Électrotechnique et électroénergétique
EMIL CAZACU 1 , IOSIF VASILE NEMOIANU
Key words: Static diamagnetic levitation, Vertical and horizontal configuration.
The paper critically analysis both static vertical and horizontal diamagnetic levitation of a tiny permanent magnet placed into a stationary magnetic field generated by a current loop, or a pair of symmetrically placed current loops. The equilibrium and stability issues are quantitatively treated, allowing an accurate predetermination of the floater’s equilibrium point coordinates and its stability area. The interactions between the levitated magnet, generated magnetic field and the diamagnetic pieces are both analytical and numerical evaluated.
1.
INTRODUCTION
One way to circumvent the terms of Earnshaw’s theorem [1], which does not allow a stable levitation to occur using only static or stationary fields, is the use of diamagnetic materials. Due to their negative magnetic susceptibility, the presence of pieces made from such materials nearby the suspended floaters can stabilize the intrinsic unstable equilibrium in a large variety of levitation configurations [2–9].
Most of the rather numerous works regarding permanent magnet levitation stabilized by diamagnetic materials treat especially vertical configurations [10–12].
Horizontal configurations, in which stabilization is done along the horizontal direction, allowing to the suspended magnet to be slightly shifted in the vertical plane, were also analysed mostly qualitatively, or for some particular cases
[13, 14]. Beside the quantitative analysis of the two levitation configurations, the paper also suggests a novel horizontal levitation setting, in which the floater is subject to a stable levitation using a particular magnetic field source that consists in a pair of current circular loops. The whole analyse is done mainly starting from the interaction between a stationary magnetic field generated by the source, and a permanent magnet as floater, in the proximity of some diamagnetic materials.
1 “Politehnica” University of Bucharest, 313 Splaiul Independenţei, 060042 Bucharest, Romania, cazacu@elth.pub.ro
Rev. Roum. Sci. Techn. – Électrotechn. et Énerg., 56 , 3 , p. 249–256, Bucarest, 2011

250 Emil Cazacu, Iosif Vasile Nemoianu 2
The vertical or the horizontal configuration could be easily incorporated in a great variety of systems where frictionless suspension is required, achieving a radical improvement of such devices performances. In this respect we could mention: new high sensibility sensors for gravitational field fluctuation
(gravimeters, seismometers) [2], vacuum sensors [6, 7], accelerometers [4], micro frictionless suspension systems [5, 9] and special high-speed motors [8].
2.
VERTICAL AND HORIZONTAL SETTINGS – EQUILIBRIUM AND
STABILITY CONDITION
The proposed levitation settings for both vertical and horizontal configurations – Figs. 1 and 2, respectively – consists of the magnetic field source, namely the current loops carrying a steady current , the levitated permanent magnets (PM), and the pieces of a strong diamagnetic material – pyrolytic graphite
(PG), as equilibrium stabilizers. The interactions between the stationary magnetic field and the PM could assure, for a certain intensity of the loop current, achieving the levitation state, when the value of the magnetic field gradient backwards directed with respect to the z -axis, develops a magnetic force that balances the force of gravity corresponding to the PM.
The stability of the equilibrium is preserved for both configuration by the diamagnetic material pieces, which being placed nearby the PM, act like a tiny servo-system providing the restoring forces needed for keeping the floaters at the equilibrium point in the stability area.
Fig. 1 – Vertical levitation configuration. Fig. 2 – Horizontal levitation configuration.

3 Vertical and horizontal permanent magnet levitation settings 251
In order to discuss the equilibrium and stability of a PM, we construct its potential energy U relationship, and impose that the second spatial derivatives to be positive when the respective first derivatives vanish [3, 5].
If M denotes the magnetic moment of the PM imbedded in the external current loop magnetic field B , the total potential energy U of the magnet can be expressed as:
U = – MB + mgz , (1) where mgz is the gravitational energy of the floater of mass m .
For both vertical and horizontal configuration, the floater will align itself along the local field direction, and therefore its energy is only dependent on the magnitude B of the magnetic field.
Thus, for the vertical configuration – Fig. 1, assuming a circular symmetry for the magnetic field B ( r , z ), the equilibrium points will belong to the z -axis of symmetry. Then, the condition that ( z
0
, 0) is an equilibrium point writes:
F = − ∇ U r z
=
=
0 z
0
= 0 ⇒ ∇ B r z
=
=
0 z
0
= − mg
M
.
(2)
Expressing the magnitude of the magnetic flux density B ( r , z ) in terms of its z -component B z
( r , z ) only, and taking into account that nearby the levitation point:
∇ · B = 0 and ∇ × B = 0, the following extension for total energy yields [11]:
U = −
Here,
M
B
0



