About stability of levitating states of superconducting myxini of plasma traps-Galateas
To develop a plasma trap with levitating superconducting magnetic coils it is necessary to carry out the search of their stable levitating states. With this purpose, based upon the superconductor property to conserve the trapped magnetic flux, in the uniform gravitational field the analytical depend...
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Moscow State Technical University of Radio Engineering, Electronics and Automation, Moscow, Russia
2015-12-10
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| Cite this: | About stability of levitating states of superconducting myxini of plasma traps-Galateas/ A.M. Bishaev, A.A. Bush, M.B. Gavrikov, A.I. Denis’uk, O.Y. D’yakonitsa, K.Y. Kamentsev, M.V. Kozintseva, T.G. Kolesnikova, V.V. Savelyev, P.G. Smirnov, M.M. Shapovalov, S.A. Voronchenko // Вопросы атомной науки и техники. — 2015. — № 1. — С. 16-19. — Бібліогр.: 6 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859649898085875712 |
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| author | Bishaev, A.M. Bush, A.A. Gavrikov, M.B. Denis’uk, A.I. D’yakonitsa, O.Y. Kamentsev, K.Y. Kozintseva, M.V. Kolesnikova, T.G. Savelyev, V.V. Smirnov, P.G. Shapovalov, M.M. Voronchenko, S.A. |
| author_facet | Bishaev, A.M. Bush, A.A. Gavrikov, M.B. Denis’uk, A.I. D’yakonitsa, O.Y. Kamentsev, K.Y. Kozintseva, M.V. Kolesnikova, T.G. Savelyev, V.V. Smirnov, P.G. Shapovalov, M.M. Voronchenko, S.A. |
| citation_txt | About stability of levitating states of superconducting myxini of plasma traps-Galateas/ A.M. Bishaev, A.A. Bush, M.B. Gavrikov, A.I. Denis’uk, O.Y. D’yakonitsa, K.Y. Kamentsev, M.V. Kozintseva, T.G. Kolesnikova, V.V. Savelyev, P.G. Smirnov, M.M. Shapovalov, S.A. Voronchenko // Вопросы атомной науки и техники. — 2015. — № 1. — С. 16-19. — Бібліогр.: 6 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | To develop a plasma trap with levitating superconducting magnetic coils it is necessary to carry out the search of their stable levitating states. With this purpose, based upon the superconductor property to conserve the trapped magnetic flux, in the uniform gravitational field the analytical dependence of the potential energy of one or two superconducting rings, having trapped the given magnetic fluxes, in the field of the fixed ring with the constant current from the coordinates of the free rings and the deflection angle of their axes from the common axis of the magnetic system has been obtained in the thin ring approximation. Under magnetic fluxes of the same polarity in coils the existence of the found from the calculations equilibrium levitating states for the manufactured HTSC rings stable relative to the vertical shifts of levitating rings and to the deflection angle of their axes from the vertical has been confirmed experimentally.
Для разработки плазменной ловушки с левитирующими сверхпроводящими магнитными катушками нужно выполнить поиск их устойчивых левитирующих состояний. С этой целью, исходя из свойства сверхпроводников сохранять захваченный магнитный поток, в однородном поле силы тяжести в приближении тонких колец получена аналитическая зависимость потенциальной энергии одного либо двух сверхпроводящих колец, захвативших заданные магнитные потоки, в поле закрепленного кольца с постоянным током от координат свободных колец и углов отклонения их осей от общей оси системы. При совпадающих по знаку потоках в кольцах существование найденных из расчетов равновесных левитирующих состояний для изготовленных ВТСП колец, устойчивых по отношению к вертикальным смещениям левитирующих колец и к отклонению их осей от вертикали, было подтверждено экспериментально.
Для розробки плазмової пастки з левітуючими надпровідними магнітними котушками потрібно виконати пошук їх стійких левітуючих станів. З цією метою, виходячи з властивості надпровідників зберігати захоплений магнітний потік, в однорідному полі сили тяжіння в наближенні тонких кілець отримана аналітична залежність потенційної енергії одного або двох надпровідних кілець, які захопили задані магнітні потоки, у полі закріпленого кільця з постійним струмом від координат вільних кілець і кутів відхилення їх осей від загальної осі системи. При співпадаючих за знаком потоках у кільцях існування знайдених з розрахунків рівноважних левітуючих станів для виготовлених ВТНП кілець, стійких по відношенню до вертикальних зміщень левітуючих кілець і до відхилення їх осей від вертикалі, було підтверджено експериментально.
