Page 103 - The Mechatronics Handbook

0066_Frame_C07 Page 13 Wednesday, January 9, 2002 3:39 PM

FIGURE 7.12 Force on a ferromagnetic bar near an electromagnet.
Example 2. Magnetic Force on an Electromagnet
Imagine a ferromagnetic keeper on an elastic restraint of stiffness k, as shown in Fig. 7.12. Under the
soft magnetic keeper, we place an electromagnet which produces N turns of current I around a soft
ferromagnetic core. The current is produced by a voltage in a circuit with resistance R.
The magnetic force will be calculated using the magnetic stress tensor developed by Maxwell and
Faraday (see, e.g., Moon, 1984, 1994). Outside a ferromagnetic body, the stress tensor is given by t and
the stress vector on the surface defined by normal n is given by ττ ττ = t ⋅ n:

1 1


[
t = ----- -- B n – B t ], B n B t = ( t n ,t t ) (7.30)
2
2
m 0 2  
For high magnetic permeability as in a ferromagnetic body, the tangential component of the magnetic
field outside the surface is near zero. Thus the force is approximately normal to the surface and is found
from the integral of the magnetic tension over the surface:
1
∫
F = -------- B n n Ad (7.31)
2
2m 0
2
and B n /2m 0 represents a magnetic tensile stress. Thus, if the area of the pole pieces of the electromagnet
is A (neglecting fringing of the field), the force is

F = B g A/m 0 (7.32)
2

where B g is the gap field. The gap field is determined from Amperes law

)
NI = R Φ, Φ = B g A (7.33)

where the reluctance is approximately given by

(
2 d 0 – )
d
)
R = ---------------------- (7.34)
m 0 A
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