. The bold arrows in Fig. 3.14b are the components of 7 parallel and perpendicular to a. Thus we can say that 7 cos 5 is the strain component in phase with the stress and 7 sin 6 is the component out of phase with the stress. We have previously observed that the elastic response is in phase with the stress and the viscous response is out of phase. Hence the ratio of... 

Fig. 17. Viscoelastic material stress ( and strain (---------) ampHtudes vs time where 5 is the phase angle that defines the lag of the strain behind the...

Fig. 18. Resolution of the complex modulus G into two vectors, G the storage modulus, and G the loss modulus the phase angle is 5.

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. 

Another concept is brushless excitation, in which an ac generator (exciter) is direc tfy coupled to or mounted on the motor shaft. The ac exciter has a stator field and an ac rotor armature which is directly connected to a static controllable rectifier on the motor rotor (or a shaft-mounted drum). Static control elements (to sense synchronizing speed, phase angle, etc.) are also rotor-mounted, as is the field discharge resistor. Changing the exciter field adjusts the motor field current without the necessity of brushes or slip rings. Brushless excitation is suitable for use in hazardous atmospheres, where conventional brush-type motors must have protective brush and slip-ring enclosures. 

Now The negalive sequence voltage is caused by an unbalance in the magnitude of voltages in the three phases, rather than in the phase angle. 

Phase-angle measuring or otit of step relay Further reading ... 

For proleclion wliere the direction of the current or the phase angle of the two ends of the line are compared to decide whetlicr ihei c is a fault on a partieiilar line. 

In TSCs the thyristors are used in anti-parallel to switch a capacitor bank ON or OFF but without any phase angle control. A TSC therefore does not by itself generate any harmonics, unlike a TCR. 

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