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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
Now let us consider the magrutude plot of each one of the terms on the right-hand
side separately. The magnitude plot of the term 20 1o9io g  = 3.5218 for all values
LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REVIEW       459
of the frequency co_ For the second term,
20 1o9io IGi( jco)l = 20 1o9,o I(J3 + 1)
= 20 ,o9,o I
For smaller values of co (low-frequency approximation), we assume
                                          20 1o9io IGi( jeo)l = 20 1o9io 1
              -O
and, for higher values of co (high-frequency approximation), we assume
                                         20 1o9io lGi( jco)l = 20 1o9io 3
Thus, for a> - 3 rad/s, 20 1o9ioIGi(jco)l=0 and, for co : 30 rad/s, 20 1o91o
 x IGi( jco)l = 20 db. -fhe slope of the high-frequency approximation of this term
is +20 db/decade. Here, one decade means a tenfold increase in frequency.
   The frequency at which the low-frequency approximation intersects the high-
frequency approximation is called the corner frequency. For the magnitude plot of
Gi( jco), the corner frequency is 3 racVs. The low-frequency approximation holds
for frequencies that are below the corner frequency, and the high-frequency ap-
proximation holds for frequencies that are above the corner frequency:Broceeding
in a similar way, we find that the magnitude plots of G3( jco) and G4( jco) have
corner frequencies of 1.0 and 2.0 rad/s, respectively. The magnitude plot of G2( ja )
is a straight line and hence has no corner frequency. Each of the magnitude plots
of G2( jco), G3( jW), and G4( jc.o) have a slope of -20 db/decade. The component
magnitude plots of Gi( jw), G2( jco), G3( jc.o), and G4( jcn) are shown in Fig. 5.9.
    The phase plot can be drawn using the relation
ZG( jco) = z g: + /Gi( jco) - ZG2( jc.o) - ZG3( jC.O) - /G4( jc.o)
Note that /~ = O. As before, let us consider the terms on the right-hand side one
by one. The phase of the second term is given by
                       ZGi(jt.o) = /(J3 +1) = ta.-  (3)
For small values of co Oow-frequency approximation), /Gi( jco) = 0. As co -y  oo
(high-frequency approximation), ZGi( jw) = 90 deg. For co = 3 rad/s, ZGi( jco)  =
45 deg and, for co  =  30 rad/s, LGi( jco)  ~  90 deg so that the slope of high-frequency
approximation is 45 deg/decade.We assume that the low-frequency approximation
holds for frequencies that are one decade below the frequency at which the phase
angle is 45 deg. For /Gi( jeo), this value is 0.3 rad/s.
   With these approximations, the phase plot LGi( jw) and those of other terms
are shown in Fig. 5.10,
-:T:
            *
460           PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
20
10
 o
201o9io jGi(ju')l  10
201o9io jG2Uu,)l
201o91o IG3(jco)l
r
[n]
10
┏━━━━━━━━━━━━━━━━━━━━━━━┓
┃~                                             ┃
┃  ~ -20db/dec                                 ┃
┃                     3 5    8       10        ┃
　
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