ON THE ELECTROMAGNETIC LEVITATION OF
SPHERICAL CONDUCTORS
ll¥
I. R. CllUC
5:38.122: 621.3.013.
Tile author obtaim lh<' approximate wlution of the pro1km of tl'.c elect1omagnetic levilalion of a conduclur in the sha1;e of very thin sphrrical shell, in the
presence of a system of i i!iform circular turns, trough which a.c. curnnt flo\vs
,md whose common axis pasS<'S tl1ruugh the center of the spherical shell. Only
<!Je case when lhc system of [urns is situated outside lhe spherical shell is dealt
with.
An exact solt;tion is also obtained for the same problem for tl;e casr of !he perfectly conducting sphere for which the statical stability upon vcrt:cal direction is likewise examined. The results obtained for this !attn cLse are c01r;pa1ed
with those for the case of the solid sphere of finite conductivity [1 ].
1. INTH.ODUCTIOX
In problems of electromagnetic levitation the forces acting on massive conducting bodies are of a particular interest, forces that appear i:iJs.
a sequence of the interaction between eddy currents induced in such bodies.
and the external magnetic field variable in time, in which the massive
conducting bodies are placed. These forces must be opposite to the gravitation forces, the case in which the resultant of t.he electromagnetic
levitation forces and that of the gravitation forces cancel each other.
Paper [l] has examined, by solving Maxwell's equation, the electromagnetic levitation of a solid conducting sphere in the presence of a
system of co-axial filiform circular turns, with spherical conductor, through
which an a.c. current is flowing. The electromagnetic levitation of a
conductor in the shape of spherical shell has not been examined. It
should be noticed that in [l] the numerical results were obtained for the
case of high frequencies only, the asymptotic expressions of the modified
Bessel functions of complex argument being used.
The :present papr~r examines the problem of the electromagnetic
levitation of a massive conductor in the shape of spherical shell in the
presence of conducting circular turns assumed to be filiform, co-axial with
the conductor, through which sinusoidal a.c. current flows.
Rev. Roum. Sci. Tecnn. - Electrotecnn. et Energ., 14, 1, 21-35, Bucarest, 1969
22
I.
R.
CIRIC
2
Further, the paper examines the electroma.g,netic levitation of a
conductor in the shape of very thin spherical shell and the electromagnetic levitation of a perfectly conducting sphere in the presence of conducting circular turns supposed to be filiform, coaxial with the spherical
conductor, placed outside it, through which sinusoidal a.c. current flows.
This latter problem iR of a particular interest due to the fact that in
electromagnetic levitation systems frequencies of the order of tens and
hundreds of thousands of Hz are used, leading to a very small penetration depth of the electromagnetic field, such that practically only a
very thin layer at the surface of the levitated conductor participates in
the interaction between the induced eddy currents and the excitation
field, case for which the theoretical model of the perfectly conducting
body can be successfully used.
2. PROBLEM OF THE ELECTROl\IAGNETIC LE\'ITATION OF A CONDli(:TOH IN
THE SHAPE OF SPHERICAL SHELL
vYe corn;ider a conducting spherical 8he11 of inner radius a and
external radius b, situated in the magnetic field of some filiform conducting circular turns co-axial with the spherical shell, through which sinusoidal a.c. currents of the same frequency
f = ~
27t
flow, and placed out-
side the spherical shell.
The system of co-ordinate axes is chosen as in Fig. J, such that the
axis z corresponds to the common axis of the circular turns; r, 6 and qi
are the spherical co-ordinates of a point, and the position of one of the
circular turns is given by the co-ordinates r=r, and 6 = 6, of its points.
Let i, = I. V2 sin cut be the electrical current which flows through turns s
in the increasing sense of the co-ordinate qi.
The material of the spherical shell is
assumed linear, homogeneous and isotropic, the
permeability µ and conductibility cs being constant and independent of the field ma.gnitude.
The media in the shell inner cavity and outside
it are non-conducting (cs=O) and non-magnetic,
with permeability practically equal to the
permeability of the vacuum µ 0 •
The general problem of the determination of the electromagnetic field corresponding
to this system has been accurately solved
in [2]. These solutions can be however hardly
used for determining the forces.
