of 9
8/10/2019 Chapter8_2_Ahmadi_notes.pdf
1/9
M|nf
4soo
-
f."Jbodr contol
Equev.q
Resporu
nsfeJ
-pl
l r .
N.1
q,rr's
f
sto l;1,'{.,,
Cn'I+nirlr-r
.
.
7s
o'"J lo
**^::
tLr
st"b;li\
.f
"1o1al-tooy sTsternr
n
freluarg
olo:nah
I
,
fr/lo,nts
delorning
tl^t
nunbqr
4
p"la
in
fl,* r'6ht-hf etav*
6f
tt"
.lrte r"t-
slJttn
ftn-
tAq
" -l"t?_
y.
Fcsponsq-
. lJnnlgez
-rxpa'innb/16
.bloiw./
{rq.
rapon*
eon
bq
usal.
-
/o
racqd {o
{ns-r
ttc
leafrba
of
4A
poles.
Jn
5ho.t
termt
3
koowig)
t1,t)l4Qa1
*'+q5-*ffi,
o"
7,a4
f
=r6,t
h,'"^V
L-{ilJ
uact'oblL
Poles
(o7r
rlcts
u
F
kQH=oa;lh
prr,t,'^er'crl
rrt).
,fte
fr//0,,'41
rfuoreln
f's
7'-
7o.-tolJ.
{o-
+ts
rt"st
'f
&
ru[ert'
/-layTtg
Throrw,,
if
Ffs)
r's
a
r^t,o
of
'f,t"
71"4'oni"lJ
in
s,
,f
/:
S
zeros
rntide a
elot"ol
conlatr
)n
tlbnl-
o^"1
P
=
#
polu
n';de
a
clatJ
coa'loat
in
s-7hl.a
'
'A"on
TAe'
'tlt
.tosd
anlour
af
Fts)
enclr.leJ
+_:W-I*-Z-P
timel
clockvb,
, "
-t
h
aarvlo
r
p
3-
Tbac
ts a
/"rays
se/ec
t/
z:/acktuse
'
Nofes
\".ir;t*)"i-'
counlor-clock-tt,b,
rwos
8/10/2019 Chapter8_2_Ahmadi_notes.pdf
2/9
T . ) - . / ' - .
l r r \
f
a t \ , ) ) t -
-
t . -
\
f 1 n , r ' i ; ,
6 i r ; f l { t ) =
I t t
,
t .
-
, - l
I
t
Figure8-47
Closedcontours n the s
plane
nd heir
corresponding
losed urves
in the F(s)
plane. vhete
F ( . r ) :
( s
+ 1 ) / ( s 1 ) .
MappingTheorem. Let -F(s) ea ratioof two
polynomialsn s.Let P be he nunr
ber
of poles ndZ be henumber f zeros f
,/?(s)hat ie nside ome losed ontour n
the s
plane,
with multiplicity f
poles ndzeros ccountedor.Let the contourbe such
that t doesnot
pass
hroughany
polesor zerosof F(s).This closed ontour n the s
plane
is
thcn
mappednto the F(.s)planeasa closed u Ne.
he otalnumberN of clockwise
encirclementsf the originof
the F(s) plane, sa rcprcsentati -
oint
{
traccs ut thc
entire
ontour n the clockwise ircction,
s equal o Z
-
P.
(Note
hatby thismapping
theorem,lhenumbersof zerosand
of polescannotbe found-only their diflerence.)
544
chapter 8
/
Frequency-ResponseAnalysis
8/10/2019 Chapter8_2_Ahmadi_notes.pdf
3/9
Ap/tkail6>
to
-rtq
,tlU
days/r
N"e/
lb
)"f;nt-
a
pfe"
a.re
avstzd
as
shoon
;
Nak.
^:,1 unst"bb
bt
o/=*ot
.f
/+Gr9
Ha)
w,ll
be
iuide
-fhis
se.,ni
ivle
conttuf
rh
S-ylane
'"c1,
+t^o*
all
fn1,^enu'at
7h;s
Fo
t
v.fe1
Z'/' h
Po4
*hroylh
ts
^i)
ca//d
l*"Nyqwif
pafi,
tu
fS
sho^ld
not
uf
+b
pot"s
or
z-e"aS
f
/+
Q0
tlts)'
On4-
64
1'tolc
l:
.>-4
4
/+
61i,.riltj"r)
is
o
s{if+e,o{
A"
4"
to
TL' ptant
A^,
1dht.
fl.,uLfa4.
onz
c4h
onl&
ficus
on
enjrdwnzafs
o/ +1"
ofen-loo7
f,
ettjal Htl,a)
b**
rhslial
cons''dtr
g^o
sa,ab
N
t6-r
p'rt
"(
+tr
1V1l"is'l-
.*h
l"cole-ol
enur.lzme/rrfs
Jr /+
4t14Htj")
Q
fl
gtaM
5
am( aJ
l+
qrr)H(jNl
8/10/2019 Chapter8_2_Ahmadi_notes.pdf
4/9
.
7hL
Pt'"n
,
f
tl-
is /A/
n/r/ar
i,lye
posihw
s,'dt
and
f^p"r*.n|
rnlus
fiof
c,w.
nctft.
"f
ov;6,ibff
qd
-
+
of (,u .
ewir6,
l '-1*oi"
nr.n
^ t ^ P
N '.-
*f
fir
hVa'li'P
\'*ncio't
-a1t'J1o
,
-f
+lt p @
rM
a
j,i
H{1d
louJ, as
to
vanet
{rovtr
encl'r-.le
lLo
pont
-'(
l< t;mt't
in
lL,l
R.rrtor(
s
1.
TAe.
q H
d o e s
zgrol
on
+
C.L.
s1rtern..
e
to
&,
mnst
C.C"W
direcio
n .
ertlenbn
c4n
b
e,
8/10/2019 Chapter8_2_Ahmadi_notes.pdf
5/9
T' /'
N= o
P - o
+
'
Uso-.'llq
q
shool /
n
."tJ
no+
6 ) o
,
7lo*
rl .)sh"
+.r
. _ L
1 1 t ^ t t
'
plan,('
Qe
I t u
,-%
a N
c f l
Z = N
)
f o .
4e-
p
lof is
6t
b