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| first_indexed | 2025-12-07T13:32:32Z |
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ISSN 1562-6016. ВАНТ. 2015. №1(95)
16 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2015, № 1. Series: Plasma Physics (21), p. 16-19.
ABOUT STABILITY OF LEVITATING STATES
OF SUPERCONDUCTING MYXINI OF PLASMA TRAPS-GALATEAS
A.M. Bishaev
1
, A.A. Bush
1
, M.B. Gavrikov
2
, A.I. Denis’uk
1
, O.Y. D’yakonitsa
1
,
K.Y. Kamentsev
1
, M.V. Kozintseva
1
, T.G. Kolesnikova
1
, V.V. Savelyev
2,3
, P.G. Smirnov
4
,
M.M. Shapovalov
1
, S.A. Voronchenko
1
1
Moscow State Technical University of Radio Engineering, Electronics and Automation,
Moscow, Russia;
2
Keldysh Institute of Applied Mathematics RAS, Moscow, Russia;
3
National Research Nuclear University MEPh, Moscow, Russia;
4
Moscow Aviation Institute (National Research University), Moscow, Russia
E-mail: kozintseva@mirea.ru
To develop a plasma trap with levitating superconducting magnetic coils it is necessary to carry out the search of
their stable levitating states. With this purpose, based upon the superconductor property to conserve the trapped
magnetic flux, in the uniform gravitational field the analytical dependence of the potential energy of one or two
superconducting rings, having trapped the given magnetic fluxes, in the field of the fixed ring with the constant
current from the coordinates of the free rings and the deflection angle of their axes from the common axis of the
magnetic system has been obtained in the thin ring approximation. Under magnetic fluxes of the same polarity in
coils the existence of the found from the calculations equilibrium levitating states for the manufactured HTSC rings
stable relative to the vertical shifts of levitating rings and to the deflection angle of their axes from the vertical has
been confirmed experimentally.
PACS: 84.71.-b, 84.71.Ba, 52.55.–s, 52.55.He
INTRODUCTION
Magnetic systems of traps-Galateas consist of
several coaxial coils [1]. They form the typical for this
type of traps configuration of the magnetic field
containing the region of zero field. In the fusion reactor
the immersed in the plasma magnetic coils (“myxini”
[1]) of such traps must levitate. The equilibrium
levitating states of superconducting myxini must have
the stability of three types: 1) relative to the vertical
shifts of their plane along the common axis; 2) relative
to the deflection angle of their axis from the common
axis of the magnetic system; 3) relative to the radial
shifts of their plane. The developed by the authors
approach is in obtaining the analytical dependence of
the potential energy of the proposed configurations from
the corresponding variables and in the search by the
calculations in Mathcad of the local minimums of such
dependence corresponding to the stable equilibrium
states [2-5].
For experiments with levitation several multiturn
short-circuited rings with different diameters (50, 60
and 80 mm) have been made from a high-temperature
superconducting (HTSC) wire of the SCS4050-i-AP 2G
HTS type (manufactured at SUPER POWER). Their
time constants measured from the curve describing the
decay of the magnetic field with the time lie in the
interval 18…35 min.
The flux trapping by HTSC rings has been carried
out by their cooling in the solenoid magnetic field to the
liquid nitrogen temperature. Then the field has been
switched off, and the coil-ring having trapped the
magnetic flux has been moved into the experimental cell
for the measuring of the magnetic induction. The
experimentally determined maximal values of trapped
fluxes are at the level 10
-3
Wb. The nonsuperconducting
coil with the constant current had the following
parameters: the middle radius was equal to 4.75 cm, the
average radius of the winding cross-section 0.6 cm,
and the number of turns 400.
1. STATEMENT OF PROBLEM
The given paper is devoted to the search of the
equilibrium states of superconducting rings which
levitate in the field of the fixed nonsuperconducting ring
with the current, stable both to the their shift along the
common axis and to the deflection of their axis from the
system common axis.
The system under consideration consists of two or
three coaxial rings lying in parallel horizontal planes.
One of the rings positions scheme is presented in the
Fig. 1. The coordinates are measured from the fixed
ring. The rings are numbered from top to bottom. It was
assumed that the radius (ak) of the cross-section of each
ring is much smaller than the middle radius (Rk) of the
corresponding ring.
The derivation of the expression for the potential
energy of the system of rings as the function of
coordinates of free rings and the deflection angle of
their axis from the common axis of magnetic system in
the analytical form is analogous to the derivation given
in [5] with the only difference that the coefficients of
the rings mutual induction Lik(x, ) are the functions not
only of the coordinate x of the levitating ring but also of
the deflection angle of its axis from the vertical.
