Time response second order

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

REFERENCE BOOKS:
1. AUTOMATIC CONTROL SYSTEM KUO & GOLNARAGHI
2. CONTROL SYSTEM ANAND KUMAR
3. AUTOMATIC CONTROL SYSTEM S.HASAN SAEED
SYED HASAN SAEED 2

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

SYED HASAN SAEED 4
22
22
2
2
)1(
2
.
1
)(
nn
nn
n
ss
sss
sC







Replace by )1()( 222
  nns
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

SYED HASAN SAEED 5
 22
)( dns  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


















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

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
tn
Put

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

Therefore at steady state there is no error between
input and output.
= natural frequency of oscillation or undamped
natural frequency.
= damped frequency of oscillation.
= damping factor or actual damping or
damping coefficient.
For equation (A) two poles (for ) are
SYED HASAN SAEED 9
n
d
n
2
2
1
1




nn
nn
j
j
10  

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

From equation (6):
 Time constant is
 Response having damped oscillation with overshoot and
under shoot. This response is known as under-damped
response.
SYED HASAN SAEED 11
n/1

UNDAMPED ( ): when the system has two
imaginary poles.
SYED HASAN SAEED 12
0

From equation (6)
Thus at the system will oscillate.
The damped frequency always less than the undamped
frequency ( ) because of . The response is shown in
fig.
SYED HASAN SAEED 13
ttc
ttc
n
n



cos1)(
)2/sin(1)(
0


For
n
n 

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

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





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

OVERDAMPED ( ): when the system has two real
and distinct poles.
SYED HASAN SAEED 17
1
Response of the
system

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

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

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
22
21





SYED HASAN SAEED 21
)15(
)1(12
1)(
22
)1( 2





 tn
e
tc
From equation (14) it is clear that when is greater than
one there are two exponential terms, first term has time
constant T1 and second term has a time constant T2 . T1 <
T2 . In other words we can say that first exponential term
decaying much faster than the other exponential term.
So for time response we neglect it, then
)16(
)1(
1
22 


n
T



Time response second order

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Time response second order