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Unformatted text preview: e frequency may have a positive feedback a t some other frequency
because of phase shift in t he transmission p ath. I n other words, a feedback system,
generally, cannot be described in black a nd w hite terms such as having positive or
negative feedback. Let us clarify this s tatement by a n example.
Consider t he case G (s)H(s) = l /s(s + 2)(s + 4). T he r oot locus of this system
appears in Fig. 6.43. This system shows negative feedback a t lower frequencies.
B ut because of p hase shift a t higher frequencies, t he feedback becomes positive.
Consider t he loop gain G (s)H(s) a t a frequency w = 2.83 (at s = j2.83).
1 G (jw)H(jw) = j w(jw + 2)(jw + 4) At w = 2.83
G (j2.83)H(j2.83) = j2.83(j2.83 + 2)(j2.83
1 _ '1800 = -e 48 J + 4) 1
=-- 48 Recall t hat t he o verall gain (transfer function) T (s) is K G(s)
T (s) = 1 + K G(s)H(s)
At frequency s = j2.83 (w = 2.83), t he gain is
T (j2.83) = KG(J·2.83)
K 6 .8 The Bilateral laplace Transform
Sit~ations involving noncausal signals a nd/or systems cannot be handled by
the (umlateraJ) Laplace transform discussed so far. These cases can be analyzed by
t he b ilateral (or t wo-sided) Laplace transform defined by
F (s) = I: f (t)e-stdt (6.101a) a nd f (t) c an be obtained from F (s) by t he inverse transformation 1 c jOO 1
27rJ + F (s)estds c -joo (6.1Olb) Observe t hat t he u nilateral Laplace transform discussed so far is a special case of
t he .bilateral Laplace transform, where t he signals are restricted t o t he c ausal type.
Basically, t he two transforms are t he same. For this reason we use t he same notation
for the bilateral Laplace transform .
. Earl~er we showed t hat t he Laplace transforms of e -atu(t) a nd of - eatu(_t)
a re Identical. T he only difference involves their regions of convergence. T he region
o f convergence for t he former is Re s > - a; t hat for t he l atter is Re s < - a as
illustrated in .Fig. 6.2. Clearly, t he inverse Laplace transform of F (s) is n ot uni~ue
unless the regIOn o f convergence is specified. I f we restrict all o ur signals to the causal
~yp:}owever, thi~ ambigui~y does n ot arise. T he inverse transform of l /(s + a)
IS e u (t). Thus, I II t he u mlateral Laplace transform, we c an ignore t he region of
convergence in determining t he inverse transform of F (s ).
We now show t hat any bilateral transform can be expressed in terms of two
unilateral transforms. I t is, therefore, possible t o e valuate bilateral transforms from
a table of unilateral transforms.
Consider t he function f it) a ppearing in Fig. 6.48a. We s eparate f it) i nto two
components, h it) a nd h it), r epresenting t he positive time ( causal) c omponent
a nd t he negative time ( anticausal) c omponent of f it) respectively (Figs. 6.48b
a nd 6.48c): 1 - 48 As long as K r emains below 48, t he system is s table, b ut for K = 48, t he s ystem gain
goes t o 0 0, a nd t he system becomes unstable. T he feedback, which was negative
below w = 2.83 (because t he phase shift has n ot reached - 180°), becomes positive.
I f t here is e nough gain ( K = 48)...
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