Page 627 - Mechatronics with Experiments

ELECTRIC ACTUATORS: MOTOR AND DRIVE TECHNOLOGY 613
magnetic field is in the direction of the fingers encircling the thumb. If the current changes
direction, the magnetic field changes direction.
The Biot-Savart law states that the magnetic field (also called magnetic flux density),
⃗ B, generated by a current on a long wire at a point P with distance r from the wire is
(Figure 8.6a and b)
⃗
i dl × ⃗ e r
d ⃗ B = (8.33)
4 r 2
where is the permeability of the medium around the conductor ( = for free space), ⃗ e r
0
is unit vector of ⃗ r. We can think of permeability as the conductivity of the medium material
(the opposite of resistance) for magnetic field. The permeability of free space is called 0
= 4 ⋅ 10 −7 Tesla ⋅ m∕A (8.34)
0
Quite often, the permeability of a material ( ) is given relative to the permeability of free
m
space,
= ⋅ 0 (8.35)
r
m
where is the relative permeability of the material with respect to free space. If the Biot-
r
Savart law is applied to a conductor over the length of l, we obtain the magnetic flux density
⃗ B at point P at a distance r from the conductor due to current i (Figure 8.6a),
⃗
l idl × ⃗ e
⃗ B = r (8.36)
∫ 2
0 4 r
The units of ⃗ B in SI units is Tesla or T.
Ampere’s Law states that the integral of a magnetic field over a closed path is equal
to the current passing through the area covered by the closed path times the permeability
of the medium covered by the closed path of integration (Figure 8.6a and b),

⃗
⃗ B ⋅ ds = ⋅ i (8.37)
∮
C
The vector relationship between the current, position vector of point P with respect to
the wire and magnetic field follows the right hand rule. It describes how electromagnetic
fields are created by a current in a given medium with a known magnetic permeability. The
magnetic flux density is a continuous vector field. It surrounds the current that generates
it based on the right hand rule. The magnetic flux density vectors are always closed,
continuous vector fields.
By using either the Biot-Savart law or Ampere’s law, the magnetic field due to
current flow through a conductor of any shape in an electrical circuit can be determined.
For instance, the magnetic field generated around an infinitely long straight conductor
having current i in free space at a distance r from it can be calculated (Figure 8.6b)
⋅ i
0
| ⃗ B| = B = (8.38)
2 ⋅ r
Similarly, the magnetic field inside a coil of solenoid is (Figure 8.6c)
⋅ N ⋅ i
0
| ⃗ B| = B = (8.39)
l
where N is the number of turns of the solenoids, l is the length of the solenoid. It is assumed
that the magnetic field distribution inside the solenoid is uniform and the medium inside
the solenoid is free space. Notice that if the medium inside the coil is different than free