3. APPROXli\IATION OF SOJ,UTIO'.\!S IN THE CASE
OF VERY THIN SPHEHICAL CONDI:CTOR
Fig. 1. - Choosing of coordinates and notations.
An approximation that simplifies the
solution without departing too much from ac-
ELECTROMAGNETIC LEVITATION
3
23
tuality is obtained in the case of the very thin spherical conductor by
considering its thickness d = b - a infinitely small and working with the
finite surface conductivity <J, = <J • d for the spherical conductor [2 ].
In such a case the conducting spherical shell will constitute a sheet of induced currents.
Due to the axial symmetry of the problem the density of the current
sheet depends only on the co-ordinate 8 and is given by [2] :
J,(8) = b"" fl:,,
P~
(1)
(costl),
n~l
where P! (cos8) is the associated Legendre function of the first kind,
order n and degree 1, and the coefficients a,. are given by the relation
y2d
~" =
_
~(n,t l_)__ £ ], sin 8, (~)
1 + _!__!!!!!___ s~l
r,
11
P! (cos 8,),
(2)
2n+l
where
(j
a+b
r0 = ---
=
v-i),
~a,..__,
(3)
b,
2
11· = µ°' µ 0 being the vacuum permeability; the underlined symbols standing for the simplified representation in complex of the respective notations ; the summation index s refers to the N filiform conducting circular
turns, having all the same axis which passes through the center of the
conducting spherical shell, all the turns being outside it.
In the case of the spherical shell of very small thickness, assumed
however perfectly conducting, one should consider
<J,
= lim
<Jd
= w,
(4)
ri->0.
Expression (2) of the coefficients a,. becomes now
~ .• =
-
2n
+1 -'b
1 z..· ___I ,sin,
.
8 ( -r o )n P,. (cos 8,).
+ 1) r
r,
2n (n
1
(2')
0 s-1
4. THE FORCE ACTI'.VG OF THE VERY TIIl'.V Sl'ERlCAL CO'.VDUCTOR OF Fl'VITE
CONDUCTIVITY
The force acting on the spherical conductor is determined having
in view that it is equal and of opposite sense with the force acting on the
system of turns through which electrical currents flow, according to the
principle of action and reaction, if we take into account that the spherical
conductor together with the circular turns form in quasi-stationary regime
an isolated mechanical system.
The force acting on the system of filiform conductors through which
the electrical current i flows is determined by applying Laplace's formula
ll.F =iii l X Bext,
(5)
24
I.
R.
CIRIC
where 6.Z is a vector whose modulus is equal to the length of the conductor element and is directed in the current sense, and Bext is the magnetic
induction produced by all the field sources, except the current in the conductor whose force is calculated.
The mean value (over a time period) of this force is
] lT
(6)
6.Fmed = - , 6.F dt =Re [1_*6.lxBextJ·
T ,o
The mean force on the length unit of the turn k is
6.F1~ec;
=
f med = _____
6.~
k
=
-~- Re{I:
rk
Sill
61,
le -
R e [I*ku-¥.
.
grad [rsin6
x
B ext ] =
•
(4 - 4.l]}r~k,\•
(7)
o~~
where u 9 is the unit vector corresponding to the co-ordinate 9, oriented
in the increasing sense of this one, and A is the resulting Yector potential
from which we subtract A., the vector potential of the field produced
by the current in the considered tmn; the expression should be taken
at the points of the considered turn.
The vector potential which depends only on the currents induced
in the conducting spherical shell is determined as in [2] and for r >b"-'r0 is:
A'
-
=
oo
1
( r
µ r ~ - - _J!.
OO.LJ9
1 r
n~1..in
+
·)n+ 1 a
-"
P~(cos
6).
(8)
The vector potential of the circular turn s , through which the electrical current I, flows, is [2]
µ 01,
A, = ___
- r,
2
. nv, ~
Sln
.Li
n~1
1
r"- pln (COS n)
v, pln ( COS n)
v ,
n(n+l) r;q
(9)
where r< and r > are the smallest and the greatest of the magnitudes
r and r,.
Owing to the axial symmetry of the system under examination the
resulting forces acting on each turn and on the shell will be oriented upon
the axis z.
As it may be easily observed, for the calculation of the resulting
force acting on the spherical shell we must sum up the forces which would
act on each turn through which the current flows assuming that the turns
lie in the magnetic field which depends only on the currents induced in
the conducting spherical Rhell, Rince the forces acting on the turns with
electrical current conditioned by the magnetic field of other turns compensate each other, beim:· in1E~n::J forces in the system of the considered
system of turns.
