Page 702 - Mechatronics with Experiments

688 MECHATRONICS

Current Amp. DC motor

a m
T
a T

FIGURE 8.64: Current amplifier plus DC motor transfer function from commanded current to
motor speed.

amplifier input (commanded current, i (t)) to the motor speed can be derived as follows
cmd
(neglecting the effect of amplifier terminal voltage saturation, Figure 8.64)
K T
̇
(s) K 1 (L a s + R a )(J T s + c) + K e K T
= ( ) (8.307)
i cmd (s) K T J T s + c
1 + K
1 K 2
(L a s + R a )(J T s + c) + K e K T K T
K 1 K T
L a J T
= ( ) ( ) (8.308)
2
s + L a c + R a J T s + K 1 K 2 J T s + K e K T + K 1 c + K 1 K 2 c
L a J T L a J T
K
= (8.309)
2
s + bs + c
K a K T
= (8.310)
( s + 1) (J s + c)
a
T
If we consider the transfer function between armature current and motor speed;
T (t) = K i(t) (8.311)
m
T
̈
̇
T (t) = J + c − T (t) (8.312)
T
l
m
K T 1
̇
(s) = ⋅ i(s) − ⋅ T (s) (8.313)
l
(J s + c) (J s + c)
T
T
8.8.3 Steady-State Torque-Speed Characteristics of DC
Motor Under Constant Terminal Voltage
Consider the electrical and electrical to mechanical power conversion relations for a DC
motor.
di
̇
V (t) = L a + R i + K (t) (8.314)
a
e
t
dt
T (t) = K i(t) (8.315)
m
T
In steady-state the effect of L will be zero. If we set L = 0 for steady-state analysis, the
a a
torque-speed terminal voltage relationship is given by,
K T K K
T e
̇
T (t) = ⋅ V (t) − ⋅ (t) (8.316)
m
t
R a R a
This is a linear relationship of the type
y =−ax + b (8.317)