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Unformatted text preview: rom t he j waxis die quickly compared to those arising
because o f poles located near t he jwaxis. In addition, t he coefficients of t he former
t erms a re much smaller t han unity. Hence, t hey are also very small to begin with
a nd d ecay rapidly. T he poles n ear t he j waxis are called t he d ominant p oles. A
c riterion commonly used is t hat a ny pole which is five t imes as far from t he j waxis
as t he d ominant poles contributes negligibly t o t he s tep response, a nd t he t ransient
behavior of a higherorder system is o ften reduced t o t hat o f a secondorder system.
I n a ddition, a closely placed polezero p air (called d ipole), c ontributes negligibly
t o t he t ransient behavior. For this reason, m any o f t he polezero configurations in
practice reduce t o two or t hree poles w ith one or two zeros. Workers in t he field have
prepared c harts for t ransient b ehavior o f t hese systems for several such polezero
combinations, which may b e used t o design most of t he h igherorder systems. j ro 6 4 K=O 4 6 1 ,=2 I
y (t) F ig. 6 .41 Designing a secondorder system to meet a given transient specifications. S2 + 8s + K (6.87b) =0 Hence t he p oles a re
S I,2 = 4 T he poles S j a nd S 2 ( the roots of t he c haracteristic equation) move along a certain
p ath in t he s plane as we v ary K from 0 t o 0 0. W hen K = 0, t he poles are  8, O.
For K < 16, t he poles are real a nd b oth poles move towards a value  4 as K
varies from 0 t o 16 (overdamping). For K = 16, b oth poles coincide a t  4 (critical
damping). F or K > 16, t he poles become complex with values  4 ± j JK  16
( underdamping) Since t he r eal p art o f t he poles is  4 for all K > 16, t he p ath of
t he poles is v ertical as illustrated in Fig. 6.41. O ne pole moves u p a nd t he o ther
( its conjugate) moves down along t he v ertical line passing thro~gh ~4.
c an
label t he v alues of K for several points along these paths, as depicted m Fig. 6.41.
Each of these p aths r epresents a locus of a pole of T (s) or a locus of a root of the
characteristic e quation o f T (s) as K is varied from 0 t o 0 0. F or this reason this set
of p aths is c alled t he r oot l ocus. T he r oot locus gives us t he info~mati~n as. to
how t he poles o f t he closedloop transfer function T (s) move as t he g am K IS v aned
from 0 t o 0 0. I n o ur design problem, we m ust choose a value of K s uch t hat t he
poles of T (s) l ie in t he s haded region shown in Fig. 6.41. This figure shows t hat t he
system will m eet t he given specifications [Eq. (6.86)J for 25 :s: K :s: 64. For K = 64, "!'Ie for instance we have P O = 16%, t r = 0.2 seconds, F ig. 6 .42 A feedback system with a variable gain K . (6.87c) ± J 16  K ts 4 =  = 1 seconds
4 (6.88) 6 .73 Root Locus T he e xample in Sec. 6.72 gives a good idea of the utility of root locus in
design o f control systems. Surprisingly, root locus can be sketched quickly using
certain basic rules provided by W .R. E vans in 1948.t W ith...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.
 Spring '13
 Bayliss
 Signal Processing, The Land

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