Page 636 - Mechatronics with Experiments

622 MECHATRONICS
The magnetomotive force (MMF) generated due to the coil and current is

MMF = N ⋅ i (8.86)
Flux circulating in the closed path along the core and through the air gap is
MMF N ⋅ i
Φ = = (8.87)
B
R R
The flux linkage to the coil is
= N ⋅ Φ (8.88)
B
= L ⋅ i (8.89)
N ⋅ i
= N (8.90)
R
N 2
L = (8.91)
R
This shows that self-inductance of a magnetic circuit involving a coil is proportional to the
square of the number of turns and inversely proportional to the reluctance of the circuit. In
actuator applications, small reluctance is desirable in order to generate more magnetic flux,
hence force or torque. However, smaller reluctance leads to large inductance, which results
in a larger electrical time constant. In electric motor design applications, this conflicting
design requirement must be balanced: for large force/torque, we want small reluctance
and for small electrical time constant we want small inductance. However, inductance and
reluctance are inversely related. As one increases, the other one decreases.

8.1.3 Permanent Magnetic Materials
Materials can be classified into three categories in terms of their magnetic properties:

1. paramagnetic (i.e., aluminum, magnesium, platinum, tungsten),
2. diamagnetic (i.e., copper, diamond, gold, lead, silver, silicon),
3. ferromagnetic (i.e., iron, cobalt, nickel, gadolinium) materials,
soft ferromagnetic materials,

hard ferromagnetic materials.

Of these, hard ferromagnetic materials are of interest to us as permanent magnetic materials.
Soft ferromagnetic materials are used as lamination material for stator and rotor frames and
transformer frames. The difference between these materials originates from their atomic
structure. Magnetic field strength, ⃗ H, and magnetic flux density, ⃗ B, relationship in a given
spatial location depends on the “permeability” of the surrounding material, ,
m
⃗ B = ⋅ ⃗ H (8.92)
m
where, = ⋅ is called the magnetic permeability of the material which is a measure
r
m
0
of how well a material conducts magnetic flux, and is called the relative permeability,
r
is the permeability of free space. The relationship between the magnetic field strength
0
( ⃗ H) and the magnetic flux density ( ⃗ B) is almost linear for paramagnetic and diamagnetic
materials. Although the same relationship can be used to describe the magnetic behav-
ior of ferromagnetic materials, the relationship is not linear and exhibits large hysteresis
(Figure 8.13). Susceptibility, , is defined as
= 1 + (8.93)
r