Due to the increase of the number of variables some
particular cases have been considered.
mailto:kozintseva@mirea.ru
ISSN 1562-6016. ВАНТ. 2015. №1(95) 17
2. LEVITATION OF MYXINE IN FIELD
OF FIXED RING WITH CURRENT
Let the top fixed ring (№1) is not superconducting
and has the constant current
1J , the bottom one (№2) is
superconducting and free, the trapped by it magnetic
flux
2
conserving. Then we may write:
1
21 1 22 2 2
12 21
;
;
.
J const
L J L J const
L L
(1)
Based upon the expression for the interaction force
between the contours with current [6] in this case:
q
L
JJ
q
U
Fq
12
21 (2)
(where q is generalized coordinate, differentiation is
carried out under constant currents) and taking into
account (1), for the potential energy of this system in
the uniform gravitational field we obtain:
gxm
L
JxL
xU 2
22
2
12122
2
)),((
2
1
),( . (3)
With the help of calculations in Mathcad system of
the dependence U(x, 2) the search of such values of the
magnetic fluxes, trapped by the ring 50 mm from
HTSC wire, and the current value in the fixed ordinal
coil has been carried out, under which the equilibrium
states of the free ring in the field of the fixed ring, stable
with respect to the shift of the levitating ring plane
along the common axis and to the deflection of its axis
from the vertical, exist.
The existence of the determined by calculations
equilibrium state for HTSC rings 50 mm under fluxes
in the rings of the same sign (Fig. 2,a) has been proved
experimentally (see Fig. 2,b).
If the bottom ring is fixed and it is not
superconducting and has the constant current
2J , the top
one is superconducting and free, having trapped
magnetic flux 1 , then the potential energy of this
system in the uniform gravitational field may be written
in form:
gxm
L
JxL
xU 1
11
2
21121
1
)),((
2
1
),( . (4)
The calculations in Mathcad system of the
distribution U(x, 1) have shown that for the fluxes in the
rings of the opposite polarity the equilibrium states of
the superconducting ring in the field of the ring, fixed
below, are stable with respect to the shift of the
levitating ring plane along the common axis and
unstable to the deflection of its axis from the vertical.
As an example, the equipotentials distributions and
photo of the corresponding to it levitating state for the
ring from HTSC wire 60 mm is shown in the Fig. 3.
The calculations in Mathcad system of the
distribution U(x, 1) have shown that for the fluxes in the
rings of the same sign the equilibrium states of the
superconducting ring in the field of the ring, fixed
below, which are stable both with respect to the shift of
the levitating ring plane along the common axis and to
the deflection of its axis from the vertical, exist. The
example of the calculation illustrating the existence of
minimums for such systems under the definite values of
parameters is given in the Fig. 4.
Fig. 1. Rings positions: №1 – fixed ring with the constant
current; №2 and 3 – levitating superconducting rings
a b
Fig. 2. a) Equipotentials distribution of the system of
two rings on the coordinate x of the free
superconducting ring and the deflection angle 2 of its
axis from the vertical (system common axis) in the
uniform gravitational field: R1=4.75 cm, R2=2.5 cm,
a1=0.6 cm, a2=0.26 cm, J1=100 A, Ф2=2 10
-4
Wb,
m2=15.3 g; b) the photo of the stable in x and 2
levitating state of HTSC ring in the field of the
supporting coil with the constant current, corresponding
to the calculated distribution in Fig. 2,a
a b
Fig. 3. a) Equipotentials distribution of the system of
two rings on the coordinate x of the free
superconducting ring and the deflection angle 1 of its
axis from the vertical (system common axis) in the
uniform gravitational field: R2=4.75 cm, R1=3.0 cm,
a2=0.6 cm, a1=0.35 cm, J2=200 A, Ф1=-1.8 10
-4
Wb,
m1=32.83 g; b) the photo of the unstable levitating state
of HTSC ring in the field of the supporting coil with the
constant current (it is fixed on the bottom of container),
corresponding to the calculated distribution in Fig. 3,a
18 ISSN 1562-6016. ВАНТ. 2015. №1(95)
3. LEVITATION OF TWO MYXINI IN FIELD
OF FIXED RING WITH CURRENT
For the system of three rings, shown in the Fig. 1,
besides the deflection angles 2 and 3 of the levitating
rings № 2 and № 3 from the vertical, it is necessary to
introduce into the consideration one more variable – the
angle . It is the angle between the projections of the
normals
2n
and
3n
to the planes of the rings №2 and №
3 onto the horizontal plane.
