Page 633 - Mechatronics with Experiments

ELECTRIC ACTUATORS: MOTOR AND DRIVE TECHNOLOGY 619
2. A changing magnetic flux is created as a result of changing inductance. In a circuit,
even though the current is constant, change in magnetic flux can be caused by the
change in the geometry and permeability of the medium (change in reluctance). This
results in induced back EMF. This phenomenon is at work in the case of solenoids
and variable reluctance motors. In general this voltage has the form (Figure 8.9b),
dΦ B
V =− (8.68)
bemf
dt
d(L(x)i(t))
=− (8.69)
dt
di(t) dL(x)
=−L(x) − ⋅ i(t) ⋅ ̇ x(t) (8.70)
dt dx
where the first term is the back EMF due to self-inductance of the circuit (L(x)),
and the second term is the back EMF induced due to the change in the inductance
(dL(x)∕dx).
3. A back EMF is induced in the conductors moving in a fixed magnetic field established
by the field magnets (permanent magnets or electromagnets Figure 8.9d). As the
conductor moves in the magnetic field, there will be a force acting on the charges in it
just as there is a force acting on a charge moving in a magnetic field. Let us consider a
constant magnetic field B, a conductor with length l moving in perpendicular direction
to magnetic field vector, and its current position is x (Figure 8.9d). The induced back
EMF because of this motion is
dΦ B d(B ⋅ l ⋅ x)
V bemf =− =− =−B ⋅ l ⋅ ̇ x (8.71)
dt dt
This is referred to as the generator action. In the case of a brush-type DC motor, the
conductor is the winding on the rotor which moves relative to stator magnets. In the
case of brushless DC motors, the permanent magnets on the rotor move relative to
the fixed stator windings. The resulting induced back EMF voltage effect is the same.
A coil of conductor has self-inductance, which opposes the change in the magnetic
field around it. Through self-inductance, L, the coil generates an EMF voltage that is
proportional to the rate of change of current in the opposite direction (Figure 8.10). If the
circuit geometry and its material properties vary (i.e., in the case of a solenoid, the air gap
varies), the inductance is not constant. The inductance is a function of the geometry (i.e.,
number of turns in a coil) and the permeability of the medium. If the core of an inductor
winding has a moving iron piece and the permeability of the medium changes as the core
moves, the inductance of the coil changes. Consider the self-inductance L of a solenoid coil
with length l and a total of N turns, cross-sectional area of A. Let us assume the medium
inside the coil is air. Let us assume that the magnetic flux inside the coil is uniform. Then,
B = ⋅ (N∕l) ⋅ i (8.72)
o
Φ = B ⋅ A = ⋅ (N∕l) ⋅ i ⋅ A (8.73)
B
o

V (t)
L
FIGURE 8.10: A coil of a conductor and its
self-inductance. Self-inductance of a coil is a function of
the number of turns in the coil, its geometry, and the
i material properties of the medium that the coil core and
L its surrounding.