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ontro oontro o
me- e ay ys emsme- e ay ys ems
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Time-Delay Element
Time-Dela
Element- l
l= me- e ayTime Responsex(t)
unit step1
0t
t
1
0 tTl
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Time-Delay SystemsExample 1
empera ure easuremen o ea - xc anger u pu
b t
Sensor
a(t)d
Heat
cold in
out v out
Sensor
Heat
cold in
v = velocitd = distance
= al
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Time-Delay SystemsExample 2
c ness easuremen o o e ee a e
Thickness
+
gauge
Time delay between
vSteel Plate c ange n c nessand measurement:
+
vT
l=
d
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Difference Between Time-Lag and Time-Delay
(a)
(b)
(c)
lsT
In case (c), transfer function of the block = ,T= time lag, Tl=
delay
( ) sTsG += 1
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ro em ue o resence o me- e ayin a Process Control Loop
Due to the presence of time-delay, any corrective action
from
the controller cannot be immediately applied to the process.
, ,
sometimes quite large, due to any load disturbance or change
in set- oint.
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Transfer Function of Time-Delay Element
t Time-Delay t = x t - T Element
Taking Laplace Transform:
( )lsTe
sX
=
Transfer Function of a Time-Delay Element2
1
Now,
12
12
12
21
12!
l
l
l
sTl
sT
sT
sTe
ee sT
+ = =
1
21
2!lsT+ +
1
21
lsT lsT
-1
21 lsT+
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Effect of Time-Delay on Process Loop Response
T.F. of a Process with Time-Delay: ( ) ( ) lsTesGsG =
where G/(s) is the transfer function of the delay free part of
the
process.Case Stu y: Proport ona Contro o a F rst-Or er
Process
with Time-Delay
controller process
+ Kp
1
le
sT
+C(s)R(s)
+_
p
T = Process Time Constant
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Case Study: Proportional Control of a First-Order Process
with Time-Delay (contd )
lsTe
controller
p1 sT++
_
The Characteristic Eqn. of the C.L. System: 1 01
lsT
pK e
sT
+ =+
Using First-Order Pade Approximation:
0
2
11
1 =
+
lpsTK
01
1
1
2
1
2
1
2 =
++
++ p
lpl KTKTT
ss
211
++l
sTsT22
ll
The standard form: 1 1T T K T +
2 22 0,n ns s + + =,
p
nlTT
=
The Natural Freq.( )2 1p lK T
=
+ +The Damping Ratio
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Case Study: Proportional Control of a First-Order Process
with Time-Delay (contd )
lsTe
controller
p1 sT++
_
The damping ratio becomes zero when:
+=
l
l
p
T
TTK
2
The system becomes unstable when:
+
> lp
TTK
2
l
Conclusion: Time-delay may cause instability even in
first-order system for large values ofKp
.
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Case Study: Proportional Control of a First-Order Process
with Time-Delay (contd )
+ Kp
1
lsTe
sT
+C(s)R(s)
+_
controller process
Loo T.F. of the C. L. S stem:eK l
sT
p
Root Locus Plot
sT+1
Usin First-Order Pade A roximation:
2
1 2
pl
pl
KsT
sK T T
=
( ) 1 21 1 2
l
l
sTsT s s
T T
+ + + + Case I:
2
lT
T >j
Kp =Kcs-plane Kc: Critical Gain
Kp
= 0 Kp
= 0
22
l
c
T
KT
+ =
0
2
lT
1
T
2
lT
2 lT T +=
Kp
= Kc
l
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Case Study: Proportional Control of a First-Order Process
with Time-Delay (contd )
+ Kp
1
lsTe
sT
+C(s)R(s)
+_
controller process
Loo T.F. of the C. L. S stem:eK l
sT
p
Root Locus Plot
sT+1
Usin First-Order Pade A roximation:
2
1 2
pl
pl
KsTs
K T T
=
( ) 1 21 1 2
l
l
sTsT s s
T T
+ + + + Case II: l
TT
