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the magnitude and phase angle of the transfer function G(s) (with s = jr.o) and
are given by
M8 - IG( jto)l
Z~g = ZG(jw)
Taking the inverse Laplace transforms in Eq. (5.109), we get
Yoo(t) = M2M [e-/M +*r +ou) + ej~ +48 +wr)]
= MtMg cos(~t + 4g + tot)
or, in phasor notation,
(5.110)
(5.111)
(5.112)
(5.113)
Mco/~ -. M,M8/(~/ + 48) (5.114)
'J
~
{
~
*
i
t,
ii
J1
~
r.
L.
:l
J
H
-.
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)
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