Energy Stored in Permanent Magnets
Energy Stored in Permanent Magnets
Energy Stored in Permanent Magnets
NOMENCLATURE
Fig. 1. Recommended definition of stored energy density in a permanent
Permeability of free space (4 10 H/m). magnet.
Relative differential permeability of magnet (as-
sumed constant). the following volume integral [1]–[4]:
Magnetic flux density (T).
Minimum experienced by the magnet (T).
Remanent of the magnet (T). (1)
Depth into page (m).
Force (N). where is a dummy integration variable.
Magnetic field strength (A/m). When force is calculated from the derivative of stored
Minimum experienced by the magnet (A/m). energy with respect to a virtual displacement, the choice
Magnet length (m). of the lower limit of integration for the energy density is
Thickness of air gap (m). mathematically arbitrary. However, for a “rigid” permanent
Thickness of magnet (m). magnet (such as a modern rare earth magnet), acting along
Volume (m . its linear demagnetization (recoil) characteristic with a slope
Work done (stored energy) (J). , we believe that it is best to choose the lower limit as
Air gap length (m). the remanence , as depicted in Fig. 1, i.e.,
(2)
I. INTRODUCTION
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506 IEEE TRANSACTIONS ON MAGNETICS, VOL. 35, NO. 1, JANUARY 1999
(Note (6) is only true in practice for a zero length air gap.)
Solving the linear system (4)–(6) gives
(7)
(9)
(10)
This section examines the simple idealized magnetic cir- Expressing the energy in terms of avoids the need to
cuit depicted in Fig. 2, containing a magnet with a linear account for the variation of and is a particularly elegant
demagnetization curve, Fig. 3, and confirms that if (2) is used expression thanks to the limits chosen for (2), namely to
for the energy in the magnet, then the force on the keeper The energy between the dotted and solid lines in Fig. 3
satisfies (3). Fringing and leakage are neglected and the iron is the energy responsible for the demagnetization and is lost
is assumed ideal, i.e., infinite permeability and nonsaturating. primarily as heat (commonly referred to as hysteresis loss).
Using subscript for magnet and for air gap then the magnet For any linear material and including, for example, when a
is assumed to satisfy component of is applied in the direction perpendicular to the
(4) magnetization direction in a magnet, the energy is of the form
(12) (inserting the appropriate approximately constant
Ampere’s law and flux conservation imply
IV. CONCLUSIONS
(5)
We have demonstrated that the stored magnetic energy per
(6) unit volume in an ideal magnet with a linear recoil line and
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LOVATT AND WATTERSON: ENERGY STORED IN PERMANENT MAGNETS 507
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