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Unformatted text preview: a ctual plot can be obtained by adding t he e rror t o t he
a symptotic p lot.
.
.
For secondorder zeros (complex conjugate zeros), t he plots are mirror Images
(about t he OdB line) of t he plots depicted in Fig. 7.6a. N~te t he reso~ance phenomenon o f t he complex conjugate poles. This phenomenon IS barely noticeable for
( > 0.707 b ut becomes pronounced as ( > O. Phase
T he p hase function for secondorder poles, as apparent in Eq. (7.19b), is For W «Wn , L H(jw) "" 0
LH(jw)::=  180°
Hence, t he phase >  180° as W > 0 0. As in t he case of amplitude, we also have a
family of phase plots for various values of ( , as illustrated in Fig. 7.6b. A convenient
asymptote for t he p hase of complex conjugate poles is a s tep function t hat is 0° for
W < W n a nd  180° for W > W n . A n error plot for such a n a symptote is shown in
Fig. 7.7 for various values o f ( . T he e xact phase is t he a symptotic value plus t he
error. 7 4 86 F requency R esponse a nd A nalog F ilters 7.2 F or c omplex c onjugate z eros, t he a mplitude a nd p hase p lots a re m irror i mages
o f t hose for c omplex c onjugate p oles. W e s hall d emonstrate t he a pplication o f t hese
t echniques w ith t wo e xamples.
• B ode P lots 4 87 50 As 45 "" ::t: 1 .'(\ : X1 i
: ", r tlll
"   . 1'
: :
Ex'\Ct 1,*',
' i ll E xample 7 .3
S ketch t he B ode plots for t he t ransfer function I
I \ I I',
ImptoUl'P~I:I '
" ' I, ',~; ' 1/11" I
, ' I ' I'
" : !:,'
, 40 I r 1
: i 00 b I) .£ H 8 _ 208(8 + 100)
( )  (8 + 2)(8 + 10) 0 35 N F irst, we w rite t he t ransfer function in normalized form
2 0x100 H (8)
= 8(1+1~O) '2><lO (1 + ~) (1 + fa) 8(1+1~O)
= 100..,..',...,...""'"...,...,(1 + ~) (1 + fa) 30 25 fflfHIHH++++ P hase P lots
We draw t he a symptotes corresponding to each of t he four factors: I ~t+HfI "
i ii I ' I ", 1 Here, t he c onstant t erm is 100; t hat is, 40 d B (20 log 100 = 40). T his t erm c an b e a dded
t o t he p lot by s imply relabeling t he h orizontal axis (from where t he a symptotes begin) as
t he 40 d B line (see Fig. 7.8a). Such a s tep implies shifting t he h orizontal axis upward by
40 dB. This is precisely w hat is desired.
I n a ddition, we have two firstorder poles a t  2 a nd  10, o ne zero a t t he origin, and
one zero a t  100.
S tep 1 . F or each of these terms, we d raw a n a symptotic plot as follows (see Fig.
7.8a):
(i) For t he z ero a t t he origin draw a s traight line with a slope of 20 d B/decade passing
t hrough w = 1.
(ii) For t he p ole a t  2, d raw a s traight line with a slope of  20 d B/decade (for w > 2)
b eginning a t t he c orner frequency w = 2.
(iii) For t he p ole a t  10, d raw a straight line with a slope of  20 d B/decade b eginning
a t t he c orner frequency w = 10.
(iv) For t he z ero a t  100, d raw a s traight line with a slope of 20 d B/decade b eginning
a t t he c orner frequency w = 100...
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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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