Primera Parte.- La autonomía municipal
4.1. La génesis de la ley y la crisis del parlamentarismo
4.1.1. Las posturas de los partidos
25 K
s s( + ) ; K = 1
read at the gain crossover frequency wg = 47 rad/sec is 28º. To realize a phase margin of 45º, the gain crossover frequency should be moved to w¢g
where the phase angle of the uncompensated system is:
– 180º + 45º = – 135º.
From the Bode plot of the uncompensated system we find that wg¢ = 25.
Since the lag compensator contributes a small negative phase when the upper corner frequency of the compensator is placed at 1/10 of the value of
¢
wg, it is a safe measure to choose wg¢ at somewhat less than 25 rad/sec, say, 20 rad/sec.
The attenuation necessary to cause w¢g = 20 to be the new gain crossover frequency is 14 dB.
The b parameter of the lag compensator can now be calculated.
20 log b = 14. This gives b = 5
Placing the upper corner frequency of the compensator a decade below wg¢, we have 1/t = 2. Therefore the lag compensator
D(s) = 0 1
2 1
.5 .5 s s
++
From the Bode plot–Nichols chart analysis we find that the compensated system has
FM = 51º; Mr = 1.2 and wb = 27.5 rad/sec.
10.11 Kv = lim
s®0 sG(s) = 5
From the Bode plot we find that the uncompensated system has a phase margin of – 20º; the system is therefore unstable.
We attempt a lag compensator. This choice is based on the observation made from Bode plot of uncompensated system that there is a rapid decrease in phase of G(jw) near wg. Lead compensator will not be effective for this system.
To realize a phase margin of 40º, the gain crossover frequency should be moved to w¢g where the phase angle of the uncompensated system is:
– 180º + 40º + 12º = – 128º
From the Bode plot of the uncompensated system, it is found that wg¢ = 0.5 rad/sec. The attenuation necessary to cause wg¢ to be the new gain crossover frequency is 20 dB. The b parameter of the lag compensator can now be calculated.
20 log b = 20. This gives b = 10
Since we have taken a large safety margin of 12°, we can place the upper corner frequency of the compensator at 0.1 rad/sec, i.e., 1/t = 0.1. Thus the transfer function of the lag compensator becomes
D(s) = 10 1
100 1 s
s ++
From the Bode plot of compensated system we find that the phase margin is about 40° and the gain margin is about 11 dB.
10.12 It easily follows that K = 30 satisfies the specification on Kv. From the Bode plot of
G(jw) = 30
0 1 1 0 1
jw(j . w+ )(j .2w+ )
we find that gain crossover frequency is 11 rad/sec and phase margin is – 24º. Nichols chart analysis gives wb = 14 rad/sec.
If lead compensation is employed, the system bandwidth will increase still further, resulting in an undesirable system which will be sensitive to noise.
If lag compensation is attempted, the bandwidth will decrease sufficiently so as to fall short of the specified value of 12 rad/sec, resulting in a sluggish system. These facts can be verified by designing lead and lag compensa-tors. We thus find that there is need to go in for lag-lead compensation.
Since the full lag compensator will reduce the system bandwidth excessively, the lag section of the lag-lead compensator must be designed so as to provide partial compensation only. The lag section, therefore, should move the gain crossover frequency to a value higher than the gain crossover frequency of the fully lag-compensated system. We make a choice of new gain crossover frequency as wg¢ = 3.5 rad/sec. The attenuation necessary to cause wg¢ to be the new gain crossover frequency is 18.5 dB. This gives the b parameter of the lag section as
20 log b = 18.5; b = 8.32, say, 10
SOLUTION MANUAL 105
Placing the upper corner frequency of the lag section at 1/t1 = 1, we get the transfer function of the lag section as
D1(s) = t
It is found that the lag-section compensated system has a phase margin of 24º.
We now proceed to design the lead section. The implementation of the lag-lead compensator is simpler if a and b parameters of the lead and lag compensators, respectively, are related as a = 1/b. Let us first make this choice. If our attempt fails, we will relax this constraint on a.
