Freq enc Domain Anal sis I
Frequency Domain Analysis I
Physical System Physical System Mathematical Model
Modeling
input output relation
Solution, Time domain Transfer Function
Laplace Transform
Stability of the
differential equation
, S-domain
Stability of the control system
Inverse Laplace Transform Direct method
Solution Solution by
state equation formulation
Feedback Control System
Concept of Frequency Response
G(s)
( ) y t ( )
u t
1 2
( ) sin
( ) n
u t P t
K s z s z s z
G
1 1 2 2
( )
( ) ( ) ( ) ( )
n n
G s s s s s s s
Y s G s u s u s P
2 2
2 2
( ) ( ) ( ) , ( )
( ) ( )
Y s G s u s u s P
s Y s G s P
s
1 2
1 2
n n
s
b b b
a a
s j s j s s s s s s
1 2
1 2
( ) j t j t s t s t n s t
ny t ae
ae b e
b e
b e
Frequency Response
( ) j t j t
y t ae ae
P P
j
2 2
( ) 2
( )
s j
P P
a G s s j G j
s j
P P
a G s s j G j
G j
j G
y
2 2
( ) s j 2
x y
a G s s j G j
s j
G j G jG
G
x
cos sin
cos sin j
G j j G j
G j j G j e
G j
j
j
G j G j e
P P
a G j G j e
Similarly,
j
2 2
2 2
j
a G j G j e
j j
P P
a G j G j e
j j
2 j 2 j
Frequency Response
( ) j t j t
j j t j j t
y t ae ae
P P
G j e e G j e e
2 2
2
j t j t
j j
j j
G j P e e
j
2 sin j
G j P t
Frequency Response of First Order Systems
G(s)
( ) y t ( )
r t
sin t steady state
( ) 1
Y s ( )
G s
Consider
sin t
sinusoidal input
steady state
( ) M
( ) ( ) 1
( ) 1
1
R s G s Ts
G j Tj
Consider,
2 2
1 ( ) | ( ) | 1
1 Tj
M j G j
T
1
1
( ) ( ) ( 1)
tan T
j G j Tj
T
1
2 2
( ) 1 sin tan
1
y t t T
T
Frequency Response of Second Order Systems
G( )
( ) y t ( )
r t sin t G(s)
sinusoidal input
A sin( t )
sinusoidal output
2
2 2
( ) , 0 1
2
( ) ( ) sin
n
n n
G s s s
R s r t t
2 2
2
2 2 2 2 2 2 2 2
( ) , ( ) sin
( ) 2 2
n
n n n n
R s r t t
s
as b cs d
Y s s s s s s s
( ) cos sin cos sin
i ( ) i ( )
n n
n n n n
t t
d d d d
d t
y t ae t b e t c t d t
A t B t
sin( ) sin( )
n
t
Ae d t B t
transient
response Steady-state
response
Steady State Frequency Response
( ) ( ) sin( )
y t A j t M ( )
( ) ( ) ( )
( )
M y t G j
r t
Magnitude ratio
1.0
( )
( ) ( )
r t
j G j
Phase
( j )
Frequency response
Steady State Frequency Response Steady State Frequency Response
( ) 2
Y s
2 2
2 2
( ) ( )
( ) 2
( )
( )
n
n n
n n
Y s G s
R s s s
Y j G j
2 2 2 2
2
2 2
( )
( ) 2 2
( )
( ) 2
n n
n n
n
R j G j j j j
M Y j j
2 2 2 2 2 2
2
( ) 2
( ) 4
1
n
n n
n n
n
M j
R j
2 2 2 2 2 2 2 2 2 2
2 2
4 1 4
n
n n
n n
( )
( ) ( )
( )
Y j M j G j
R j
Steady State Frequency Response
( ) M
2
Steady State Frequency Response
M 1.0
1
2 2
1
( ) tan 2
lim ( ) tan 0 180
n n
j j
M
m( j )
m
lim ( j ) tan 0 180
j
90
2 2
180
2 2
n
2
n
Steady State Frequency Response Steady State Frequency Response
2 1
( )
M 2 2
2 2
2
2 2
( )
1 4
n n
M
2
2
2 2 2
2
2
2 1 2 8
( ) n n n 0
dM d
2 2 2 2
2
2 2
1 4
n n
d
2 2
2 2
2 2 2 2
2 1 2 8 0 , 1 2 0
n n n n
2
2
1 2 , 1
2 1
n m M m
Unit Step Response VS Frequency Response Unit Step Response VS Frequency Response
( ) y t ( )
r t
22 2
n
2 2
2
n ns s
( ) 1 ( ) 1 r t u
t
2 1
2
( ) 1 1 sin 1 cos
1
n
t
y t e n t
U it t F
( ) M j
M
m( ) y t M
pUnit step response Frequency response
1.0
mt
mt
1 exp 2 p 1
M
2
2
1 2 , 1
2 1
m n M m