Assuming that the magnetic fluxes trapped by the
superconducting rings (№2 and 3) remain unchanged,
and the current in the nonsuperconducting ring (№1) is
maintained constant, by analogy with [5]:
1
21 1 22 2 23 3 2
31 1 32 2 33 3 3
; ;
,
.
ik kiJ const L L
L J L J L J const
L J L J L J const
(5)
Now the potential function, formally coinciding with
the expression in [5], is the function both the
coordinates of the levitating coils x2 and x3, and the
angles 2, 3, :
2 2
2 3 2 3 22 2 23 2 3 33 3 2 2 3 3
1
( , , , , ) 2 .
2
U x x L J L J J L J m gx m gx (6)
In formulae (1)-(6)
i
the magnetic flux trapped by ith
ring;
ikL coefficients of the self- and mutual induction
of the rings; Jk – currents in the rings; mi – mass of ith
ring.
In order to carry out the calculations of the potential
energy distributions the formulae for the coefficients of
the rings mutual induction ),(12 xL for the system of
two rings and ),,,,(
kikiik
xxL for the system of three
rings, correspondently, have been deduced (formulae
are not represented because they are too cumbersome
ones).
The search of the stable levitating states of two
myxini in the field of the fixed ring with the current is
carried out step by step. The example of this search for
the system of HTSC ring with 31 mm [4] and the ring
from HTSC wire 60 mm in the field of the supporting
coil with the constant current for the fluxes of the same
polarity is represented in Fig. 5. Calculations have been
carried out under the following values of parameters:
R1=4.75 cm, R2=3.0 cm, R3=1.15 cm, a1=0.6 cm,
a2=0.35 cm, a3=0.4 cm, J1=1 10
2
A, 2=7 10
-4
Wb;
Ф3=3.18 10
-5
Wb, m2=32.83 g, m3=9.7 g. At first,
having specified the values of the physical parameters,
the field of values of the variables x2 and x3 is defined,
where the equilibrium levitating states of
superconducting rings, stable relative to the vertical
shifts of their plane along the common axis, are realized
(see Fig. 5,a). Then this field of parameters is tested for
the stability in 2, 3 and respectively (see Figs. 5,b,
c, d). The photo of the stable in x and levitating state
of the system of HTSC ring with 31 mm and the ring
from HTSC wire 60 mm in the field of the supporting
coil with the constant current, corresponding to the
calculated distribution in Fig. 5,a, is presented in
Fig. 5,e.
Fig. 4. Equipotentials distribution of the system of two
rings on the coordinate x of the free superconducting
ring and the deflection angle 1 of its axis from the
vertical (system common axis) in the uniform
gravitational field: R1=20 cm, R2=25 cm, a1=1.5 cm,
a2=0.6 cm, Ф1=5 10
-3
Wb, J2=5 10
4
A, m1=2 10
4
g
a b
c d
e
Fig. 5. a) the determination of the field of the stability
on coordinates x2 and x3; b) the testing of the defined
field of stability in coordinates x2 and x3 for the stability
relative to the deflections of the axis of the top levitating
ring by the angle 2; c) the testing of the defined field of
stability in coordinates x2 and x3 for the stability relative
to the deflections of the axis of the bottom levitating ring
by the angle 3; d) the testing of the field of the stability
in x2, x3, 2, 3 for the stability relative to the angle ;
e) the photo of the stable in x and levitating state of
the system, corresponding to the calculated distribution
in Fig. 5,a
ISSN 1562-6016. ВАНТ. 2015. №1(95) 19
CONCLUSIONS
Realized earlier experiments in levitation of HTSC
rings in the field of the fixed rings with the constant
current under the same sign of the fluxes in rings [5]
have shown that levitating states both one and two rings
in the field of ordinary coil with the constant current,
founded by the dependence U(x) of the potential energy
only from the coordinate, were stable not only to the
shift of their plane along the vertical (along the common
axis), but also to the deflection angle of their axis
from the vertical. The analytical dependencies U(x, )
confirm this fact and allow due to their generality to
determine the field of stability in x and under any
values of the physical parameters.
This study was supported by the Ministry of
Education and Science of the Russian Federation and in
part by the Russian Foundation for Basic Research
(project nos. 13-08-00717 and 12-01-00071).
