Controllers and Final Control Elements.ppt

Controllers and Final Control Elements

1. Lecture: 41
CONTROLLERS AND FINAL
CONTROL ELEMENTS
Ideal Transfer Function of
Control Valve

2. CONTROL SYSTEM
Fig-1:
Schematic diagram
of control system

3. LUMPED BLOCK DIAGRAM
Transducer Controller Converter
Measured
Variable
x ma ma psig
p
(a)
“Controller”
x p
(b)
Fig-2:Equivalentblockfortransducer,controllerandconverter

4. IDEAL TRANSFER FUNCTIONS
OF CONTROL VALVE
Apneumaticvalvealwayshassomedynamiclag,
whichmeansthatthestemmotiondoesnotrespond
instantaneouslytoachangeintheappliedpressurefrom
thecontroller.
Fromexperimentsconductedonpneumaticvalves,it
hasbeenfoundthattherelationshipbetweenflowand
valve-toppressureforalinearvalvecanoftenbe
representedbyafirst-ordertransferfunction;thus
WhereK
visthesteady-stategain,i.e.,theconstantof
proportionalitybetweensteady-stateflowrateandvalve-
toppressure,
andτ
v
isthetimeconstantofthevalve.1)(
)(


s
K
sP
sQ
v
v

………………………………(1)

5. IDEAL TRANSFER FUNCTIONS
OF CONTROL VALVE
Inmanypracticalsystems,thetimeconstantofthe
valveisverysmallwhencompared withthetime
constantsofothercomponentsofthecontrolsystem,
andthetransferfunctionofthevalvecanbe
approximatedbyaconstant
Undertheseconditions,thevalveissaidtocontribute
negligibledynamiclag.
Tojustifytheapproximationofafastvalvebya
transferfunction,whichissimplyK
v,considerafirst-
ordervalveandafirst-orderprocessconnectedin
series,asshowninFig-3.v
K
sP
sQ

)(
)(
………………………………(2)

6. IDEAL TRANSFER FUNCTIONS
OF CONTROL VALVE
Ifweassumenointeraction,thetransferfunctionfrom
P(s)toY(s)is
Foraunit-stepchangeinP,)1)(1()(
)(


ss
KK
sP
sY
Pv
Pv

…………………………(3)
Valve Process
P
Y1s
K
v
v
 1s
K
P
P

Fig-3:Blockdiagramforafirst-ordervalveandafirst-orderprocess)1)(1(
1


ss
KK
s
Y
Pv
Pv

…………………………(4)

7. IDEAL TRANSFER FUNCTIONS
OF CONTROL VALVE
Theinverseofwhichis
Ifτ
v<<τ
P,thisequationisapproximately:
Eq.(6)istheunit-stepresponseofthetransferfunction
sothatthecombinationofprocessandvalveis
essentiallyfirst-order.
Thisclearlydemonstratesthat,whenthetimeconstant
ofthevalveismuchsmallerthanthatoftheprocess,
thevalvetransferfunctioncanbetakenasK
v.

















Pv
t
v
t
PPv
Pv
Pv eeKKtY


 //11
1)()(
……(5))1()(
/
P
t
Pv eKKtY


…………………………..(6))1()(
)(


s
K
K
sP
sY
P
P
v

…………………………..(7)

8. IDEAL TRANSFER FUNCTIONS
OF CONTROL VALVE
Atypicalpneumaticvalvehasatimeconstantof
theorderof1sec.
Manyindustrialprocessesbehaveasfirst-order
systemsorasaseriesoffirst-ordersystems
havingtimeconstantsthatmayrangefroma
minutetoanhour.
Forthesesystemswehaveshownthatthelagof
thevalveisnegligible,andweshallmakefrequent
useofthisapproximation.