Taking into acco·m' expression (7), for the mean resulting force
acting on the conrluetin:c:· :;r1herical :;;hell, one may use the expres8ion
'£ r
~; COil e __!/.- (r sin64') - 8in6 _!!__ (8in6 A')j,
{lO)i
1.~1 l
iJr
ae
r~r;:
where k is the unit vector co1TeRponding to the axis z.
""
Fmed
=
-k 2r::Re
e~o
ELECTROMAGNETIC
LEVITATION
25
·when performing the calculation account should be taken of the
recurrence relations for the associated Legendre functiom. Considering
the orientation on the negu,tive sense of the axi::; z (i.e. upwards), we get
for the mean resulting force acting on the conducting spherical shell the
following expression
ll ~I.·
ll ~k
..f~·llll\l--= -·])~I*.
'"hl ce l...1 - • Slnvk L.i
·' ::;1n v, L.i _,.
,~.1
k=l
""~!
(~)n'l(!:_i_)"r1(
.,ll)V(
8)
•
I ..
n co.~v, LL,, cos
.t ,.
1.
\ 1,,
(11
where the currents I" and I, are considered directed in the sense of the
tangential vector u'<, while k,. ancl X, (cos 0J have the expressions
2dr
0
k cc-=~
y
" - (2n+l) + y 2 dr 0
X,.(cos ek)== }_ f
(12}
(2 ---1- ) cose. P~(COfl8i}-P~,
I (cos 8Jj.
(13}
n+l
·when there exists a single turn p, through which the electrica,l current
JP flows, in the presence of the conducting spherical shell, the summatiom over k and s are no longer made and we get
n ·'
(14)
where rp and ep are the corresponding spherical co-ordinates of the points
of the circular turn p.
It may be observed that in relations (ll) and (14) the only one
complex magnitude is~. (see (12) ) whose real part is
d 2 r~
1 )2
d2 rg
n++----(
2
(15}
i)4
where i3 is the penetration depth of the electromagnetic field in the
=
V
conducting spherical shell material, i3
_1ix =
w~o cr ·
Further, we assume that the same electrical current flow;;; through
the turns in series
l.1
=
I2
=
...
=IN =I.
(16)
It should be considered however that for the :,;o-called "stabilization" turns which are mounted either above the spherical conductor, or
below it, through which the electrical current flows in opposite direction
with respect to the sense of the electrical current in the so-called "main"
turns which are always mounted under the spherical conductor, the forces act in the sense opposite to the forces due to the "main" turns. In order
to take this into account we introduce a coefficient e:"( e:.) which has the
26
I.
R.
6
CIRIC
value + 1 for each "main" turn and - 1 for each ''stabilization" turn.
Expressions (11) and (14) become
Fmed =
7t[Lol2k~l sine,, s~l E.,e,sine, n~l q. l-:~-r+I( :~
r
P:(cos e,)X. (cos 6,)
(11')
.and
(14')
respectively.
5. FOHCES
ACTI~G O~
THE PERFECTLY COX DUCTING SPHERE
In this case in relations (11') and (14') we replace qn = 1 and r 0 = b
and we get
Fmed
=
7t[Lol2 .tl sine.
tl
Ekes
sine, n~I (-:,,
r+I (~. r p;,
(cos 0,) X,, (cos
o.)
(11")
and
Fmed =
7tµ 0I2 sin 2 0p
~ (__!!_·)?.n+
n==-..01
1
P;, (cosO,,) X" (cos 6p),
(14")
rv
respectively.
00
The infinite series
1: in formulas
(11") and (H") may be summated
n~I
if we observe that the magnetic field induced by the electrical turns with
electrical current and placed in the exterior of a perfectly conducting
sphere, co-axial with the circular turns, is identical with the magnetic
field induced outside the sphere by the given turns through which electrical current flows and by their "imagm;" with respect to the surface
of the given sphere.
