Log modulus

When expressed in terms of the zeros of %, the sin-cos coefficients of the log modulus and of the phase are, respectively. 

By substituting these expressions into Eq. (55), one can see after some algebra that ln,g(x, t) can be identified with lnx (t) + P t) shown in Section III.C.4. Moreover, In (f) = 0. It can be verified, numerically or algebraically, that the log-modulus and phase of In X-(t) obey the reciprocal relations (9) and (10). In more realistic cases (i.e., with several Gaussians), Eq. (56-58) do not hold. It still may be due that the analytical properties of the wavepacket remain valid and so do relations (9) and (10). If so, then these can be thought of as providing numerical checks on the accuracy of approximate wavepackets. 

In the excited states for the same potential, the log modulus contains higher order terms mx(x, x, etc.) with coefficients that depend on time. Each term can again be decomposed (arbitrarily) into parts analytic in the t half-planes, but from elementary inspection of the solutions in [261,262] it turns out that every term except the lowest [shown in Eq. (59)] splits up equally (i.e., the/ s are just 1 /2) and there is no contribution to the phases from these temis. Potentials other than the harmonic can be treated in essentially identical ways. 

The relations pertain to the fine, small-scale time variations in the phase and the log modulus, not to their large-scale changes. 

In general, the complete system frequency response is obtained by summation of the log modulus of the system elements, and also summation of the phase of the system elements. 

Sinee 1 / 2 is —3 dB, the exaet modulus passes 3 dB below the asymptote interseetion at /T rad/s. The asymptotie eonstruetion of the log modulus Bode plot for a first-order system is shown in Figure 6.8. 

Flenee the absolute log modulus at a = is 20 dB). The Bode diagram is given by Figure 6.14. Note in Figure 6.14 that the phase eurve was eonstrueted by reading the phase from Figure 6.11(b), an oetave either side of ain. 

FIGURE 34.22 Log (strain ratio) versus log (modulus) for field and oven-aged SUV/Minivan-A tires. Both sets are linear indicating oxidative aging, but the slopes are different indicating a change in composition of the wedge rubber. 

Maximum closed-loop log modulus A plot of the magnitude vs. frequency of the closed-loop transfer function. 

Then semilog graph paper can be used to plot both phase angle and log modulus versus the log of frequency, as shown in Fig. 12.12. There are very practical reasons for using these kinds of graphs, as we will find out shortly. 

The units of log modulus are decibels (dB), a term originally used in communications engineering to indicate the ratio of two values of power. Figure 12.13 is convenient to use to convert back and forth from magnitude to decibels. 

A. GAIN. If G ) is just a constant K, G,j ) = K, and phase angle = arg G(Ib) = 0- Neither magnitude nor phase angle vary with frequency. The log modulus is... 

Bode plots of phase angle and log modulus versus the logarithm of frequency. 

Both the phase angle and log modulus curves are horizontal lines on a Bode ni as shown in Fig-1Z14. 

The high-frequency asymptote intersects the L = 0 line at breakpoint frequency. The log modulus is "flat (horizontal) out to this point and then begins to drop off. 

Note the very unique shape of the log modulus curves in Fig. 12.19. The lower the damping coefficient, the higher the peak in the L curve. A damping coefficient of about 0.4 gives a peak of about +2 dB, We will use this property extensively in our tuning of feedback controllers. We will adjust the controller gain to give a maximum peak of +2 dB in the log modulus curve for the closedloop servo transfer function X/X. ... 

G. GENERAL TRANSFER FUNCTIONS IN SERIES. The historical reason for the widespread use of Bode plots is that, before the use of computers, they made it possible to handle complex processes fairly easily. A complex transfer function can be broken down into its simple elements leads, lags, gains, deadtimes, etc. Then each of these is plotted on the same Bode plots. Finally the total complex transfer function is obtained by adding the individual log modulus curves and the individual phase curves at each value of frequency. 

Therefore the log modulus curves and phase-angle curves of the individual components arc simply added at each value of frequency to get the total L and d curves for the complex transfer function. 

Note that the total phase angle drops down to -180° and the slope of the high-frequency asymptote of the log modulus line is —40 dB/decade since the process is net second-order. 

The final plot that we need to learn how to make is called a Nichols plot. It is a single curve in a coordinate system with phase angle as the abscissa and log modulus as the ordinate. Frequency is a parameter along the curve. Figure 12.24 gives Nichols plots of some simple transfer functions. 

The numerical case given is fgr a 20-tray column with 10 trays in the stripping section. A constant relative volatility of 2 1 used. The column steadystate profile is given in Table 12.3, together with the values of coefficients and the transfer functions in terms of log modulus (decibels) and phase angle (degrees) at frequencies from 0 to 10 radians per minute. The values at zero frequency are the steadystate gains of the transfer functions. 

IKL Write a digital computer program that gives the real and imaginary parts, log modulus, and phase angle for the transfer functions ... 

Naturally we also can show closedloop stability or instability on Bode and Nichols plots. The ( — 1, 0) point has a phase angle of — IS0° and a magnitude of unity or a log modulus of 0 decibels. The stability limit on Bode and Nichols plots is, therefore, the (0 dB, —180°) point. At the limit of closedloop stability... 

The maximum closedloop log modulus does not have these pioblems since it measures directly the closeness of the G B curve to the (—1,0) point at all frequencies. The closedloop log modulus refers to the closedloop servo transfer function ... 

The feedback controller is designed to give a maximum resonant peak or hump in the closedloop log modulus plot. 

Typical log modulus Bode plots of these two closedloop transfer functions are shown in Fig. 13.10a. If it were possible to achieve perfect or ideal control, the two ideal closedloop transfer functions would be... 

Equation (13.30) says that we want the output to track the setpoint perfectly for all frequencies, and we want the output to be unaffected by the load disturbance for ail frequencies. Log modulus curves for these ideal (but unattainable) closed-loop systems are shown in Fig. 13.10b. 

In most systems, the closedloop servo log modulus curves move out to higher frequencies as the gain of the feedback controller is increased. This is desirable since it means a faster closedloop system. Remember, the breakpoint frequency is the reciprocal of the closedloop time constant. 

A commonly used maximum closedloop log modulus specification is 4 2 dB. The controller parameters are adjusted to give a maximum peak in the closedloop servo log modulus curve of -1-2 dB. This corresponds to a magnitude ratio of 1.3 and is approximately equivalent to an underdamped system with a damping coefficient of 0.4,... 

A Nichols chart is a graph that shows what the closedloop log modulus and closedloop phase angle d, are for any given openloop log modulus Lq and openloop phase angle 0q. See Fig. 13.11a. The graph is a completely general one... 

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