B
0
=
+



B '
0
B z
( 0 , z )
− mg
M 


( z r z =
= z
0
0
; B '
0
−
= z
0
) +
∂ B z
∂
(
1
2
B ''
0
( z z
0 , z )
− z r
=
= z
0
0
; z
0
) 2
B ''
0
+
=
1
4 


2
B
B
'
0
0
− B ''
0


 r 2
∂ 2 B z
∂ z
( 0 ,
2 z ) r z =
= z
0
0
.
+ "




.
(3)
Similarly, for the horizontal configuration – Fig. 2, where at the equilibrium point, the magnetic flux density exhibits only a horizontal x - component B = B x the condition that point (0, 0, z
0
( z ),
) is an equilibrium one requires a null value of the total force F exerted onto the floater, which can also be expressed in terms of energy gradient:
F = − ∇ U x y z
=
=
= z
0
0
0
= M
∂ B x
∂ z
( z ) x z y
=
=
= z
0
0
0 k + mg k = 0 ⇒
∂ B x
∂ z
( z ) x z y
=
=
=
0
0 z
0
= − mg
M
,
(4)
B = B x
( z ) i , where i , k are the unit vectors of the x and z -axis, respectively.

252 Emil Cazacu, Iosif Vasile Nemoianu 4
Along with satisfying the equilibrium conditions, stability is to be achieved.
This goal mainly requires a positive concavity of the energy surface graph around the equilibrium point, regardless the chosen direction. Therefore, the stability restrictions for both levitation settings can be written as:
D x
=
∂ 2 U
∂ x 2 x z y
=
=
=
0
0 z
0
> 0 ; D y
=
∂ 2 U
∂ y 2 x z y
=
=
=
0
0 z
0
> 0 ; D z
=
∂ 2 U
∂ z 2 x z y
=
=
=
0
0 z
0
> 0 .
(5)
The left-hand side quantities of the inequalities (5), denoted by D x
, D y
and D z are also known as discriminants of stability [11, 12]. In order to achieve a stable static levitation for the suspended floater in both configurations, all three stability discriminants at the equilibrium point must be positive, simultaneously.
Fig. 3 – Stabilizing the PM in vertical direction. Fig. 4 – Stabilizing the PM in horizontal direction.
In order to obtain stability for the unstable equilibrium of the floater in either vertical or horizontal direction, two pieces of diamagnetic material are placed in the proximity of the levitated magnet [3, 10] – Figs. 3 and 4. Consequently, new terms C ( z − z
0
) 2 and Cx 2 , respectively, are to be added to the right-hand side of
(1), as a quantitative reflection of the diamagnetic material contribution to the total potential energy U , for both vertical and horizontal configurations.
The diamagnetic influence factor, denoted above by C , can be evaluated using the dipole approximation of the floater and its corresponding mirror images within the two diamagnetic stabilizing pieces. An analytical expression for C is given in [3, 10]:
C =
6
π (
χ
2 c
µ
0
+
M w
2
) 5
, (6)

5 Vertical and horizontal permanent magnet levitation settings 253 where: µ
0
= 4π×10 -7 H/m represents the vacuum magnetic permeability, χ is the magnetic susceptibility of the diamagnetic plates, c is the separating distance between the floater and each diamagnetic plate, and l , w , h are the dimensions of the floater, as depicted in Figs. 2 and 4, respectively.
Without the presence of the diamagnetic materials, by setting C = 0 the curvature of floater potential energy along the zaxis (for vertical configuration), or x -axis (for horizontal configuration), becomes negative at the equilibrium point, and thus the system is unable to provide stability when either D z
< 0, or D x
< 0.
3.
NUMERICAL EVALUATIONS
The problem of determining the coordinates of the equilibrium point and its stability area for the suspended permanent magnet in both analyzed configurations leads to the evaluation of the magnetic flux density, produced by the current loops at different points located in the vicinity of the equilibrium point – Fig. 5.
Fig. 5 – Magnetic flux density vector for a current loop.
This goal is achieved by using the analytical expression for the on and offaxis magnetic flux density of a single circular current loop [15]:
B r
=
µ
2
0
π
I
B z
=
µ
2
0
π
I z
( a + r
)
2 + z 2
1
( a + r
)
2 + k =
2



