Online proceedings - EDA Publishing Association

A. Continuous domain
Starting from a given configuration of permanent magnets
in terms of shape, dimensions and polar orientation, the
magnetic flux density can be calculated in the region of
space surrounding the magnets as a 3-dimensional vector
field. The diamagnetic proof mass is parallel to the plane of
magnets and is situated in the same space region; its midplane
is initially positioned at the distance from the
magnets surface. The diamagnetic force in vertical direction
can be estimated for all the points of the proof
mass, assuming that each contribution of the force acts on
the infinitesimal volume d d d d. The total vertical
diamagnetic force at the height of the mid-plane is
d
(11)
where is the proof mass volume. The static equilibrium of
the diamagnetic proof mass is given by the following
relation:
(12)
where is the gravity force acting on the proof
mass, is the proof mass density and is the acceleration
of gravity. Depending to the vertical position of the mass,
the force balance may not to be verified and a recursive
calculation is needed. In the further step of the calculation,
the magnetic force should be evaluated at the height . The
next value of the height must be assumed according to the
following cases:
if , then d
if , then d
The recursive calculation ends when
at the levitation distance L .
B. Discrete modeling
In this case, starting from the configuration of permanent
magnets, the magnetic flux density is calculated in the
region of space surrounding the magnets as a 3-dimensional
discrete vector field. This means that the value of is
calculated only in specific points that are comparable to the
nodes of a finite element model. The distribution of can
be calculated in the discrete domain, starting from a given
configuration of the permanent magnets, for example by
using a commercial FEM simulator.
The diamagnetic proof mass is parallel to the plane of
magnets and is situated in the same space region; its midplane
is initially positioned at the discrete distance from
the magnets surface. The diamagnetic force in vertical
direction can be estimated for all the discrete points
of the proof mass, assuming that each contribution of the
force acts on the discrete volume ∆ ∆ ∆ ∆. The
central finite difference method can be used for this
calculation. The total vertical diamagnetic force at the
height of the mid-plane is
∑∆
(13)
As described before, the vertical equilibrium between
diamagnetic and gravity forces has to be considered with a
recursive calculation; the height of the mass at the further
11-13
May 2011, Aix-en-Provence, France
step must be assumed according to the following cases:
if , then ∆
if , then ∆
The recursive calculation ends when
at the levitation distance L .
C. Approximations
The discrete modeling approach can be simplified by the
following assumption: in case of thin proof mass, the
variation of the diamagnetic force along the thickness can
be neglected. This means that const. over the
thickness. Thanks to this approximation, only the mid-plane
of the proof mass can be considered in the modeling.
The diamagnetic force in vertical direction can
be estimated only for the discrete points of the mid-plane,
assuming that each contribution of the force acts on the
discrete volume ∆, which is placed across the mid-plane.
The total vertical diamagnetic force at the height of the
mid-plane is
∆
(14)
∆ ∑
V. FEM SIMULATION OF MAGNETIC FIELD
The distribution of the magnetic field around the
permanent magnets in the described configuration ( 1)
was calculated with a 3D FEM simulation by the
commercial tool ANSYS 13.0. The elements solid96 were
used to model the magnets and the surrounding air; the
mesh size was 0.5mm and the coercive force
750 kA⁄ m was assumed for the magnets. The first
magnetization curve represented in Fig. 4 was used for
NdFeB. The FEM model is shown in Fig. 5.
The magnetic field distribution in the surrounding air was
calculated; the value of at the vertical height 1mm is
reported in the diagrams of Fig. 6. Due to the magnets
orientation, the simulation results show the following
symmetries of the magnetic field: ,
, | | | |.
Fig. 4. First magnetization curve of NdFeB magnets.
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