Digital Signal Processing Reference
In-Depth Information
Fig. 8.8. Armature-controlled dc
motor. (a) Cross-section; (b)
schematic representation.
(a)
+
R
a
L
a
angular
velocity
w
(
t
)
torque
T
m
+
V
em
−
dc
motor
v
a
(
t
)
Load
J
i
a
(
t
)
k
f
w
(
t
)
viscous
friction
−
pt
(b)
8.3.1 Mathematical model
The linear model of the armature-controlled dc motor is shown in Fig. 8.8(b),
where a load
J
is coupled to the armature through a shaft. Rotation of the arma-
ture of the dc motor causes the desired motion in the attached load
J
. Moving
a conductor within a magnetic field also generates a back electromagnetic field
(emf) to be induced in the dc motor. The back emf results in an opposing emf
voltage, which is denoted by
V
emf
in Fig. 8.8(b). In the following analysis, we
decompose the motors into three components: armature, motor, and load. The
equations for the three components are presented below.
Armature circuit
Applying Kirchhoff's voltage law to the armature circuit,
we obtain
d
i
a
d
t
L
a
+
R
a
i
a
+
k
f
ω
(
t
)
=
V
a
(
t
)
,
(8.29)
V
emf
(
t
)
where
V
a
(
t
) denotes the armature voltage and
i
a
(
t
) denotes the armature current.
The electrical components of the armature circuit are given by
L
a
and
R
a
, where
L
a
denotes the self inductance of the armature and
R
a
denotes the self resistance
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