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Unformatted text preview: d t
L HUm) jm Re .... 0 m.... 0 -Ct. d' 0 mo m.... - 11 :i:
( c) (b) (a) t
Im jmo t
IH(jw)1 = d (7.30) where t he e xact value of constant K is n ot i mportant a t t his point. As w increases
from zero up, d decreases progressively until w reaches t he value woo As w increases beyond wo, d increases progressively. Therefore, according t o Eq. (7.30),
t he a mplitude response IH (jw) 1 increases from w = 0 u ntil w = wo, a nd i t decreases
continuously as w increases beyond wo, as i llustrated in Fig. 7.l4b. Therefore, a
pole a t - 0 + jwo r esults in a frequency-selective behavior t hat enhances t he gain a t
t he frequency Wo (resonance). Moreover, as t he pole moves closer t o t he i maginary
axis (as 0 is reduced), this enhancement (resonance) becomes more pronounced.
This is because 0 , t he d istance between t he pole a nd jwo (d c orresponding t o jwo),
becomes smaller, which increases t he g ain K id. In t he e xtreme case, when 0 = 0
(pole on t he i maginary axis), t he gain a t Wo goes t o infinity. Repeated poles further
enhance t he frequency-selective effect. To summarize, we c an enhance a gain a t a
frequency Wo by placing a pole opposite t he p oint jwo. T he closer t he pole is t o jwo,
t he higher is t he gain a t wo, a nd t he gain variation is more rapid (more frequencyselective) in the vicinity of frequency woo Note t hat a pole must be placed in t he
L HP for stability.
Here we have considered t he effect of a single complex pole on t he s ystem gain.
For a real system, a complex pole - 0 + jwo m ust accompany its conjugate - 0 - jwo.
We can readily show t hat t he presence of t he c onjugate pole does not appreciably
change t he frequency-selective behavior in t he vicinity of woo T his is because t he
gain in this case is K I d d', where d' is t he d istance of a point jw from t he c onjugate
pole - 0 - jwo. Because t he c onjugate pole is far from jwo, t here is no d ramatic
change in t he length d ' as w varies in t he vicinity of woo T here is a g radual increase
in t he value of d ' as w increases, which leaves t he frequency-selective behavior as it
was originally, with only minor changes.
Gain Suppression by a Zero jm
11 Re .... 0 -Ct. 499 t he a mplitude response IH (jw)1 for a certain value of w, we c onnect t he pole t o t he
p oint j w (Fig. 7 .l4a). I f t he l ength of this line is d, t hen IH(jw)1 is p roportional t o L H(s)ls=p = (<PI + <P2 + . .. + <Pn) - (/11 + /12 + . .. + /1 n ) t
1m 7.4 Filter Design by Placement of Poles a nd Zeros 0 COo ro .... 0 00 . ... r'
,:~. (d) (e) ( t) F ig. 1 .14 The role of poles and zeros in determining the frequency response of an LTIC system.
Gain Enhancement by a Pole To u nderstand t he effect o f poles a nd zeros on t he frequency response, consider
a hypothetical c ase of a single pole - 0 + jwo, as depicted in Fig. 7.l4a. To find Using t he s ame argument, we observe t hat zeros a t - 0 ± jwo (Fig. 7 .l4d) will
have exactly the opp...
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