By "image" of a filiform circular turn through which the electrical
current iv flows, with respect to the surface of a sphere of radius r=b,
the circular turn being co-axial with the considered sphere and having
the spherical co-ordinates of its points r=rv and 6= 6p, we also understand a filiform circular turn co-axial with the sphere r=b, through which
however the electrical current flows i~ = - i !Lin a sense opposite to the
p
b
sense of the electrical current ip, and having the spherical co-ordinates
of its points
r;
= ...!!:... and
e;
= Op •
rp
The interaction force between two filiform circular turns, having
the same axis and the electrical currents ip and i'J> is given by the for-
ELECTROMAGNETIC LEVITATION
7
--------
--------------------
27
mula [3]
where K and E are complete elliptical integral of the first and second kind,
respectively, having the mode k whose square has the expression
- - - -4bpb~
-----.
2
(bp
z2
1.2 -
/\,
(18)
+b;) +
The interaction force P between these two co-axial turns is oriented
upon their common axis, the turns being attracted if the currents ip and
i; have the same sense and reJected if the two currents have opposite
seme. In expreR:;;ions (17) and (18), µ is the permeability of the medium
in which the two turns are situated, l is the distance between the parallel
planes of the two turns and bp and b; are the radii of the two circular
turns.
Expression (17) may be 8et under the following form, mon: convenient for calculations
P = u.i i'
~_Z __ k [-K(k 2 )
' p p 2(bp b~)'1,
+
2
2
-k
2(1-k 2 )
E(k 2 ) ] .
(17')
In view of the observMiom m::iide above and taking into account
the Rimple geometrical relations between the elements shown in Fig. 1,
we get for relations (Jl") and (14") the following expressiom
µ~I_ 2 ~
F meu _
L.J
2b
2
1.-~1
,, ; .
1 k l..J
~ ~ r,r, cos 8, - b
CJ,,; CS
,
(r,.r,
«oi
Slll
8k
2
e,.
C08 __
•
Sln
OJ''
1•
[-K( 1 .~
fiSk
.
h!S,.J
-l-I
2
+ -9 - k',.
2(1 -
E ( k; )] ,
k;,)
(19)
k
where
4r,r, sin 8k8in e_,
=-(
" '')
.,
b l+-k - ' -2r,r,cos(8,+8J
b2 b2
.
r~
(20)
r~
2
and
Fmed
=
µ 0 l2
2b2
(r! -
b ) cos8v k [-K(k 2 ) -f- 2 - k; E(k 2 )·1.
. 8
v
v
9.(l--k2)
v
Slll p
P.
2
(21)
J
in which
4r; Rin 2 8v - - -
7.~
fup
b2l'1
+ _r!)
b4
- 2r cos 28
2
p
p
(22)
28
I. R.
CIRIC
8
An important conclusion which may be drawn is that for the case
in which the system of inducting turns lies in the interior of a spherical
cavity made in a perfectly conducting medium, the same current flowing
through the turns which have the same position as the above "images",
the mean force acting on the cavity walls and the losses by the Joule
effect in the walls of this cavity, assuming that they have a finite surface conductivity, are the same as those met with in the case in which the
system of inducting turns is mounted in the exterior of the perfectly
conducting sphere which would occupy just the eavity volume.
6. LOSSES IN THE \iERY THIN CONDUCTI'.\G SPHERICAL SHELL
Under the approximation of the very thin spherical conductor
of finite conductivity (see section 3), the losses by the Joule effect may
be directly estimated
-- 2 nr02 p, r· ~
.l..J 2n(n+l) I' an.12] ,
n=l 2n+l
-
where
p,
(p,
is the resistivity
~')
=
and the coefficients
(23)
~.
have the
expression (2). It may be ovserved tJ:at
I i:!n [2
I: [
=
r5
2
1 lfn [ lf
(~)n P! (cos0.)]
2
n+l
2n(n
+ 1)
f
2
RinO,
r,
s=1
2
(24)
It results
P
=
n
I 2 p,
~
2
n+l q. [
n=ln(n+l)
~
sinO,
(~)" P! (cose,)]
S=l
r,
2
,
(25)
in which we have replaced
d 2 r~
i ~. 12
o4
= _(_ _l_)_2_a_2_r~.,--,
n+2
=
q.
(15')
+o4
according to relation (15).
!n the case in which there is a single tnrn through which the
electrical current flows in the presence of the spherical conducting she!l~
expression (25) becomes
P
=
"" -2n+l
r )
nI 2 p, sin 2 0p ~
- - - q. ( ___<!_
n=l n(n + 1)
rp
2
"
[
P!( cos Op) 2].