The frequency at which the lag-section compensated system has a magnitude of – 20 log (1/ a ) = – 10 dB is 7.5 rad/sec. Selecting this frequency as gain crossover frequency of the lag-lead compensated magnitude curve, we set
1
a t2 = 7.5
The transfer function of the lead-section becomes D2(s) = t
The analysis of the lag-lead compensated system gives FM = 48º and wb = 13 rad/sec.
In order to meet the specification of Kv = 40, K must be set at 1440. From the Bode plot of
G(jw) = 144000
36 100
jw w(j + )(jw+ )
we find that the gain crossover frequency is 29.7 rad/sec and phase margin is 34º.
The phase lead required at the gain crossover frequency of the compensated magnitude curve = 48º – 34º + 10º = 24º
a = 1 24
1 24
-+
sin º sin º = 0.42
The frequency at which the uncompensated system has a magnitude of – 20 log ( /1 a) = – 3.77 dB is 39 rad/sec. Selecting this frequency as gain crossover frequency of the compensated magnitude curve, we set
1
at = 39
The transfer function of the lead compensator becomes D(s) = 1440 0 04 1
0 0168 1
( . )
. s s
+ +
It can easily be verified by Nichols chart analysis that the bandwidth of the compensated system exceeds the requirement. We assume the peak time specification is met. This conclusion about the peak time is based on a second-order approximation that should be checked via simulation.
10.14 K = 2000 meets the requirement on Kv. We estimate a phase margin of 65º to meet the requirement on z.
(a) From the Bode plot we find that the uncompensated system with K = 2000 has a phase margin of zero degrees. From the Bode plot we observe that there is a rapid decrease of phase at the gain crossover frequency. Since the requirement on phase lead is quite large, it is not advisable to compensate this system by a single-stage lead compensator (refer Fig. 10.15).
(b) Allowing 10º for the lag compensator, we locate the frequency at which the phase angle of the uncompensated system is:
– 180º + 65º + 10º = – 105º. This frequency is equal to 1.5 rad/sec. The gain crossover frequency wg¢ should be moved to this value. The necessary attenuation is 23 dB. The b parameter of the lag com-pensator can now be calculated.
20 log b = 23. This gives b = 14.2
SOLUTION MANUAL 107
Placing the upper corner frequency of the compensator one decade below wg¢, we have 1/t = 0.15. Therefore the lag compensator
D(s) = 6 66 1 94 66 1
. .
s s
+ +
The phase margin of the compensated system is found to be 67º.
(c) The bandwidth of the compensated system is found to be wb = 2.08 rad/sec.
10.15 Approximate relation between fm and z is z » 0.01fm
z= 0.4 ® fm = 40º Let us try lag compensation.
The realize a phase margin of 40º, the gain crossover frequency should be moved to wg¢ where the phase angle of the uncompensated system is:
– 180º + 40º + 10º = – 130º. From the Bode plot of the uncompensated system we find that, wg¢ = 6. The attenuation necessary to cause wg¢ to be the new gain crossover frequency is 9 dB.
20 log b = 9. This gives b ~- 3
Placing the upper corner frequency of the lag compensator two octaves below wg¢, we have 1/t = 1.5.
The lag compensator D(s) = 0 67 1
2 1
. s s
+ +
Nichols chart analysis of the compensated system gives wb = 11 rad/sec.
Using second-order approximation
wb = wn 1 2 2 2 4 2 4 4 - z + - z + z 1 2/
we get wn = 10
Therefore ts = 4/zwn = 1 sec
Note that the zero-frequency closed-loop gain of the system is nonunity.
Therefore, the use of – 3 dB bandwidth definition, and the second-order system correlation between wb and z & wn may result in considerable error.
Final design must be checked by simulation.
10.16(a) From the Bode plot of uncompensated system we find that FM = 0.63º
(b) Phase margin of the system with the second-order compensator is 9.47º. There is no effect of the compensator on steady-state performance of the system.
(c) The lead compensator D(s) = 1 0 0378
1 0 0012 + +
. .
s s
meets the requirements on relative stability.
10.17 Kv = 4.8
CHAPTER 11 HARDWARE AND SOFTWARE