Overview
Overview
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sinunusosoidid ofof papa titicuculala frfreqequeuencnc ReRecacallll ththatat sisinunusosoididalal unun titionon (c(comomplplexex exexpoponenentntiaial)l) isis anan eieigegenfnfunun titionon of
of CTCTsysyststemem memeananinin ifif
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lDtlDt gogo n,n,KeKeJlD1JlD1 cocomeme ouout,t, heherere isis cocompmplele nunumbmb r.r. plplvs,
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The
ououtptputut thth bebecocomeme Aej(lJ)r+~),Aej(lJ)r+~), whwhicic hahaor
ordsds sisi ( C O t )( C O t )
-+
-+
sinsin ( C O t( C O t~)
~)
vs.vs. cocoandand vsvs Simple,Simple, eh?eh?Pl
Pleaeasese nonotete ththatat ththiningsgs plplotottete veversrsusus (radians/sec)(radians/sec) notfnotf (H(Herertztz oror cycyclcleses/s/secec).). IfIfneneedededed thth coconvnverersisionon is
is 2rr.f.2rr.f.
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Th Bi
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e~t LTLT SySyststemem
1-...
1-...
H(s)eH(s)estst HereHere H(s)H(s)
ul
ul plpl otot oror ofof hehe sfsf H(s)H(s) evaluatedevaluated
lo
longng thth jcojco-.a-.axisxis (s=(s=jcojco).).
uo
uousus-t-timim sysyststemem fofo stst adady-y-ststatat sisinunusosoididalal ininpuputsts Bo
Bodede plplotot H(s)H(s) evevalaluauatete alalongong jejeo-o-axaxisis (s=jco)(s=jco)
Z.
tH(ieo~ LH(jeo) 2010g .JH(ieo)t
s+ 10 10
magnitud ofH(jw) magnitude of numerator O.099S -S.71 deg ·20.04 dB
magnitude of denominato 0.0894 ·26.S7
phas ofH(jw) phas of nume ator phas of denominato
SO . I magnitud of bj 100 CHAPTERS H(s) ho pl nt oan mp he ud nd ph HUeo). hasi ha ov de bi of he ve on ni pl ni de be bb vi dB It de 2010g Furthermore, th va hi on og
is
H(jeo) vs. rithmi axesLinear Scal 'D Logarithmi Scal
....
." _11 0.. --- -DO DG to 200,...,t ...,...,
,no too _01~ ' ~ ~ - - - , ~ ~ - - - - ~ . ~ ~ - - - - . ~ ~ - - ~ - - , ~
--io
pr xi ns ui ut ve ur hand of he odphas plot change slop at .lxbre kpoint an lOxbre kpoint At th breakpoint itself th actu magnitud (11
45° poin sinc I H ( j e o ) 1 2
ollowing exampl usin th same H(s) as above
CHAPTER6
lots
Asymptoti Approximations
H(s)Magnitude
s+ 10Phase
"'"'"7 H U " ". .! .. . -20dBro O.1xpole
"'"'"7 H U 10 10 "'"'"7 -20dB/decadeo>
xp le
"'"'"7 H U m ) -90 Jm "'"'"7 H U m )pole
"'"'"7 H U m ) 0 ~ ,' ... -10 ->0 I-I'
...
"'-:0-' w! .0 -e' ,. ,.' '...-.cYlfIIdI~ F~(r.D'_1.4 Relati ns ip to th PolelZ
pl
H(s).Imagin ta ting atro:O
- 1
00 he raphin th de pl t.
00=10 rad/sec. Betwee r o = O and 00=10, th lo -mag itud th ta
ully th followin iagram il ak thin clea
lm{s} jm
slid betwee ro=O and +- H(s) 10 H U m ) Jm~IO
Re{s}
0,)increases, 10 increases I H U m ) 1 decreases. 0,)increases, 0 ) g 0 ° LHUm) goes [0_90°.
-10
Bo
an Si fa e' co at ex at as at ea ze fa or o) 1000 (s+ 10)2 ~I(] ro 2()dB ·2I1dBldec 2010gIH(jro)1= LH(jro)= '~!I.~I'-II"KI-ro
2010g jH(jro)1 LH(jm)Answer:
fa 10 H(s) causal stabilit will erPlottin Magnitud
IH(jm)1at re c.o=O (0,like]. ]. al ak s= s= ec each zero at or gi ). Co ea
lo
decrease increase slope
unle ther ar no more brea points left
ii HUm)
plot Inhere ar 0 0 = 0 is th simplest choice Continue labellin
j-axis
usin slopes of asymptotesas
guides.7. +/~ t.