(26)
In the case of the perfectly conducting sphere one may take into account
for the caleulation of losses a certain finite surface conductivity of the
ELECTROMAGNETIC LEVITATION
9
spherical conductor cr;
= -
1
p,;
,
using
however
2!)
the
distribution
of
electrical current from the ideal case of the perfectly conducting sphere.
By using expression (2') for the coefficient '!:n we get
00
p
=
7tl2p;
~
n=l
2n + 1
n(n+l)
l
N
~
S=l
sin 6, ( - b
r,
)n P!
(cos 6.) ]2
(27)
and for the case in which there is a single turn in the presence of the
perfectly conducting sphere
(28)
7. APPLICATIOXS
7.1. The very thin conducting spherical shell. The sums appearing
in the expression of the mean force acting on the conductor in the shape
of very thin spherical shell (11'), (24') as well as in the expressions of losses
due to the Joule effect in this conductor (25), (26), cannot be carried out
exactly.
The calculations of the mean force and losses by the Joule effect
can be made approximately by retaining a sufficiently great number of
terms in the sums that intervene.
In the expressions of the mean force and of losses by the Joule effect,
1'0
rn
r0
d
2N + 2 dimensionless parameters~ , ... ~ , -,
...
,-.
r1
r,...
b1
bN
a
a
intervene, N being the number of co-axial turns of the inducting
enwraping.
The surface conductivity intervening in the calculation of losses must
be considered
-·
G, = Gd
G,
= cr€l
for
for
d
Cl
< €),
> a.
For the very thin conducting spherical shell d
G,
=
(29)
< a, and accordingly
rrd.
r·
The series intervening in expressions (11 '), (l 4 '), (25) and (26)
have the convergence ensured by the subunitary factors ( ::
which
decrease rapidly with n. The convergence of these series worsen when
the ratio
r,
approaches unity.
30
I.
R.
cmIC
1(}
Calculations have been made for the case of low frequencies, for
2
r5. negligible
. . with re8pect to (1 + 1 ) 2 = 2.2.5, when one can
which d~IS
2
approximate (see expression (15))
d 2 r~
q,.
. -ro
C ons1"d ermg
~ b
= -12
0!.
~
-(--]-)on + -2-"
1
. .
.
an d re t a1n1ng
o., - 6 terms in the sums mter-
"
vening in expressions (14'), we obtain the following results :
30°
Fme<I
1)4
-----
µOJ2
1.887~10- 3
45°
60°
33.85x10- 3
48. 75.00- 3
d2ro2
For example, for a copper spherical shell having r 0 =~ J:_ cm, d =b-a=
2
=
J:_
2 mm, p
= 2.10-s
n cm in the presence
of a single
. tnrn with bp
= 1
cmt
the angle 6,, being of 60°, we get in the MKSA unit system
d2
0
r(j
?)4
F me<l
=
=
0.2435:oo- 6j2,
1. 493xl0- 14 J2 1 2 •
7.2. The perfectly conducting sphere. We observe that for the body
made of perfectly conducting material, the expressions obtained for the
distribution of the induced current, for the mean force acting on the body
and for the losses by the Joule effect in the body in which a finite surface
conductivity has been assumed, are the same both for the solid sphere
and for the spherical shell.
It may be also observed that the results obtained in the case of
the perfectly conducting sphere may be also used for the sphere of finite
conductivity at high frequencies, when the penetration depth is very small
with respect to the sphere radius and the distribution of the induced current
at the periphery of the conducting sphere is very close to that from the
case of the perfectly conducting sphere. Accordingly, at very high frequencies one may use for the calculation of losses by the Joule effect in
the sphere of finite conductivity the expressions (27) and (28) for the case
of perfectly conducting sphere, in which the surface conductivity is taken
"'
cr,I = 1- = cr6.
p;
For the calculation of the mean force acting on the perfectly conducting sphere we use formulas (19) and (21).
31
ELECTROMAGNETIC LEVITATION
11
In expressions (19), (21), (27) and (28) 2 N dimensionless parameters
intervene..!_, ... ..!_, ..!_, ... ..!_ , N being the number of co-axial turns
r1
b1
rN
bN
of the inducting enwraping. A useful parameter for drawing the graphs
and for comparing the results with those obtained in [1] is
h
(ri - bi)112
=
= cotg el,
bl
(30)
ru el and bl = 1\ sin el refer to the lowest turn of the turn system considered (see Fig. 2).
