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Overview

Overview

"Q"

"Q"

an

an re

reso

sona

nanc

nce.

e. Fi

Fina

nall

ll

how

how

to

to

de

dete

tenn

nnin

in

fr

freq

eque

uenc

nc re

resp

spon

onse

se ex

expe

peri

ri en

enta

tall

ll

r.r. frfreqequeuencnc rerespspononsese us

us m.m. haha nsns dede bebe hehe stst popo

si

sinunusosoidid ofof papa titicuculala frfreqequeuencnc ReRecacallll ththatat sisinunusosoididalal unun titionon (c(comomplplexex exexpoponenentntiaial)l) isis anan eieigegenfnfunun titionon of

of CTCTsysyststemem memeananinin ifif

elel

lDtlDt gogo n,n,KeKeJlD1JlD1 cocomeme ouout,t, heherere isis cocompmplele nunumbmb r.r. plpl

vs,

vs, 0000 SinceSince mpmp prpr ntnt hehe ueue popo b-b- phph

an

an phphasas veve susu 0000 hihi asas

Ae1+.

Ae1+. The

The

ououtptputut thth bebecocomeme Aej(lJ)r+~),Aej(lJ)r+~), whwhicic haha

or

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.

Th

Th Bi

Bi Pi

Pict

ctur

ur

In

In gengeneraeral,l, e"e" isis anan eieigegenfnfuncunctitionon (s(s (J(J jco)jco)

e~t

e~t LTLT SySyststemem

1-...

1-...

H(s)eH(s)estst Here

Here 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.

(2)

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 axes

Linear 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 od

phas 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

(3)

CHAPTER6

lots

Asymptoti Approximations

H(s)

Magnitude

s+ 10

Phase

"'"'"7 H U " ". .! .. . -20dB

ro O.1xpole

"'"'"7 H U 10 10 "'"'"7 -20dB/decade

o>

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

(4)

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 er

Plottin Magnitud

IH(jm)1

at re c.o=O (0,like]. ]. al ak s= s= ec each zero at or gi ). Co ea

(5)

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 asymptotes

as

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 te

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 label

y-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)

(6)

'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 is

C) 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

(7)

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 S2

0.2s

1.01 4.5 denominator 2as roots 0.1 ±j 3.5 -a±j3j

12.5

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).

(8)

si (1 01 65°) U(I)

---tl.~I__

H_(S_)_":----II.~ 711

CHAPTER6

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!

References

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