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As shown by Eq. (16.13, the frequency response for G(s) = e- is AR=1 c$ = -or radians or C#I = -57.2958 or degrees
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In this expression, o is in radians and 57.2958 is the number of degrees in one radian. There is no need to plot the AR since it is constant at 1 .O. On logarithmic coordinates, the phase angle appears as in Fig. 16.9, where WT is used as the abscissa to make the figure general. The transportation lag contributes a phase lag, which increases without bound as w increases. Note that it is necessary to convert or from radians to degrees to prepare Fig. 16.9.
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Phase characteristic of transportation lag.
Propmtlonal
Controller
A proportional controller with transfer function K, has amplitude ratio K, and phase angle zero at all frequencies. No Bode diagram is necessary for this component .
Proportional-Integral
Controller
This component has the ideal transfer function G(s) = K, 1 + $ i 1 Accordingly, the frequency response is given by AR = IG<jo>l = K, 1 + 1 &I =KcJ+&
Phase=4G(jo)=4(l+~)=tan- (-~) The Bode plot of Fig. 16.10 uses (~1) as the abscissa. The constant factor K, is included in the ordinate for convenience. Asymptotes with a corner
g s4 lo \\ E 1 1 A 0.1
1 10 =G-
100 FIGURE 16-10 Bode diagram for PI controller.
IWQLENCY RESPONSE
frequency of 0, = l/r1 are indicated. The verification of Fig. 16.10 is recommended as an exercise for the reader.
Proportional-Derivative Controller
The transfer function is G(s) = K,(l + T~S) The reader should show that this has amplitude and phase behavior that is just the inverse of the first-order system 1 7s + 1 Hence, the Bode plot is as shown in Fig. 16.11. The corner frequency is W, = l/To. This system is important because it introduces phase lead. Thus, it can be seen that using PD control for the tank temperature-control system of Example 16.3 would decrease the phase lag at all frequencies. In particular, 180 of phase lag would not occur until a higher frequency. This may exert a stabilizing influence on the control system. In the next chapter, we shall look in detail at designing stabilizing controllers using Bode diagram analysis. It is appropriate to conclude this chapter with a summarizing example.
Example 16.5. Plot the Bode diagram for the open-loop transfer function of the control system of Fig. 16.12. This system might represent PD control of three tanks in series, with a transportation lag in the measuring element. The open-loop transfer function is lO(0.5~ + l)e-s lo G(s) = (s + 1)2(0.1s + 1)
qi+ .Ol 0.1
1 wr, -F
FIGURE 16-11
Bode diagram for PD controller.
INTRODUCIlONTOFREQUENCYRESF'ONSE
Kc=10 B
iD= vi e-+r
FIGURE 16-12
Block diagram of control system for Example 16.5.
The individual components am plotted as dashed lines in Fig. 16.13. Only the asymptotes are used on the AR portion of the graph. Here it is easiest to plot the factor (s + 1)-2 as a line of slope -2 through the comer frequency of 1. For the phaseangle graph, the factor (s + l)- is plotted and added in twice to form the overall curve. The overall curves am obtained by the graphical rules previously presented. For comparison, the overall curves obtained without derivative action [i.e., by not adding in the curves corresponding to (0.5s + l)] am also shown. It should be noted, that, on the asymptotic AR diagram, the slopes of the individual curves are added to obtain the slope of the overall curve.
\ .c
0.05 FIGURE 16-13
Bode diagram for Example 16.5: (a) amplitude ratio; (b) phase angle.
220 FREQUENCY
90 45
RESPONSE
-225. -270.
FIGURE 16-13 (Continued)
Bode diagram for Example 16.5: (a) amplitude ratio; (b) phase angle.
PROBLEMS
16.1. For each of the following transfer functions, sketch the gain versus frequency,
asymptotic Bode diagram. For each case, find the actual gain and phase angle at o = 10. Nore: It is not necessary to use log-log paper; simply rule off decades on rectangular paper. @) (10s +l& + 1) @) (s + l,(K + 1)2 (c) (0.1s , l$lOS + 1) cd1 (0.1s +&los (e) (10s + 1)2 + 1)
cf) (10 + sj2 16.2. A temperature bath in which the temperature varies sinusoidally at various frequencies is used to measure the frequency response of a temperature-measuring element B. The apparatus is shown in Fig. P16.2. A standard thermocouple A, for which the time constant is 0.1 min for the arrangement shown in the sketch, is placed near the element to be measured. The response of each temperature-measuring element is recorded simultaneously on a two-channel recorder. The phase lag between
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