Calculations have been carried out for two systems of inducting
..
'h t h e rat10
. - b = -1 an d t h e other
turns. One consists
ma.
smg1e turn wit
bp
2
consists of three co-axial turm placed on the surface of a cone with the
vertex downwards, the angle between the generatrix and the axis being (3
(see Fig. 2), for the following particular cases ..!_ = _! , tg{3 = 0; 0.2;
2
bl
0.4 and 0.8.
In the case of the system with a single turn we have used formula (21)written under the form
in which h
=
b2)1 '2
( 2
rp
-
p
bp
=
cotg
op'
in compliance with relation (30)t"
as well as expression (22) written under the form
9
k;
4
~ ( ~ )\1+h'J' + ( :J-2(h'-1)
(32}
The results obtained are given in Fig. 3.
In the case of the system of three co-axial turns, placed as shown in
Fig. 2, we have used formula (19) written under the form
Fmeu =
µ
012
2
t t (!!J__) ({3z+h~)
2
k~l s~l
b
h •
r-
( ..!_
hk
· b1
~~
( {3k {3,)
1 + M k,.[-K(k;k)+
2 -k;k- E(k;k) ] ,
+ --
(33)
k;k)
2(1 -
where
{3k = 1
k-1
+- tg{3,
'
4
hk
=
h -
k-1
4
'
'k = 1,2,3'
(34}
32
I.
R.
12
CIRIC
in compliance with relation (30), as well as expression (20) written under
the form
(:J
4~k~s
2
[
1
1
+ (~
r(~~+M) (~;+h;)]-2(hkh,-~.~,).
(35)
The results are given in Figs. 3 and 4.
,....-L --·- - __...
__,.,
_.__ --____
- - - -r--___
Fmed
I
µ012
W
Fmed
(a single
turn)
(
µo/2-
(three turns)
O.~l----J;,.,_~-~-~--1
1.2
I
I
Fig. 2. - System of three inducling turns
(dashes show the possible position for the
stabilization turn).
Fig. 3. - Dependence of the magnitude
Fmed
- - on the parameter h :
µ0/2
- - - a sigle turn, - - - - three
1
b
=
bp
b
-
;
2
1
turns = - tg ~=0.4.
' b1
2'
fmedr--.-~~~-~~
JlofZ
lr-----1-~-.t---t--+---1
· de F med
. 4 . - Dcpen d ence o f t h e magmtu
- on th e
F 1g.
µ0/2
parameter h for the three turn syslem: -
1
b
=
-
:
2
(/) tg ~ = O; (II) tg ~ = 0.2; (III) tg ~ = 0.4;
(IV) tg ~ = 0.8.
bl
0.11 -
c.z _l_--+-~~.......-1
I
The height at which the conducting sphere is to be placed in order
that the electromagnetic levitation force be maximum corresponds to the
values h for which the maximum of the functions plotted in Figs. 3 and
4 is obtained.
ELECTROMAGNETIC LEVITATION
13
33
The minimum current for levitation is obtained by equalizing the
mean force acting on the sphere to its weight, when the sphere is at the
height corresponding to the electromagnetic levitation maximum force.
For instance, for a copper sphere having the radius b
=
_.!._ cm
2
and the weight G = 0.0457 N in the presence of a single turn of radius
bp = 1 cm, we obtain from Fig. 3 a current of I = 913 .A necessary for
levitating the sphere at 1 cm above the turn (h=l).
For the same sphere, in the presence of a system of three inducting
turns as shown in Fig. 2 (without "stabilization" turns) with b1 = 1 cm,
tg [3 = 0.4, we obtain from Fig. 3 a current of only I = 255 .A for levitating the sphere at the same height of only 1 cm above the turn (h=l).
7.3. Static stability. The static stability of the considered systems is
achieved below only for the case of the perfectly conducting sphere and
only upon vertical direction.
