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