Phase angles, definition

With the restriction of harmonicity (Equation 9.118) and the equivalence (Equation 9.119) between pulsations, the phase angle definition in Equation 9.85 can now be written as... 

To make the phase angle plot, we simply use the definition of ZGp(joo). As for the polar (Nyquist) plot, we do a frequency parametric calculation of Gp(jco) and ZGp(joo), or we can simply plot the real part versus the imaginary part of Gptjco).1 To check that a computer program is working properly, we only need to use the high and low frequency asymptotes—the same if we had to do the sketch by hand as in the old days. In the limit of low frequencies,... 

We can also use the Bode plot of G to do phase margin calculations. From the textbook definition, we are supposed to find the phase angle ( > = ZG0L where G0L = 1. If the phase margin is 45°, should be -135°. It appears that we need to know Kc beforehand to calculate GOL, but we do not. 

The amount of decrease of the resonance width may be simply estimated in the following way 50). Let the motion of the spins be characterized by a time tc, that is t is the average time a spin stays in a definite environment or the correlation time for the motion. This environment will cause a difference 5w in the precessional frequency of the spin which may be positive or negative from some average value to. During the time Tc the spin acquires a phase angle 60 = TcSu in addition to that acquired by the uniform precession at to. If we consider the motion to be a random walk process (51), after n such intervals during a time t the mean square phase acquired will be... 

Examination of the passive control conditions in Eqs. (5.26)-(5.29) shows that there are two values of the sum of phase angles for which zero transfer occurs. In principle, then, one can simultaneously block the transfer of, say, the energy and select the direction of the transfer of the population. One particularly interesting case is the definition of the phase angles for zero total power absorption. Since no energy is absorbed or emitted from the field these conditions define laser catalysis [44]. 

Since is a real number, it alone cannot fully characterize the crack tip field. In order to specify K, which has two real components, the phase angle ip defined by Eq. (5) is also required. Using this definition, a point in the (( , ip) plane is uniquely related to a point in the (JCj, Ku ) plane. Following the concept of failure locus introduced by Rice [20] for a given phase angle ip, the interface crack will... 

The methods are very complicated and require a large number of discrete items of information to describe a particle shape reasonably well they actually produce the signature of the particle, particularly if the coefficients are coupled with their respective phase angle. Meloy found that, in spite of a considerable scatter of data points, the log-log plot of coefficient amplitudes yields a straight line. He named this the law of morphological coefficients and defined the concept of random particles whereby, by definition and under certain assumptions, a random particle has a straight line as its signature. 

Fig. 12. Definition of a phase angle with respect to an imaginary wave scattered at the unit-cell origin.

The separation between the resonance frequencies is very small for very small Rs while it becomes obvious and cannot be neglected for higher Rs. The major challenge for sensor electronics design arises from the fact that the acoustically relevant frequency for sensing purposes is/s, whereas oscillators work at a certain phase angle, usually (p = 0, i.e., at/r in case of an ideal series oscillator. Applying the respective definitions one finds ... 

A Bode plot (Figure 7.60) displays the impedance Z against the frequency. The absolute value of the amplitude A (o>) and the phase angle rp(co) of the frequency response are separately plotted over the frequency co. A co) and co are usually displayed on a logarithmic scale whereas rp(co) is plotted linearly. The dimension of A(co) is in decibels (dB), and by definition A()dB = 20 log A co). Consequently, the logarithmic representation of A has a linear scale called the magnitude. 

Clearly, a more definitive and uniform procedure would consist in reporting the value of the pseudo-rotational phase-angle, and we plan to do this in subsequent issues. 

The anisotropic viscoelastic properties in shear of the meniscus have been determined by subjecting discs of meniscal tissue to sinusoidal torsional loading [35](Table B2.8). The specimens were cut in the three directions of orthotropic symmetry, i.e. circumferential, axial and radial. A definite correlation is seen with the orientation of the fibers and both the magnitude of the dynamic modulus IG I and the phase angle 8. 

Fig. 7.12 Frequency dependence of the real solid line) and imaginary dash curve) parts of the dielectric permittivity of an isotropic liquid (a) and the definition of the phase angle < ) (b)...

Table 5.1 Torsion phase angles used to relate >couplings to the definition of the dihedral backbone angle (f>, according to lUPAC/lUB definition [84] Coupling Torsion phase S) (°)...

Putting together the definition of the phase angle (Equation 9.85) and the differential equation of the wave function (Equation 9.96) provides a differential equation... 

GRAPH 9.38 Insertion into the Formal Graph of a spatial oscillator of halfway intermediate variables ( semienergies ) allows the definition of the derivation with respect to the phase angle cp. 

To find the dependence of the wave function and of the phase angle on the space vector r, the technique is the same as for the temporal oscillator. It consists of the definition of the wave vector operator in a similar manner as for the wave pulsation operator, that is, by taking a derivation operator, with a proportionality factor chosen for obtaining a real solution ... 

From the definition in Equation 9.142 of the phase angle and from the differential equation 9.146 of the wave function, one may write... 

From definition 11.128 of the phase angle this operator can be identified with the composition of reduced capacitances... 

This means that the phase angle 0 is zero. For a capacitor, we need to apply the definition of capacitance, C, from the relation between the potential, E, across the plates to the charge q residing on the plates ... 

Of particular interest is the phase angle ll of the stress state at the edge of the straight-sided buckle. In light of the results obtained in (5.22) and (5.25), the definition of phase angle (4.28) leads to the result that, in the case of the straight-sided buckle. 

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