Time response second order

TIME RESPONSE OF SECOND ORDER SYSTEM

1. TIME RESPONSE
OF
SECOND ORDER SYSTEM
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URL: http://shasansaeed.yolasite.com/
1SYED HASAN SAEED

2. REFERENCE BOOKS:
1.AUTOMATICCONTROLSYSTEMKUO&GOLNARAGHI
2.CONTROLSYSTEMANANDKUMAR
3.AUTOMATICCONTROLSYSTEMS.HASANSAEED
SYED HASAN SAEED 2

3. SYED HASAN SAEED 3
Block diagram of second order system is shown in fig.
R(s) C(s)
_
+)2(
2
n
n
ss

 s
sR
sR
sssR
sC
A
sssR
sC
nn
n
nn
n
1
)(
)(
2)(
)(
)(
2)(
)(
22
2
22
2










For unit step input

4. SYED HASAN SAEED 422
22
2
2
)1(
2
.
1
)(
nn
nn
n
ss
sss
sC







Replace by)1()(
222
 
nn
s
Break the equation by partial fraction and put)1(
222
 
nd 1
)3(
)()(
.
1
2222
2





A
s
B
s
A
ss
dndn
n

 )2(
)1()(
.
1
)(
222
2





nn
n
ss
sC

5. SYED HASAN SAEED 5 
22
)(
dn
s 
Multiply equation (3) by and put)2()(
))((
)(
2
2
2
nnn
nd
dndn
dnn
dn
n
n
dn
ssB
sj
jj
j
B
j
B
s
B
js


















6. Equation (1) can be written as
SYED HASAN SAEED 6)4(
)(
.
)(
1
)(
)(
1
)(
2222
22















dn
d
d
n
dn
n
dn
nn
ss
s
s
sC
s
s
s
sC








Laplace Inverse of equation (4))5(sin.cos.1)( 








tetetc
d
t
d
n
d
t
nn




 2
1 
nd
Put

7. SYED HASAN SAEED 7 tt
e
tc
ttetc
dd
t
dd
t
n
n








sincos.1
1
1)(
sin.
1
cos1)(
2
2
2














 )sin(
1
1)(
1
tan
cos
sin1
2
2
2
















t
e
tc
d
t
n
Put

8. SYED HASAN SAEED 8)6(
1
tan)1(sin
1
1)(
2
12
2




















t
e
tc
n
t
n
Put the values of d
  &)7(
1
tan)1(sin
1
)(
)()()(
2
12
2





















t
e
te
tctrte
n
t
n
Error signal for the system
The steady state value of c(t)1)(

tcLimite
t
ss

9. Thereforeatsteadystatethereisnoerrorbetween
inputandoutput.
=naturalfrequencyofoscillationorundamped
naturalfrequency.
=dampedfrequencyofoscillation.
=dampingfactororactualdampingor
dampingcoefficient.
Forequation(A)twopoles(for )are
SYED HASAN SAEED 9n
 d
 n
 2
2
1
1




nn
nn
j
j 10

10. Depending upon the value of , there are four cases
UNDERDAMPED ( ): When the system has two
complex conjugate poles.
SYED HASAN SAEED 10 10

11. Fromequation(6):
Timeconstantis
Responsehavingdampedoscillationwithovershootand
undershoot.Thisresponseisknownasunder-damped
response.
SYED HASAN SAEED 11n
/1

12. UNDAMPED ( ): when the system has two
imaginary poles.
SYED HASAN SAEED 120

13. From equation (6)
Thus at the system will oscillate.
Thedampedfrequencyalwayslessthantheundamped
frequency()becauseof.Theresponseisshownin
fig.
SYED HASAN SAEED 13ttc
ttc
n
n



cos1)(
)2/sin(1)(
0



For n
 n
 

14. SYED HASAN SAEED 14
CRITICALLY DAMPED ( ): When the system has
two real and equal poles. Location of poles for
critically damped is shown in fig. 1

15. SYED HASAN SAEED 15)(
11
)(
)(
)(
2
.
1
)(
1
2
2
2
2
22
2
n
n
nn
n
n
n
nn
n
ssssss
ss
sC
sss
sC



















For
After partial
fraction
Take the inverse Laplace )8()1(1)(
1)(




tetc
etetc
n
t
t
n
t
n
nn





16. SYED HASAN SAEED 16
From equation (6) it is clear that is the actual
damping. For , actual damping = . This actual
damping is known as CRITICAL DAMPING.
The ratio of actual damping to the critical damping is
known as damping ratio . From equation (8) time
constant = . Response is shown in fig.n
 1 n
  n
/1

17. OVERDAMPED( ):whenthesystemhastworeal
anddistinctpoles.
SYED HASAN SAEED 171
Response of the
system

18. From equation (2)
SYED HASAN SAEED 18)9(
)1()(
.
1
)(
222
2





nn
n
ss
sC )1(
222

nd
Put )10(
)(
.
1
)(
22
2



dn
n
ss
sC


We get
Equation (10) can be written as)11(
))((
)(
2



dndn
n
sss
sC



19. After partial fraction of equation (11) we get
SYED HASAN SAEED 19
Put the value of d
   
  
)12(
112
1
112
11
)(
22
22





dn
dn
s
ss
sC

   
  
)13(
)1(112
1
)1(112
11
)(
222
222







nn
nn
s
ss
sC

20. Inverse Laplace of equation (13)
From equation (14) we get two time constants
SYED HASAN SAEED 20)14(
)1(12)1(12
1)(
22
)1(
22
)1(
22







 tt
nn
ee
tc n
n
T
T


)1(
1
)1(
1
2
2
2
1





21. SYED HASAN SAEED 21)15(
)1(12
1)(
22
)1(
2





 t
n
e
tc
Fromequation(14)itisclearthatwhenisgreaterthan
onetherearetwoexponentialterms,firsttermhastime
constantT
1andsecondtermhasatimeconstantT
2.T
1<
T
2.Inotherwordswecansaythatfirstexponentialterm
decayingmuchfasterthantheotherexponentialterm.
Sofortimeresponseweneglectit,then)16(
)1(
1
2
2 


n
T
 

22. THANK YOU
SYED HASAN SAEED 22