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gain margin indicates how much the open-loop gain can be increased before the
closed-loop system becomes unstable. For example, a gain margin of30 db implies
that the open-loop gain can be increased by a factor of 31.6228 before the closed-
loop system becomes unstable. On the other hand, if the gain margin is -30 db,
then the closed-loop system is already unstable, and the gain has to be reduced by
a factor of 31.6228 to make the closed-loop system stable.
    The phase margin is defined as the amount of additional phase lag at the gain
crossover frequency that can be introduced in the open-loop system to make the
closed-loop system unstable. The gain crossover frequency 002 iS that frequency
when the magnitude of the open-loop transfer function G( jeo) is uruty.
LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REVIEW      483
    The phase margin is usually denoted by #M and is expressed in degrees and is
given by
4M - 180 + 4
(5.144)
M(db)
+
z
Phasc(deg) -90
  -180
Positive Gain Margin
   ~}                
┏━━━━━━━━━┓
┃  ~   ~o          ┃
┣━━┳━━━━━━┫
┃-~- ┃        Pi  ┃
┃   }┃     oa     ┃
┣━━┻━━━━━━┫
┃Positive Phase    ┃
┃Margin            ┃
┗━━━━━━━━━┛
M(db)
+
Z
(deg) -90
 -180
                           
  ~   / ~:y:e'ai'          
┏━┳━━━━━━━━━━┓
┃  ┃                    ┃
┣━╋━━━━━━━━━━┫
┃~ ┃.\~ o,.             ┃
┃  ┃ .b   .             ┃
┃~-﹢Fig. 5.22    Gain and phase margins using Bode plots.
Margiri
484            PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
5.8   Relations Between Time-Domain and Frequency-Domain
Parameters
   Generally, the performance requirements for control systems are specified in
terms of time-domain parameters like rise time Tr, settling time Ts, time for peak
amplitude Tp, and percent overshoot  Os. In the following, we present some rela-
tions between these time domain parameters and frequency-domain parameters.
These relations will be useful in the analyses and design of control systems using
frequency-domain methods.
   Consider a second-order system whose open-loop and unity feedback closed-
loop transfer functions are given by
         G(s)=- to~        (5.145)
                                                              = s~s + 2-l ton)
                     T(s)=  +2 .. +                  (5.146)
Let M denote the magnitude of the closed-loop frequency response. rfhen,   '
                   60~
            (5.147)
    M-.IT(/w)l=~ r+4C2tO~C02
    A typical plot of M vs c.o is shown in Fig. 5.23.
201o91o
┏━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃   10 ' ┃                                                                                                  ┃
　
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