Page 699 - Mechatronics with Experiments

ELECTRIC ACTUATORS: MOTOR AND DRIVE TECHNOLOGY 685

L
a
T

t (s) _ (s)
+
e m e
FIGURE 8.61: Block diagram of the relationship between motor terminal voltage and load
torque to motor speed for a typical DC servo motor.

The transfer function describing the effect of terminal voltage and load torque on the
motor speed can be found as (Figure 8.61)
K T (L s + R )
a
̇
a
(s) = V (s) − T (s)
l
t
(J s + c)(L s + R ) + K k (J s + c)(L s + R ) + K k
t a a T e t a a T e
The transfer function from motor terminal voltage to motor speed is given by
̇
(s) K T
= (8.288)
V (s) (J s + c)(L s + R ) + K K
T e
t
a
t
a
K T
= (8.289)
J L s (L c + J R )s + (cR + K K )
2
T a a T a a T e
K T
= ( ) ( ) (8.290)
2
J L s + L a c+J T R a s + cR a +K T K e
T a
J T L a J T L a
The poles of the transfer function are given by
( ) ( )
L c + J R cR + K K
a
T e
a
T a
2
s + s + = 0
J L J L
T a T a
Normally, this equation has two complex conjugate roots.
Special Case: DC Servo Motors DC servo motors have very low inductance
(L small), and very low damping (c small). Using these facts for the DC servo motors case
the transfer function can be approximated as
K T
̇
(s) J T L a
≃ ( ) ( )
V (s) s + R a s + K T K e
2
t
L a J T L a
where the poles are given by
√
( ) 2 ( )
R a − 4 K T K e
R a L a J T L a
p 1,2 =− ± (8.291)
2L a 2
√
( 2 )
a
4K k J R J T − L a
T e T
R a 4K T K e
=− ± (8.292)
2L a 2L J
a T
( )
R J 1 − 2L a K T K e
R J a T R J T
2
a T
a
≃− ± (8.293)
2L J 2L J
a T a T