REFERENCES
1. A.I. Morozov, V.V. Savel’ev. On Galateas –
magnetic traps with plasma-embedded
conductors // Physics-Uspekhi. 1998, v. 41, № 11,
p. 1049-1089.
2. A.M. Bishaev, A.A. Bush, et al. The Potential Energy
of a Superconducting Ring System Locking Magnetic
Flows in a Gravity Field // Technical Physics Letters.
2012, v. 38, № 10, p. 880-883.
3. A.M. Bishaev, M.V. Kozintseva, et al. About
Levitation of Superconducting Rings for Magnetic
System of Multipole Plasma Trap // Problems of Atomic
Sciences and Technology. Series “Plasma Physics”
(19). 2013, № 1, p. 48-50.
4. A.M. Bishaev, M.V. Kozintseva, et al. Equilibrium of
a System of Superconducting Rings in a Uniform
Gravitational Field // Technical Physics. 2013, v. 58,
№ 5, p. 684-691.
5. A.M. Bishaev, M.V. Kozintseva, et al. Levitating
States of Superconducting Rings in the Field of a Fixed
Ring with Constant Current // Technical Physics. 2014,
v. 59, № 6, p. 940-943.
6. L.D. Landau and E.M. Lifshitz. Course of Theoretical
Physics. Vol. 8: Electrodynamics of Continuous Media.
Pergamon, New York, 1984.
Article received 10.12.2014
ОБ УСТОЙЧИВОСТИ ЛЕВИТИРУЮЩИХ СОСТОЯНИЙ СВЕРХПРОВОДЯЩИХ
МИКСИН ПЛАЗМЕННЫХ ЛОВУШЕК-ГАЛАТЕЙ
А.М.
Бишаев, А.А.
Буш, М.Б.
Гавриков, А.И. Денисюк, О.Ю. Дьяконица, К.Е.
Каменцев,
М.В. Козинцева, Т.Г. Колесникова, В.В. Савельев, П.Г. Смирнов, М.М. Шаповалов, С.А. Воронченко
Для разработки плазменной ловушки с левитирующими сверхпроводящими магнитными катушками
нужно выполнить поиск их устойчивых левитирующих состояний. С этой целью, исходя из свойства
сверхпроводников сохранять захваченный магнитный поток, в однородном поле силы тяжести в
приближении тонких колец получена аналитическая зависимость потенциальной энергии одного либо двух
сверхпроводящих колец, захвативших заданные магнитные потоки, в поле закрепленного кольца с
постоянным током от координат свободных колец и углов отклонения их осей от общей оси системы. При
совпадающих по знаку потоках в кольцах существование найденных из расчетов равновесных
левитирующих состояний для изготовленных ВТСП колец, устойчивых по отношению к вертикальным
смещениям левитирующих колец и к отклонению их осей от вертикали, было подтверждено
экспериментально.
ПРО СТІЙКIСТЬ ЛЕВІТУЮЧИХ СТАНІВ НАДПРОВІДНИХ
МІКСИН ПЛАЗМОВИХ ПАСТОК-ГАЛАТЕЙ
А.М. Бішаєв, А.А. Буш, М.Б. Гавріков А.І. Денісюк, О.Ю. Дьяконіца, К.Є. Камєнцев,
М.В. Козінцева, Т.Г. Колеснікова, В.В. Савєльєв, П.Г. Смірнов, М.М. Шаповалов, С.А. Воронченко
Для розробки плазмової пастки з левітуючими надпровідними магнітними котушками потрібно виконати
пошук їх стійких левітуючих станів. З цією метою, виходячи з властивості надпровідників зберігати
захоплений магнітний потік, в однорідному полі сили тяжіння в наближенні тонких кілець отримана
аналітична залежність потенційної енергії одного або двох надпровідних кілець, які захопили задані
магнітні потоки, у полі закріпленого кільця з постійним струмом від координат вільних кілець і кутів
відхилення їх осей від загальної осі системи. При співпадаючих за знаком потоках у кільцях існування
знайдених з розрахунків рівноважних левітуючих станів для виготовлених ВТНП кілець, стійких по
відношенню до вертикальних зміщень левітуючих кілець і до відхилення їх осей від вертикалі, було
підтверджено експериментально.