Plotting Phas LHUm)
3. LHfjm=O)
~900
ls doesn' change anything
as flat line unti reaching 0. xbreakpoint.
S. pole subtracts 90 1 x p
added. atch ou or multiple pole /zeros
6.
is alculated) fo more ac ur te6°
at O.lxbreakpoint an at 10xbreakpoint. Example 2OJogIH(ioo)1 ·2fklBlde<: r o un d c o m er s by 3d 20dB -4OdB =loo(s+lO) s2 lOoos H ( s ) 0.1 10 plug in 00=100 to labely-axis on magnitude plo
factor ). so etimes sefu co ve io is 20 dB/d ca e" dB/o tave It can be derive as follows: 2010g (x)
'8
Bode
Plots
":,..
can't
rules fo drawin an accurat plot ,I accuracy
', ., "-,(,'t.! needed plug number into H ( s ) us MAlLAB. Nevertheless yo should stil be able to ro id er ro gh
be able
Just to type
H ( s ) s""::2:--+-0-::"."":"4-s+ " " : " 1 - : . 0 4 ~ .2 Ij 20
---~"""''''''''-:-:''':":''-:--:-'''''''''''''''''''
'i
bump peak whe~ th product of distanbes topoles isC) is approximately equal to th imaginary
part of thepol location if they ar clos toth jfo-axis
th o s h e p o a r th
jOl-axis, the larger thebump heighl
-180
~ - - - - ~ - - - ~ - - ~ ~ ~ - - - ~ ~ ~ - - ~ - - ~
(rad/sec)
Sketchin th Comp ex Po elZero od pl t:
methods, first draw th pole/zer diagram. toobserv th fo lowing exac re at onship ha
tr when raphin an freque cy response
IH(jro)l (distanc from poleto jei)
i= h=
(anglefrompole{tojro)
.. .l
race ou fi ge al ng th osltiv te-a ls an enta ly appr xi at he ab ve rela ions I' et in cl se
be tothe positive
de lots
ni ho nt ho th be sh d.
H(8) valuated s=jw?
th
-00).
with
< O p e a J F P
ow 2a.Anothe wa of describing"Q m. nanc
curv is show below:
R e so na nc e C ur v
H(s) If S20.2s
1.01 4.5 denominator 2as roots 0.1 ±j 3.5 -a±j3j12.5
P .
if ::I(in general. " \ > c o t will be slightly
1.15
I eS ic ha n b ac h
~",J!.
Am 0.ifa«~
halfpowe pain
J 2 - - - .
..,._...
AOPZo:
half I X l w e r bapdwidth):r op eak :
0.1 FreqIMfI(:y (..diMe).8 Sa
le
lo
ti
here ar gene ally ou type of Bode plot problems
H(s) real
magnitud {Y-axi values will be
hi ot po ut he va un own)
H(s) hi ki oi
H(s); in othe
he ue ni ud hi ne
gets to prac ic drawin Bode plots, th othe person practi es econst ucting H(s).
si (1 01 65°) U(I)
---tl.~I__
H_(S_)_":----II.~ 711CHAPTER6
Bode
ot
H( This type
with co plex oles an zero atch fo ubtletie ik lope of ines -axi alue location of resona peak etc.
an
wha information it ro ides An exampl ro le show elow
What is th teady-stat output of he fo lowing sy te H(s)
Method J: th H(s) ta th nverse
ransform an et -+-.
2 s 1001 U(/», then
th ilateral ap ac tran form woul ot ex st Method
eval at ng ts agnitude an
al IH(s)1 an adding LH(s) to th phase.
Method 2IH(lOOj~ si (lOO 65° re it H(s) using MATLAB. IH(jO)1 LH(jro) Jags (i delaye ersi f) th input. Continue record ng he ag itudes an hase response
of
en Re Os
(0 2nj). ef
1,2, method (1,2,5,10,20,50,100,200,500,1000 etc. produces roughl equall spaced points on logarithmi
al itto yo oss!