Page 634 - Mechatronics with Experiments

620 MECHATRONICS
di dΦ B
V bemf =−L =−N (8.74)
dt dt
L ⋅ i = N ⋅ Φ B (8.75)
N ⋅ Φ B
L = (8.76)
i
N ⋅ ⋅ (N∕l) ⋅ i ⋅ A
o
L = (8.77)
i
2
⋅ N ⋅ A
o
= (8.78)
l
N 2
L = (8.79)
R B
which shows that the inductance is a function of the coil geometry and permeability of the
medium. If the coil core is iron, then would be replaced by for iron which is about
o
m
1000 times higher than the permeability of air. Hence, the inductance of the coil would be
higher by the same ratio.
In electric actuator design, it is desirable to have a small magnetic reluctance, R ,
B
so that more flux (Φ = MMF∕R ) is conducted per unit magnetomotive force (MMF).
B
B
On the other hand, it is desirable to have small inductance (L) so that the electrical time
constant of the motor is small. These are two conflicting design requirements. A particular
design must find a good balance between them that is appropriate for the application.
Let us consider the transformer shown in Figure 8.11. A transformer has two windings,
a primary and secondary winding, and a laminated iron core which magnetically couples
them. The laminated design, as opposed to solid metal piece design, reduces eddy current
losses. The iron core material of the laminations has a large magnetic permeability and a
large flux saturation level, which helps conduct the generated magnetic flux through the
circuit without saturation. In other words, it provides an efficient electromagnectic coupler
between the two coils.
A transformer works based on Faraday’s induction principle, that is, voltage is induced
on a conductor due to a change in the magnetic field. In the case of transformers, the change
in magnetic field is due to the alternating current (AC) nature of the source at the primary
winding. An ideal transformer can be viewed as having pure inductance, although in reality
there is some resistance and capacitance.
Faraday’s law states that the voltage across the primary winding is proportional to
the rate of change of the magnetic flux and opposes that change,
(t) = V ⋅ sin t (8.80)
1 1
dΦ B
(t) =−N 1 (8.81)
1
dt

N 1 N 2

V (t) N N V (t) V (t) V (t)
1 1 2 2 1 2
FIGURE 8.11: (a) An ideal
transformer with primary and
secondary coil windings,
Iron core laminated soft iron core.
(b) circuit diagram for a
(a) (b) transformer.