Page 698 - Mechatronics with Experiments

684 MECHATRONICS

a a
i
(t) m (t)
(t) (t)
e
m l l

FIGURE 8.59: DC motor dynamic model.

̇
where k (t) term is the voltage generated by the back electromotive-force (back EMF)
e
as a result of the generator action, L , R are inductance and resistance of the winding,
a a
respectively. The electrical to mechanical power conversion (current to torque) is given by
T (t) = K i(t)
T
m
where, K is the torque gain, and T is the torque generated by the motor. Finally the
T
m
mechanical relation between torque, inertia, and an other load is given by
̈
̇
T (t) = (J + J ) + c (t) + T (t)
m
l
l
m
where J is rotor inertia, J is load inertia, c is damping constant, T is load torque.
m l l
From these three basic relationships, we can derive various transfer functions, such as the
transfer function between terminal voltage to motor speed or motor position, or the transfer
function from armature current to motor speed. Physically, an amplifier manipulates the
motor terminal voltage in order to control the motor motion. This voltage control may be
based on current feedback, voltage feedback or both.
Let us derive the transfer function from motor terminal voltage to motor speed. Taking
the Laplace transform of the above equations for zero initial conditions,
̇
V (s) = (L s + R )i(s) + k (s) (8.286)
a
e
a
t
1
̇
→ i(s) = [V (s) − k (s)] (8.287)
e
t
L s + R a
a
T (s) = K i(s)
m
T
̇
(J s + c) (s)y = T (s) − T (s)
l
m
T
where J = J + J .
T
m
l
l
(t) + m _ (t)
_ +
a a
e
DC motor model
FIGURE 8.60: Block diagram of the DC motor dynamic model.