Log modulus plot

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

The lines of constant closedloop log modulus L, are part of the Nichols chart. If we are designing a closedloop system for an L specification, we merely have to adjust the controller type and settings so that the openloop B curve is tangent to the desired line on the Nichols chart. For example, the G B curve in Fig. 13,11b with X, = 20 is just tangent to the +2 dB line of the Nichols chart. The value of frequency at the point of tangency, 1.1 radians per minute, is the closedloop resonant frequency aif. The peak in the log modulus plot is clearly seen in the closedloop curves given in Fig. 13.12. 

Flo. 7.56. Log modulus plots of the open-loop transfer function GjXs) in Example 7.8 ... 

Fig. 7.64. Log modulus plots of open-loop transfer function for Example 7.10 curve (i)—uncompensated system curve (ii)—compensated system...

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. 

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. 

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

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. 

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. 

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... 

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... 

A proportional controller merely multiplies the magnitude of at every frequency by a constant. On a Bode plot, this means a proportional controller raises the log modulus curve by 20 logiQ decibels but has no effect on the phase-angle curve. See Fig. 13.13n. 

On a Bode plot (Fig, 13.16), the log modulus curve of G B must pass through the 0-dB point when the phase-angle curve is at —135°. This occurs at tu = 1 radian per minute. The log modulus curve for K,. = 8 must be raised - -9 dB (gain 2,82). Therefore the controller gain must be (8X2.82) = 22.6. 

The Bode plot of is given in Fig. 13.20 for D = 0.5. The ultimate gain is 3.9 (11.6 dB), and the ultimate frequency is 3.7 radians per minute. The ZN controller settings for P and PI controllers and the corresponding phase and gain margins and log moduli are shown in Table 13.2 for several values of deadtime D. Also shown are the values for a proportional controller that give +2-dB maximum closedloop log modulus. 

A process has and openloop transfer functions that are first-order lags and gains Tm. tj, and X. Assume -r is twice Tj,. Sketch the log modulus Bode plot for the closedloop load transfer function when ... 

If a transfer-function model is desired, approximate transfer functions can be fitted to the experimental curves. First the log modulus Bode plot is used. The low-frequency asymptote gives the steadystate gain. The time constants can be found from the breakpoint frequency and the slope of the high-frequency asymptote. The damping coefficient can be found from the resonant peak. 

The peak in the plot of over the entire frequency range is the biggest log modulus... 

Generation of Master Curves. Modulus and loss factor data were processed into a reduced frequency plot in the following manner modulus curves at different temperatures were shifted along the frequency axis until they partially overlapped to obtain a best fit minimizing the sum of the squares of a second order equation (in log modulus) between two sets of modulus data at different temperatures. This procedure was completely automated by a computer program. The modulus was chosen to be shifted rather than the loss factor because the modulus is measured more accurately and has less scatter than the loss factor. The final result is a constant temperature plot or master curve over a wider range of frequency than actually measured. Master curves showing the overlap of the shifted data points will not be presented here, but a typical one is found in another chapter of this book (Dlubac, J. J. et al., "Comparison of the Complex Dynamic Modulus as Measured by Three Apparatus"). 

Mechanical measurements do not range over many decades of frequency, so that the well-established principle of time-temperature equivalency (49) is used to fill in the gaps. The continuous plot of log modulus versus log frequency obeys a distinctive profile for a liquid (Systems 1 and 2), for lightly gelled solid (Systems 3 and 4), and for the hardened film (System 5). 

---