Page 700 - Mechatronics with Experiments

686 MECHATRONICS
where we used the approximation
√ x
1 − x ≃ 1 − ; for x ≪ 1
2
The poles are
K K
T e
p =− (8.294)
1
J R
T a
( )
−2R J + 2L a K T K e
a T
p =− R a (8.295)
2
2L J
a T
R a K K
T e
=− + (8.296)
L a J R
T a
Since R a ≫ K T K e the second pole is approximately
L a J T R a
R a
p ≃− (8.297)
2
L
a
The motor terminal voltage to motor speed transfer function can be approximated as
( )
K T
̇
(s) J T L a
= (8.298)
V (s) (s − p )(s − p )
t
1
2
( )
1
K e
= (8.299)
( s + 1)( s + 1)
m e
where
1 J R
T a
=− = mechanical time constant (8.300)
m
p K k
1 T e
1 L a
=− = electrical time constant (8.301)
e
p R
2 a
and the mechanical time constant is much larger than the electrical time constant for
m e
most motors. The motor speed is also influenced by disturbance or load torque. The transfer
function including the load torque can be derived as follows
( )
K T L a s+R a
̇
(s) = ( ) J T L a ( )V (s) − ( ) J T L a ( )T (s) (8.302)
t
l
2
2
s + R a s + K T K e s + R a s + K T K e
L a J T L a L a J T L a
1 1 K T
( s + 1)R a
e
K e K T J T L a
= V (s) − ( ) ( )T (s) (8.303)
t
l
( s + 1)( s + 1)
2
m e s + R a s + K T K e
L a J T L a
1 1 R a
( s + 1)
e
K e K e K T
= V (s) − T (s) (8.304)
t
l
( s + 1)( s + 1) ( s + 1)( s + 1)
m e m e
1 1 R a
̇
(s) = K e V (s) − K e K T T (s) (8.305)
l
t
( s + 1)( s + 1) ( s + 1)
e
m
m