Fiyst ooder Spter
the
fya otder ssitm becaye
Thi
epagsenes
byfye
Sysiem
Can
be
Yder diffexentialeuaton
Mexcuythemcm
Lquid
level
Mixin
tank
eler
syAtem
spornxe from
Tangient
Osde
oder Syaiem
G1l&AOl
F i l 35tance
MeiCu thetmomeHI:
SuxYoundin
&m
>Film eastane
.
1fluid
Fuid
MevcuN
Fim
Gle unll
estance
mercu
meYCu
All xesigtane heat
Actua
the cluid
Au pho
w
Allke
ygsstonce
the balb
fusoandin
at
any
(.e,
the
xexik tanCa ofoved by
meYCuM nleced).
l a nd
i A the
to eot tamzfer Koi des in the flm
thevma
capacity n
the meYLu
ipskant, the mercu
oyhuns
the
Fsthe mo
a unifsim
mpeature
thioh out
i) The Gla walcontainin
Cotiact duing
the
tay ent
I t A aAuned that the
Thi
meagy that
temPevatue
besove
t h time.
mesCur dot not espand o)
xes pone
thcmomeje i a t iitially &are
time Zeo, there A is
no
Change
in
time r o , the
At
Some
Ome
theYmbmeteY usill easubjeced
Chovae n the usioundinj
By aPplyin
the
4emperatue z(t).
usiedtole eneray balan@
Tnput Nate Out Put Taje = Rate of accumulokion
hA Ca)
dt
wher
A
Ssoce
oxea
for
af b
heat
tiaytfelf)
Cheot Capacrty ofmeCu
m1 = mas omeyuY in bulb
t
ime
h
=
film Co-efictent fot
heat bsaef
*Aboe equakonStoes that the, Kate of flous f
heot through the flm exstance fuvounding the bulb
Cases the inBevna enexy of the mercury to ineyee
at
at
the
ame Boe
The incvese in
manifeAtd bj a change
in
nternal
tmperatuNe and
Coieponeling pangion o mercu, hch
me cubey
h
Column ov)
e
deend
heqy
ene
a
cauzes the
eading o
on the flou Jate nd P1ofey ties of
the&us5Ounding flm and
the dimeryiops of the bulb
Fosea&tae Cohditfon, eg"0 Can be vitth aA,
hA
(-s) =0
t<0
9
Noud
esO - evO
dt
mc (9-1s
ha-7--1
dt
Cayant)
ot
Let y
defne Vaviables,
dev'otion Vayialbles: differ ende
dy
blus Vaviales and ther
Steady Atoe yalue.
X = -s
hA
X-Y)
=
d
me
mC
X-Y
=
Time contan t
X-Y= T dY
dt
trafstrya on both id,
APPinaplo.ce
L(X-Y) =TLdy
At ime =o
4x)- YE)- * s Y)- yo)
xs) -
yC)
out Put Value
zeYd
YCo)
= TS Y$)
TS YS) + yS)
x ls)= (1+ TS) YtS)
XS)
Out Put
inPut
=
Y)
Xs) TFTS
s
xonfer
+1
YIS) LLC-1)
( o )o XS L(7-1
fanckon of the Syem
Phsica Syptem fo hich the xelaton
aplace tsaafims of 7p and o/p deviation
Any
torm of
The
eg"
called
ato the
a
betoeen
the
fysi ovder 48km
laplace tvanAfosm of
the
op
to
the YP
calles taptfex fanc hon
o y o.ye called deviaaon Vasiales. The eaßon
foX the e
Vovialles
*
ofdevaston Vaviae
becae, thee
03 fee fom boundavy Condihong
taled time
Cortant
for
mecuTy
thevmomek
mC
T A
()Resprye
oYder procep for drfeent for Cn
of
unctgs
S standard foY of 1 orde
Y
tragejer tEROun Cton
X)
O
S
input
ult) =
us)
T4
CResFopge
o
t<0
Kp 14)
S
at20
.
(
fist
Fox
ovdeY
System,
y)
S+
X(S)
yIs) XS) FS+
YS)
=
TS+
S
YS)= A
TS
Y) ANous taking
S+
inVeyse la.pla.ce tros fo
Ye) =A -e
t20
Theunchon yt) seachgn 3.2X
itultmale value. hen t=T
Se, time Cortont Can be defned a
athe
time
sequived for
the
seGch
632
1Ponse, Yt)
to
oitultimae tme Value.
