response of first order system in various input

7/25/2019 Response of First Order System in Various Input

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M A L L A R I , D O N N A J O Y R .C H E - 4 2 0 2

Response of First Order Systemin Various Input


2/23

WHAT IS A FIRST-ORDER SYSTEM?

A first-order system is one !ose output" y#t$" ismode%ed &y a first order" %inear differentia% e'uation

If a( ) (

a% dy* a o y ) &f#t$ dt

a% dy* y ) &f#t$ aodt a o

W!ere f#t$ is t!e input#for+in, fun+tion$


3/23

Define

and

p is .no n as t!e time constant of t!e pro+ess/ p is +a%%ed t!esteady state ain or static ain orsimp%y t!e ain o! t"e #$ocess.

a% ) pao

& ) / pao

p dy* y ) / p f#t$ dt


4/23

T!e Transfer Fun+tion of a first order pro+ess is ,i0en &y1

It is defined on%y for %inear time-in0ariant system2 For non-%inear systems" t!e mat!emati+a% mode% must first &e%ineari3ed around a steady state2 W!en definin, t!e transferfun+tion" a%% initia% +onditions must &e set to 3ero2T!e time +onstant is re%ated to t!e speed of response of t!epro+ess on+e t!e pro+ess starts to responds to a for+in,fun+tion2 T!e s%o er t!e pro+ess t!e %ar,er is t!e 0a%ue of 2 A first-order pro+ess it! a Transfer Fun+tion is a%so .no nas4 first-order %a," %inear %a," e5ponentia% transfer %a,2

6#s$) Y#s$ ) /p 7#s$ s*8


5/23

Response of t!e First Order Systems to VariousInputs

Standard 9ro+ess Inputsa2 Step Input &2 Ramp Input+2 Impu%sed2 Re+tan,u%ar 9u%see2 Sinusoida% Input


6/23

Step Input

T!is +orresponds to a sudden and sustained input +!an,es x(t) = 0 t < 0 A

x(t) = A t 0 x(t)( t

:ero time is t!e time at !i+! t!e sudden +!an,e in

ma,nitude A o++urs2E52 a step +!an,e in f%o rate +an &e o&tained &y t!esudden openin, of a 0a%0e2


7/23

Ramp Input

T!is +orresponds to inputs t!at drift" t!at is" t!ey+!an,e re%ati0e%y s%o %y up ard or do n ard forsome period it! a rou,!%y +onstant s%ope2

x(t) = 0 t < 0 slope=a x(t) = at t 0 x(t)

t


8/23

Impu%se

Impu%se fun+tions are not en+ountered in usua% p%antoperations" nor are t!ey easy to ,enerate p!ysi+a%%y &y t!e en,ineer2 To o&tain an impu%se input it is

ne+essary to in;e+t a finite amount of ener,y ormateria% into a pro+ess in an infinitesima% %en,t! oftime2 x(t) = A(t)

(t)=lim u#t$-u#t-&$ &

8


9/23

Re+tan,u%ar 9u%se

T!is is a sudden &ut un-sustained type ofdistur&an+e2

5#t$ ) A =Au#t-t $

A

( t t


10/23

Sinusoida% Input

These inputs vary periodically.

x(t) = 0 t < 0 x(t) = ASint t 0

W!ere4 A ) amp%itude of t!e sinusoida% fun+tion = t!e radian fre'uen+y 1 >) @ 9) period is re%ated &y 9) @

>


11/23

Response to Step Input of t!e First Order System

T!e step input fun+tion is ,i0en &y

5#t$)

!ere A is t!e ma,nitude of step +!an,e2 Ta.in, t!eBap%a+e transform" t!e input in t!e s domain is 5#s$) A s

T!e response of t!e system to t!e first order system is ,i0en

&y t!e e'uation

6ettin, t!e in0erse Bap%a+e transform of y(s) !ives

( t "0

A t#0

y#s$) A ) /p s s*8

y#t$) A /p #8- e -t< $


12/23

W!en t) " t< )8 and t!e 0a%ue of t!e response oft!e system is y# $) A/ p #8- e-8 $) 2C A/ p2


13/23

9ro&%em

E%am#&e '.(. A t!ermometer !a0in, a time +onstant of (28 min is

at a steady state temperature of (GF2 At time t ) ("

t!e t!ermometer is p%a+ed in a temperature &at!maintained at 8((GF2 Determine t!e time needed fort!e t!ermometer to read 2

Ans er 4 (2 8C8 min


14/23

9ro&%em

9ro&%em J2 A t!ermometer !a0in, a time +onstant of8 min is initia%%y at J(GK2 It is immersed in a &at!maintained at 8((GK at t ) (2 Determine t!etemperature readin, at t ) 82 min2

9ro&%em J28( In pro&%em J2 " if at t ) 82J min" t!et!ermometer is remo0ed from t!e &at! and put in a &at! at LJ K" determine t!e ma5imum temperature

indi+ated &y t!e t!ermometer2 W!at i%% &e t!eindi+ated temperature at t ) ( min?


15/23

Impu%se Response of t!e First Order System

Impu%se fun+tion is ,i0en &y t!e e'uation"T!e Bap%a+e Transform of t!e impu%se

fun+tion is 5#s$)8 2

T!e response of t!e first order of t!e system toimpu%se +!an,e is 2

Ta.in, t!e in0erse Bap%a+e Transform yie%ds

y#s$) 8 s*8

y#t$) e-t * 82 T!is is a% ays sma%%er t!an 82 We oftenstate t!is &y sayin, t!at t!e si,na% is attenuated.2 T!e output %a,s &e!ind t!e input &y an an,%e N2 It is+%ear t!at %a, o++urs" for t!e si,n of N is a% ays

ne,ati0e2


21/23

Effe+t of Varyin, t!e Amp%itude of t!e Sinusoida% InputFun+tion to t!e Response of a First Order System


22/23

9ro&%em

E5amp%e J2 2 A mer+ury t!ermometer !a0in, a time+onstant of (28 min is p%a+ed in a temperature &at! at8((GF and a%%o ed to +ome to e'ui%i&rium it! t!e &at!2 At time t ) (" t!e temperature of t!e &at! &e,ins to 0ary sinusoida%%y a&out its a0era,etemperature of %OOPF it! an amp%itude of GF Ift!e fre'uen+y of os+i%%ation is 8(


23/23

QT!e most important su&;e+t to %earnin %ife is to %earn !o to %o0e

= 9ope Fran+is

-- T!an. you

response of first order system in various input

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