Page 628 - Mechatronics with Experiments

614 MECHATRONICS
space, such as steel with ≫ , then the magnetic flux density developed inside the coil
m
0
would be much higher.
Magnetic flux (Φ ) is defined as the integral of magnetic flux density ( ⃗ B) over a
B
cross-sectional area perpendicular to the flux lines (Figure 8.6c),
Φ = ⃗ B ⋅ d ⃗ A Tesla ⋅ m 2 or Weber (8.40)
B ∫ ps
A ps
where d ⃗ A ps is differential vector normal to the surface (A ). The area is the effective
ps
perpendicular area to the magnetic field vector. It is important not to confuse this relationship
with Gauss’s law.
Gauss’s law states that the integral of a magnetic field over a closed surface that
encloses a volume is zero (integration over the closed surface A showninFigure8.6c),
cs
⃗
⃗ B ⋅ dA = 0 (8.41)
∮ cs
⃗
where dA is a differential area over a closed surface (A ), not a cross-sectional perpen-
cs cs
dicular area to flux lines. This integral is over a closed surface. In other words, net magnetic
flux over a closed surface is zero. The physical interpretation of this result is that magnetic
fields form closed flux lines. Unlike electric fields, magnetic fields do not start in one
location and end in another location. Therefore, the net flow-in and flow-out lines over a
closed surface are zero (Figure 8.6c).
Let us define the concept of flux linkage. Consider that a magnetic flux (Φ )is
B
generated by a coil or permanent magnet or a similar external source. If it crosses one turn
of conductor wire, the magnetic flux passing through that wire is called the flux linkage
between the existing magnetic flux and the conductor,
=Φ ; for one turn coil, (8.42)
B
If the conductor coil had N turns instead of one, the amount of flux linkage between the
external magnetic flux Φ and the N turn coil is
B
= N ⋅ Φ ; for N turn coil, (8.43)
B
Magnetic field strength ( ⃗ H) is related to the magnetic flux density ⃗ B with the permeability
of the medium,
⃗ B = ⋅ ⃗ H (8.44)
m
Magnetomotive force (MMF) is defined as,
MMF = H ⋅ l (8.45)
where l is the length of the magnetic field strength path.
Reluctance of a medium to the flow of magnetic flux is analogous to the electrical
resistance of a medium to the flow of current.
The reluctance of a medium with cross-sectional area A and thickness l can be defined
as
l
R = (8.46)
B
⋅ A
m
where is the permeability of the medium, such as air, iron. Permeance of a magnetic
m
medium is defined as the inverse of reluctance,
1
P = (8.47)
B
R
B
Reluctance is analogous to resistance, and permeance is analogous to conductivity.