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        <=,~7~2  .     (5.150)
Using these relations, given the value of Mp, we can determine the percent over-
shoot Os of the given closed-loop system and vice versa.
      The bandwidth C-OBW iS another important characteristic of the closed-loop fre-
quency response. The bandwidth is defined as that frequency at which the magni-
tude M drops to 0.707 or 1/1~ ofits value at w - O. This is also equivalent to
a drop by 3 db. From Eq. (5.147), we find that M - 1 when a  : O. Therefore,
substituting M  N  1]^~ and co = O)B W  in Eq. (5.147), we obtain
                       N
         COBW-tOn,/-C: 2-2~+ ~     (5.151)
The settling time Ts and time for peak amplitude Tp given by Eqs. (5.77) and (5.72)
are reproduced in the following:
Ts = < 4                                        (5.152)
                              Tp = -.                              (5.153)
                           7r
                                -- -                              (5.154)
                                                10 ,~-7_
Using Eq. (5.151), we can rewrite these relations in terms of bandwidth COBW as
follows:
  -_-
   Ts~(4)   V  7/  cs.iss)
                                           N
  Tp=(~.,./T7,),~~2-2~+ ~ (5.156)
     The above equations give relations between the time-domain parameters Ts and
Tp and the frequency-response parameter 60BW for second-order systems. These
relations contain the two basic system parameters, < aricl CDn.
F. .j
486           PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
   Another important parameter of.the frequency-domain design method is the
phase margin 4M.A relation between phase margin 4m and the damping parameter
< can be obtained as follows,
    Let co : coi  when the magnitude of the open-loop frequency response is unity,
i.e., tG( jco)l = 1, or
so that
Gj(toi )l
t01 - tOn
l(-co~
The phase angle of G( jto) at co = wi is given by
/G(jt.oi) = -90 - tan-l   o)l
      2<w,
~ -90 - tan-l
The phase margin is given by
     1
4M - 180+/G( jcoi) = 90 - tan-l
-. tan-l
2<
The variation of @,y with < is shown in Fig. 5.24.
(5.157)
(5.15 8)
(5.15 9)
(5.160)
) ,5.161)
(5.162)
5.9   Design of Compensators
     The response character:istics of a given system depend on the inte:mal physical
nature of the system and may not meet the specified performance requirements.
A simple modification of the plant dynamics is-an obvious first choice to meet
the performance specifications. However, such a thing may not be possible in
practice because the plant can be quite complex so that it may not be'easy to
affect the necessary modifications.ln such cases, the adjustment of the gain is the
next obvious step. However, in many cases, this alone may not be sufficient, and
one may have to redesign the entire plant. Such a process can be quite expensive
　
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