According to Biot-Savart
law, the flux density along the axis of a permanent magnet can
be calculated with the dimensions and residual induction (Br)
of the material.
Cylindrical
Magnet Magnetized Axially
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Rectangular
Magnet Magnetized Along Its Length
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Cylindrical
Tube Magnet Magnetized Axially
Thus, the flux
density for the tube can be calculated by subtracting the flux
density of a cylinder of inside diameter (2ri)
from the flux density of a cylinder of outside diameter (2r0),
The same logic can be applied to other symmetrical hollow shapes:
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Flux
In The Air Gap
Between
Two Axially Aligned Magnets
B
= B1 + B2
B2
= the flux density for a rectangular block at distance (g-d).
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Between
Two Magnets In A C-Shaped Yoke
B
= B1 + B2
B1
= the
flux density for a rectangular block with length (2l)
at a distance (d).
B2
= the flux density for a rectangluar block
with length (2l) at a distance (g-d).
For
An Ideal Magnetic Circuit
Illustrated as Below:
Assuming no loss of
magnetomotive force in the steel, The Ampere's law tell:
Hmlm
= Hglg
Hm = MMF
of the magnet, lm =
length of the magnet,
Hg = MMF across the gap,
lg = length of the gap.
Assuming no flux leakage
(i.e. all the flux from the magnet passes through the gap),
we can also write the following relationship:
BmAm
= BgAg
Bm =
the induction of the magnet, Am
= the cross-sectional area of the magnet,
Bg = the flux density
within the gap, Ag =
the cross-sectional area of the gap.
So, we have the permeance
coefficient (PC) :
Bm
/ Hm
= ( BgAglm
) /
(
HgAmlg
)
B/H can
be used to define the permeability of a material. In CGS units,
the permeability of air is unity, so Bg=Hg.
The equation for permeance coefficient then becomes:
Bm
/ Hm
= ( Aglm
) /
( Amlg
)
Note that this is purely dependent
on circuit geometry, not magnetic properties. The permeance coefficient
is the slope of the load line, drawn from the origin. The intersection
of the load line and the second quadrant demagnetization curve is
called the operating point, where Bm and Hm
for that geometry are determined. From Bm, we
can calculate the flux density in the gap, (Bg):
Bg
= ( BmAm
)
/ Ag
Consodering the flux
leakage and MMF losses in the circuit in a practical condition,
the leakage factor (σ)and
the reluctance factor ( f
) are introduced into the above equations for greater
accuracy.
Hmlm
= f Hglg
BmAm
= σ
BgAg
Bm
/ Hm
= ( σ
Aglm
) /
(f
Amlg
)
The leakage factor
is the ratio of total available flux to gap flux. A empirical
permeance formula is :
σ
= Pt
/
Pg
Pt =
the sum of the permeance of all leakage paths (including the gap)
of the circuit;
Pg = the permeance of the
gap.
The reluctance factor
(f)
accounts for the loss in MMF through the flux carrying
components of the circuit (i.e. steel) and small gaps between
parts. It is also empirically determined, and ranges from 1.1
to 1.5 for most circuits. Higher values of f are measured for
circuits where the steel is in saturation. The reluctance factor
is:
f
= Ht
/
Hg
Ht =
the total MMF; Hg =
the MMF across the gap.
A commonly
used equation to estimate holding force is:
F =
0.577B2A
B
= flux density in kilogauss, A = the area of a
pole face in square inches.
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For The Holding Assembly
As pictured
below:
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F
= 0.577(B12A1+B22A2)
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For
A Single Magnet In Air Separated By A Distance
A good
approximation for B can be calculated using the
current sheet equations.
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For
A Magnet In Direct Contact With Steel
F = 0.577Br2lmA
Br =
residual induction; lm
= length of magnet; A
= pole area.
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For
Thicker magnets
Using
current sheet calculations produces a more accurate approximation.
Assuming that a magnet on steel is roughly equivalent to a magnet
of twice the length, we can calculate the flux density in the
center of a magnet twice the length, and use it in the equation:
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