Phase-angle equation

In Example 8.12, we used the interacting form of a PID controller. Derive the magnitude and phase angle equations for the ideal non-interacting PID controller. (It is called non-interacting because the three controller modes are simply added together.) See that this function will have the same frequency asymptotes. 

Any periodic wave can be considered as a sum of cosine and sine waves [amplitudes A[hkl) and B hkl), respectively]. The ratio of the amplitudes of the two waves gives a measure of the phase angle (Equation 6.1), and the sum of the squares of the amplitudes gives the intensity (Equation 6.2), which is the square of the amplitude. 

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... 

Table 5.1 ThetoppartofthefinaloutputofDavidSayre sphase-determination program (year 1967). Note that for a centrosymmetric structure phase angles, equation 5.30, can only be 0 or 180° so the phase of a structure factor is just a plus or a minus sign...

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 ... 

To determine the pipeline potentials, the resultant induced field strengths have to be included in the equations in Section 23.3.2. Such calculations can be carried out with computers that allow detailed subdivision of the sections subject to interference. A high degree of accuracy is thus achieved because in the calculation with complex numbers, the phase angle will be exactly allowed for. Such calculations usually lead to lower field strengths than simplified calculations. Computer programs for these calculations are to be found in Ref. 16. 

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... 

Inertia foree + Damping foree + Spring foree + Impressed foree = 0 From the previous equation, the displaeement lags the impressed foree by the phase angle 6, and the spring foree aets opposite in direetion to... 

Using equations (6.18) and (6.21), values for the modulus and phase angle may be ealeulated as shown in Table 6.1. The results in Table 6.1 may be represented as a Polar Plot, Figure 6.4(a) or as a reetangular plot. Figures 6.4(b) and (e). Sinee the reetangular plots show the system response as a funetion of frequeney, they are usually referred to as frequeney response diagrams. 

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. 

The above equation is known as a Fourier series, which is a function of time or /(/). The amplitudes (A , A2, etc.) of the various discrete vibrations and their phase angles (cpi, cj>2, 3 ) can be determined mathematically when... 

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... 

If the signal decay is a single-exponential curve, equations 16 and 17 result in values for X that are in agreement with each other. Dissimilar values indicate multiexponential decay, which usually means that the sample contains more than one fluorophore. Multiexponential decay can be resolved by using a phase fluorometer with phase sensitive detection. A time-independent, direct-current signal is produced that is proportional to the cosine of the difference between the phase angle of the detector ( D) and the phase angle of the fluorescence ( ) ... 

A CSTR is operated in series with a PFR, with a fraction ce of the flow in bypass around the CSTR. Equations are to be found for the Gain and phase angle of a frequency response. 

It is important to note that the velocity of the wave in the direction of propagation is not the same as the speed of movement of the medium through which the wave is traveling, as is shown by the motion of a cork on water. Whilst the wave travels across the surface of the water, the cork merely moves up and down in the same place the movement of the medium is in the vertical plane, but the wave itself travels in the horizontal plane. Another important property of wave motion is that when two or more waves traverse the same space, the resulting wave motion can be completely described by the sum of the two wave equations - the principle of superposition. Thus, if we have two waves of the same frequency v, but with amplitudes A and A2 and phase angles

resulting wave can be written as ... 

This behaviour, depicted in Fig. 5.11 with the values calculated using Equation 5.3, represents the phase-angle variations exhibited by Ru/Ti electrodes containing 20-40 at.% Ru, which show low Rt values. 

A complication that occurs on a low at.% Ru electrode is that, owing to the low Faradaic currents (low Ru content) and hence large Rt value, currents due to other trace redox reactions, e.g. oxygen reduction, become more detectable. This reveals itself in a phase-angle of 45° as co 0 as trace oxygen reduction would be diffusion-controlled. The impedance corresponding to this situation can be shown to be the same as that in Equation 5.3, with U(p) expressed by the relationship ... 

Fig. 5.12 Theoretical phase-angle plots using Equations 5.3 and 5.4 for Re = 10 2 and Rt = 100, along with the relevant co values ...

Fig. 5.14 Theoretical phase-angle plots simulating the experimental Bode plots during deactivation using Equations 5.3 and 5.8. Note that indicates Q for these cases is equal to the value noted divided by (a + 1). 

In the upper panel of Figure 13.6, the emission is drawn assuming a modulation frequency of 30 MHz and a lifetime of 9 nsec. Using the equations above, the phase angle is 59.5° and the demodulation factor is 0.5. (For further details, the reader is referred to Lakowicz(66)). Additionally, multifrequency phase and modulation instruments that operate over a range of frequencies have been described(67, flS) and simple instruments are possible if only one or several discrete frequencies are required (Figure 13.6, lower panel). 

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