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                                 = MtMg cos(~t + 4g + tot)
or, in phasor notation,
(5.110)
(5.111)
(5.112)
(5.113)
Mco/~   -. M,M8/(~/ + 48)                         (5.114)
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Mt =~
4/ = -t  -':~
458            PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
~&,  with frequency is called the frequency response of the system. In other words,
the ffequency response of a system whose transfer function is G(s),s -  jw iS
nothing but the variation of Mg and 4g with frequency co.
    One of the most widely usc~'methods of obtainin~ the frequency response of
a transfer function is the Bode plot It consists of two parts: the magnitude plot
in decibels where one decibel of M  = 20 1o9io M and the phase plot in degrees,
both plotted against frequency co, which is usually expressed in radians/second.
      Gevnerally, the Bode plotis drawn for open-loop transfer function G(s). Further-
more,if the transfer function contains a variable gain k, then the Bode plotis made
for k -.  1. For any other value of k, the corresponding Bode plot can be easily
obtained by shifting the entire Bode plot by 20 1o9io k. vfhe plot shifts upward if
k > O and downward if k < 0. We willillustrate the'method of drawing a Bode
plot with the help of Example 5.1.
                                             Example 5.1
Draw the Bode plot for a system given by
                                         G(s) = (, ,k/(S~S3l_ 2)
  So/ution.  The first step is to assume k-l and rewrite the given transfer
function in the following form:
      ~(3+1)
                                               G(s) = s(s +1)(2 +1)
             '1(S)
                         = G,(,)GG,(()G,(,)
where
                                        Gi(s) = (3+1)
                                           G2(S) :s
                                                                   G3(s) -. (s + 1)
                                      G4(S) = (2+1)
Substituting s  =  jc.o, taking logarithms on both sides, multiplying by 20 to convert
to decibels, and taking absolute values, we get
               20 lo91o IG( jc,o)] = 20 1o91o lz + 20 lo91o lGi( jco)l
                                -. 20  1o91o IG2( jco)l  - 20 1o91o IG3( jco)l  - 20  1o9io IG4( jco)l
　
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