Stvni lar y,
Y t)
t
2T
t3T
t
=
o:
632 A
Y t ) = 0865 A
Yt) = o:15A
t)o.98 A
o-632
impulpe poryse ResFane
inpuleP
X= A e
X t) = e
X S) =
XS)
A
A
(unit impule input
to
Ys)
X)
TSt
YS) = XC)
Y
TSt
= S
bytaking inVese loplace tsamfosmation,
Ytt)=
tT|
0
Ko
XS)
and
TS+1
TYe)
x (s)= A
YS)
XS)
K
Ys)=KA (
YIS)YIS) = KAt/
T
)
yC) + pA
unit tmpulse rspore
fos
a
oider Syatem
Romp mput CRe Poyae Ramp input
Th uncton incxean% ines
depcribed b
ith time ond
euatio
the
t0
t20
X = bt
Slope-b
Puse
t
input
X(S) =X()edt
A
t
Ae
Jt
t0
tw
-S
Ae
5
A
X()=A -
-3)
Sts
tt
Stnugoidal ReAponAe
w
Vevy
ThyYery
impovtont
the
Stabiltyf
eful
ond
in
find in
Control Aystem
t40
At)
At)
Asin t
X(S)
A
20
A
S
1 cyele
A Amplitude
6
Fe guency, sad/s
Ditonce
Peviod
blu tao peaks
21T
TS+
XS)
Y)
YS)
=
XS)
TStIJ
=A
TS+
bCommonepesaHon method
TS4
P
A
(S+)(Ts+1)
A
(s
Tst)
TSt
S+R
st
P(S)+ (0$tR) (Yst
)(rst)
the
P(s++(9$+R) Es*) CYS+)
AG
A = Ps'+Pw+
TS+ 95t
RTS R
A = s(Pt9T)+ s(p+TR)+ P+R
P+t=O
P=-9T
P-9T
-TR
+TR=o
A IT
A= P+R
R
A
P
P=A T
AW - RT
R
A
R
It
A wTY
Y S) =
-
A9T
S
ltus (Sh)
ts
Yt)
Yt)
)
A
As
By tkn
A
nveyae lapla.ce tafoYmahon,
AT
A
c9s t
4A
Sin t
I+
AwT-t/7ATcoA
A
ut t A i n 9t
in
tono medric eloon
sin (P+)
+
Sin (+ 8)
uhere
tan (4)
dentity above eq"
above
Yt)
SinLt+
T
wheve,
Find
A
A 9T
wrteh4
be
can
Total out Put
= to(-sT)
fo inysoida)
APonse
Value af y (tt)
StRo.dy Atat
hen
A
i n in ( - t t )
The out Put
a
egual to that f
Awplite Raho
Ae aVe sith frequerc s
input Agnal
The
xato
of
out
Put
toinput anplitude.
AR =
(3 Phot oad
pheaa a
f
g>o
if 2o
Cmpitude
Lag
Lead
Liuid
(
Level
Rotape Syskm):-
(1 order
Tank
q9-
Th AA4em Corn
f a tank
afUnioím C9s-Bechon al area
A
hich
to
attached
3etance ' . Such
a pipe, ()
a
9
a
eit
M
e
Aume
Out
let
E
9,t)
tho+
the
though the vAana s
bthe
E Ro
ffat)-S It) =
byvidin
t)
lin2o
R
bolan de
Inle
flo
Valve
Volume sic flous Ya4eVoltime
seaeloled to the heod
If
ht)
ee
accunu lahan
AL)
t
th
bo-th Aide
. ) =A dh
dt
elatonakie.
above eqh
SubstituR. 9
A dh
dt
Som
Stodj Aate
Roe
of aCuiulation=o]
R
btta
Su
fom
e
e
O
hh-hs
(1-3)-(-) - A
Ch-hs)
d
+A
=-hs)
dt
9-4)
9
Ad
tokin
loplace trorfom,
c0
wV
)
+ALsH)-
R
96)
=LH)+AS
H()
tARS
R
HS)
(S)
HS)
R
()
SAR+)
R
TSt)
he
-AR
M
*
R
coled
Find
ultimate
Nolue
of
Stead
Value
H+),
im
t ate
ain.
o HE). e., {ead Zto
lims
H$)>
So
H t)
fos unitA4ep input,
t0
9t)
t20
PS)=
Sulstiue e n
Ht)
t
Subtitute Qs)
to
a
pply
ti
im
Ht)
mit
S
es
lin S
S->0
in te
R
(TS+-1 o)
aboe e
im
S>o
(TS +1) S
S-So,
Ht)=R
when, ) = 5
This Show thot ulote change in H
chonge
in
9
inply R.
for un't
Cote 2
Fsom e
-
Fo Steodyy
Stoye
then
HS)
*-9%s
G)
St
where ,
if
Ch-hs)
R
H S)
=
.
0
R
cone O.
From e
T'St1
R
TSt1
s)
Minin
R
HS
s)TSt1
(1* otdes 8m):
Tank
W
the mixing PyoCeM
Conider
of Solution
flos
Contaning diolved alt
in bich
at
a stream
Volumetic
floa
eV
vate
Corytant
atonk af ConAtat
into
Volume v .
hold
*the
Conentyo4tion
moyf Salt
Volum Volume
o Salt
a
in enterinStream,
rth ime.
Aimi
desxed
It
elon
out
to
to
detetmine
ConCentration
let
tsasfev func on
y
tothe
inlet
Con Cen a i o n
Ma balance eq
9-93=
Fo
dt
Sttady t o ,
sSbtract
oth the
quos*
dt
x
9Y =v
dt
x-y
-Y
NO
dy
dt
takin
lapla.ce rard fosm,
Xs)-Ys) = T |S Ye)-o
Xs)=T'S Y(S) -+ Ys)
xS) = (rs+1)
Ys)
x
S+|
Y)
3whee, T=
Sumnayy o
mc
T=
0
hA
T
the momeRY,
level
fov liguid
AR
to mixin
T
izinq
fo
RC
tonX ene
RC
poce,
pyoCC%
CiCut
balane
Sttam (o")
Elecici
Heatn Yoce
A Stream at tmpeY akureT
T fd
a
dded
t» the tonk. Heot
to
the tank
by meoa
T
electic keae
The
af the
tonk
ellmixed
etit S+Se am
Flous
KoR
an
the tem pexahure
T
bh
Yat of
of
ener flo
enegs flb
in
out
Teey flow
in f%om
heatr
Ra of
aCcumu latoh
oeneyn
ton k
+
1
T
2
1
1)
s+)
whee,
TS