From Figs. 3 and 4 one may observe that the space above the
system of inducting turns is divided into two by the value h = hm corresponding to the maximum value of the function F'med • For h < hm , the
µJ2
levitation is statically unstable since a small displacement of the sphere
upon the vertical direction leads to a further displacement of the sphere
in the samA sense. For h > hm the statically stable levitation is possible
since after any small displacement from the equilibrium position upon
vertical direction, the sphere tends to be restored to equilibrium.
The lateral stability (in the horizontal plane) is practically obtained
by conveniently disposing the "main" turns, e.g., such that they be placed on a cone with downwards directed vertex (see Fig. 2) and by using
some "stabilization" turns mounted usually above the conducting sphere,
through which the electrical current flows in the sense opposite to that
in the "main" turns.
8. COXCLUSIONS
In this paper, we have obtained the approximate solution for the
problem of the electromagnetic levitation of a conductor in the shape
of very thin spherical shell in the presence of a system of co-axial circular
turns, which are placed in the exterior of the conducting spherical shell
and through which sinusoidal a.c. currents of the same intensity and
frequency flow.
Likewise, we have obtained the exact solution of the same problem
for the case of the perfectly conducting sphere.
The turns of the inducting enwraping have been assumed filiform.
The results obtained are practically valid provided the cross-section radius
of the inducting turn conductor be negligible with respect to the radius
of these turns and with the radius of the levitated conducting sphere.
From Fig. 4 one may observe the effect of the angle [3 at the vertex
of the cone surface on which the system of the three inducting turns is
3-c. ll M
34
I.
R.
CIRIC
14
mounted, the dimensions being indicated in Fig. 2 ; with the increase of
the angle
~
the ratio 1J'med decreases. The maximum electromagnetic
fLol2
levitation force is obtained if the inducting turns are enwraped on the
surface of a cone with upward directed vertex of a certain angle. However,
for technical reasons related to the levitation of a liquid state conductor
as well as to the stability of the respective systems, it is required that the
inducting turns be mounted on the surface of a cone with downward
directed vertex, of a certain angle.
From formulae (11'), (14'), (25), (26), as well as from the application
7.1, it results that the mean force and the losses due to the Joule effect
on spherical conductors for sufficiently low frequencies are practically
proportional with the frequency square. This result is similar to that
obtained in paper [6] for the case of cylindrical conductors.
For sufficiently high frequencies, as it results from paper [1 ], the
mean force acting on the induced conductors is practically independent
of frequencies, and the losses by the Joule effect in the same conductors
are practically proportional to the frequency square root.
In all cases the mean force and the losses by the Joule effect are
proportional to the square of the intensity of electrical current in the
inducting enwraping.
If the numerical results obtained for the perfectly conducting sphere
are compared with those obtained in [1], when the solid sphere of finite
conductivity is considered, we observe that they are practically similar,
no differences being noticed at frequencies of the order of hundreds o
thousands Hz, for which the calculations are performed in [1 ].
Consequently, for frequencies (of the order of hundreds of thousands
of Hz) which intervene in problems of electromagnetic levitation, the
electromagnetic levitation force, the electrical current, necessary for
achieving the electromagnetic levitation, the height reached by the conducting sphere in electromagnetic levitation as well as the regions in which
the stable static equilibrium is possible may be calculated with high
accuracy by using the theoretical model of the perfectly conducting sphere.
Likewise, the losess by the Joule effect in the sphere of finite conductivity are practically the same in the frequency domain indicated above
as the losses corresponding to the perfectly conducting sphere assumed
to have a finite surface conductivity.
The main conclusion which may be drawn from this paper, as well
as from paper [6], is that for solving problems of electromagnetic levitation one may use successfully, for massive bodies of finite conductivity,
the theoretical model of the same bodies having however perfect conductivity.
Received July 3, 1968
Institute of Power Engineering,
Academy of the Socialist Republic of
Romania
ELECTROMAGNETIC LEVITATION
15
35
REFERENCES
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A. TuGULEA. I. R. Crn1c, Cimpul electromagnetic al unor spire parcurse de curent alternativ
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\V. R. SMYTHE, Static and dyr.amic electricity. New York, Toronto, London, 1950.
,.
Tables of Associated Legendre Functions. Columbia University Press, New York,
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E. JANKE, F. EMDE, F. LoscH, Tafeln hOherer Funktionen. B.G. Tcubner Verlagsgesellschaft,
Stuttgart, 1960.
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1. \V.
2.
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