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Unformatted text preview: 6.43. T he loci cross over t o R HP
for K > 48. For K < 48, t he s ystem is stable. Let us consider t he case K = 24.
F igure 7.10b shows Bode plots when K = 24. Let t he frequency where t he angle
plot crosses - 180° b e w p ( the p hase c rossover f requency). Observe t hat a t W p ,
t he gain is 0.5 or - 6 dB. This shows t hat t he gain K will have to double (to value
48) to have unity gain, which is t he verge of instability. For this reason we say t hat
t he system has a gain margin a M = 6 dB. On t he o ther hand, if W g is t he frequency
where t he gain is u nity or 0 dB, (the g ain c rossover f requency), t hen a t t his
frequency, t he open-loop phase is - 157.5°. T he phase will have to decrease from 7 Frequency Response a nd Analog Filters 492 (a) 24 12 o
OJ - - -12 -24
( b) -90" +-____ __ __ __ -18~ ~_ ___ ~ ~ ~ --~+-~~---------4---------- OJ - - -27~ 7.3 C ontrol System Design Using Frequency Response this value to - 180° before t he s ystem becomes unstable. Thus, the system has a
phase margin of (J M = 22.5°. Clearly, t he gain a nd phase margins are measures of
relative stability of t he system.
Figure 7.lOc shows t hat t he Nyquist plot of K G(s)H(s) crosses t he real axis
a t - 0.5 for K = 24. I f we double K t o a value 48, t he m agnitude of every point
doubles (but t he p hase is unchanged). This step expands t he Nyquist plot by a
factor 2. Hence, for K = 48, t he N yquist plot lies on t he real axis a t - 1; t hat
is, K G(jw)H(jw) = - 1, a nd t he s ystem becomes unstable. For K > 48, t he p lot
crosses a nd goes beyond t he p oint - 1. T hus, t he critical point - 1 lies inside the
curve; t hat is, t he curve encircles t he critical point - 1. W hen t he Nyquist plot
of a n open-loop transfer function encircles t he critical point - 1, t he corresponding
closed-loop system becomes unstable. This statement, roughly speaking, is t he wellknown N yquist c riterion i n a simplified form.t For the Nyquist plot in Fig. 7.lOb
(for K = 24), the gain will have t o d ouble before the system becomes unstable.
Thus, t he gain margin is 2 (6 dB)in this case. In general, if t he Nyquist plot crosses
t he negative real axis a t - a"" t hen t he gain margin is l /a",. Similarly, if - 7r + (Jm
is t he angle a t which t he N yquist plot crosses t he u nit circle, t he phase margin is
( Jm' In t he p resent case, (Jm = 22.5°.
In order t o p rotect a system from instability because o f v ariations in s ystem
parameters (or in t he e nvironment), t he s ystem should be designed with reasonable
gain a nd p hase margins. Small margins indicate t hat t he poles of t he closed-loop
system are in t he LHP, b ut t oo close to t he j w axis. T he t ransient response of such
systems will have a large overshoot ( PO). On t he o ther hand, very large (positive)
gain a nd phase margins may indicate a sluggish system. Generally, a gain margin
higher t han 6 dB a nd a phase margin of a bout 30° t o 60° a re considered desirable.
Design specifications for transient performance are ofte...
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