− K + a 2 + r 2
( a − r
)
2
+
+ z 2 z 2 z 2


K
1 ar
[
( a + r
)
2
+ a 2 − r 2
( a − r
)
2
−
+ z 2 z 2
+ z 2
] .
E
E
( )




,
,
(7)

254 Emil Cazacu, Iosif Vasile Nemoianu 6 where, K and E are the complete elliptic integrals of the first and second kind, respectively.
Then, necessary first and second order spatial derivatives of the flux density may also be computed.
Aiming also to get some quantitative results for both vertical and horizontal structures – Figs. 1 and 2, particular geometrical data and material properties for the models were considered. Thus, for the current loops we assume the distances
2 d = 2 a = 50 mm, and for the direct current carried by them I
0 was chosen a tiny flat NdFeB right prism ( w = 2 mm and l = h = 4 mm) with the magnetic moment M = 0.051 Am 2 and mass m
= 550 A. The floater
= 0.39 g. The two pyrolytic graphite plates (of magnetic susceptibility χ = 450ּ10 –6 ), placed initially at c = 1 mm, symmetrically to the levitated magnet, stabilize the equilibrium.
With the foregoing numerical data graphical representations show the coordinate of the equilibrium point determination and the magnetic flux density variation B ( z ) and B x
( z ) – scaled by a factor of 10 for a better readability– on the symmetry axis, for vertical – Fig. 6, and horizontal configuration – Fig. 7.
Fig. 6 – Equilibrium point and the magnetic flux density variation for vertical configuration.
Fig. 7 – Equilibrium point and the magnetic flux density variation for horizontal configuration.
The vertical levitation configuration gives an equilibrium point located at z
0 c
= 46.13 mm. The maximum value of the gap distance (stability area) is max
= 1.76 mm. This corresponds to a factor C max
= 0.121 J/m 2 a positive value for the vertical discriminant of stability D z
, which still ensures
at the considered levitation point. Thus, the stability area is restricted to the interval (0 –1.76) mm.

7 Vertical and horizontal permanent magnet levitation settings 255
Performing a similar computation, for the horizontal levitation configuration , the equilibrium point is located at z
0
= 31.95 mm. The maximum air gap width is c max
C max
= 1.30 mm, with a corresponding diamagnetic influence factor value
= 0.233 J/m 2 .
Performing a similar computation, for the horizontal levitation configuration , the equilibrium point is located at z
0
= 31.95 mm. The maximum air gap width is c max
C max
= 1.30 mm, with a corresponding diamagnetic influence factor value
= 0.233 J/m 2 .
In order to perform a quantitative analyse of the two levitation configuration, all the significant numerical results are gathered in Table 1.
Table 1
Comparison between vertical and horizontal levitation configuration
Vertical configuration Horizontal configuration x
0 y
0 z
0
C c max
0
0
46.13 mm
0.375 J/m 2
1.76 mm
0
0
31.95 mm
0.375 J/m 2
1.30 mm
C max
0.121 J/m 2 0.233 J/m 2
One can easily notice that the vertical configuration offers a larger stability area for the floater, with almost 36 % greater, compared to the horizontal case.
Furthermore, is important to notice that the equilibrium point is located at different distances away from the symmetry axes in each of the two configurations. Thus, for the vertical setting the equilibrium point coordinate is placed by 44 % greater distance off the symmetry axis, compared to the horizontal setting case. These differences can be easily modified by properly adjusting the intensity of electrical current trough the loops.
4.
CONCLUSIONS
Aspects regarding the diamagnetic stabilization of permanent magnets levitation in stationary fields are treated. The comparative study presents two main levitation settings: vertical and horizontal, in order to highlight each configuration limits and performances.
The analytical and numerical computations relay on the electromechanical interaction between the external magnetic field generated by a source and the

256 Emil Cazacu, Iosif Vasile Nemoianu 8 suspended permanent magnet in a nearby presence of diamagnetic materials, which stabilizes the equilibrium.
The whole analyse is carried out both qualitatively and quantitatively by aiming to accurately predetermine the equilibrium point coordinates and its stability area. This could facilitate an easy design process for a specific configuration, each requiring particular magnetic field sources or levitated magnets.
Received on 8 February 2011
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