|
| id | nasplib_isofts_kiev_ua-123456789-82103 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T13:32:32Z |
| publishDate | 2015-12-10 |
| publisher | Moscow State Technical University of Radio Engineering, Electronics and Automation, Moscow, Russia |
| record_format | dspace |
| spelling | Bishaev, A.M. Bush, A.A. Gavrikov, M.B. Denis’uk, A.I. D’yakonitsa, O.Y. Kamentsev, K.Y. Kozintseva, M.V. Kolesnikova, T.G. Savelyev, V.V. Smirnov, P.G. Shapovalov, M.M. Voronchenko, S.A. 2015-05-25T06:28:20Z 2015-05-25T06:28:20Z 2015-12-10 About stability of levitating states of superconducting myxini of plasma traps-Galateas/ A.M. Bishaev, A.A. Bush, M.B. Gavrikov, A.I. Denis’uk, O.Y. D’yakonitsa, K.Y. Kamentsev, M.V. Kozintseva, T.G. Kolesnikova, V.V. Savelyev, P.G. Smirnov, M.M. Shapovalov, S.A. Voronchenko // Вопросы атомной науки и техники. — 2015. — № 1. — С. 16-19. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 84.71.-b, 84.71.Ba, 52.55.–s, 52.55.He https://nasplib.isofts.kiev.ua/handle/123456789/82103 To develop a plasma trap with levitating superconducting magnetic coils it is necessary to carry out the search of their stable levitating states. With this purpose, based upon the superconductor property to conserve the trapped magnetic flux, in the uniform gravitational field the analytical dependence of the potential energy of one or two superconducting rings, having trapped the given magnetic fluxes, in the field of the fixed ring with the constant current from the coordinates of the free rings and the deflection angle of their axes from the common axis of the magnetic system has been obtained in the thin ring approximation. Under magnetic fluxes of the same polarity in coils the existence of the found from the calculations equilibrium levitating states for the manufactured HTSC rings stable relative to the vertical shifts of levitating rings and to the deflection angle of their axes from the vertical has been confirmed experimentally. Для разработки плазменной ловушки с левитирующими сверхпроводящими магнитными катушками нужно выполнить поиск их устойчивых левитирующих состояний. С этой целью, исходя из свойства сверхпроводников сохранять захваченный магнитный поток, в однородном поле силы тяжести в приближении тонких колец получена аналитическая зависимость потенциальной энергии одного либо двух сверхпроводящих колец, захвативших заданные магнитные потоки, в поле закрепленного кольца с постоянным током от координат свободных колец и углов отклонения их осей от общей оси системы. При совпадающих по знаку потоках в кольцах существование найденных из расчетов равновесных левитирующих состояний для изготовленных ВТСП колец, устойчивых по отношению к вертикальным смещениям левитирующих колец и к отклонению их осей от вертикали, было подтверждено экспериментально. Для розробки плазмової пастки з левітуючими надпровідними магнітними котушками потрібно виконати пошук їх стійких левітуючих станів. З цією метою, виходячи з властивості надпровідників зберігати захоплений магнітний потік, в однорідному полі сили тяжіння в наближенні тонких кілець отримана аналітична залежність потенційної енергії одного або двох надпровідних кілець, які захопили задані магнітні потоки, у полі закріпленого кільця з постійним струмом від координат вільних кілець і кутів відхилення їх осей від загальної осі системи. При співпадаючих за знаком потоках у кільцях існування знайдених з розрахунків рівноважних левітуючих станів для виготовлених ВТНП кілець, стійких по відношенню до вертикальних зміщень левітуючих кілець і до відхилення їх осей від вертикалі, було підтверджено експериментально. This study was supported by the Ministry of Education and Science of the Russian Federation and in part by the Russian Foundation for Basic Research (project nos. 13-08-00717 and 12-01-00071). en Moscow State Technical University of Radio Engineering, Electronics and Automation, Moscow, Russia Вопросы атомной науки и техники Магнитное удержание About stability of levitating states of superconducting myxini of plasma traps-Galateas Об устойчивости левитирующих состояний сверхпроводящих миксин плазменных ловушек-Галатей Про стійкiсть левітуючих станів надпровідних міксин плазмових пасток-Галатей Article published earlier |
| spellingShingle | About stability of levitating states of superconducting myxini of plasma traps-Galateas Bishaev, A.M. Bush, A.A. Gavrikov, M.B. Denis’uk, A.I. D’yakonitsa, O.Y. Kamentsev, K.Y. Kozintseva, M.V. Kolesnikova, T.G. Savelyev, V.V. Smirnov, P.G. Shapovalov, M.M. Voronchenko, S.A. Магнитное удержание |
| title | About stability of levitating states of superconducting myxini of plasma traps-Galateas |
| title_alt | Об устойчивости левитирующих состояний сверхпроводящих миксин плазменных ловушек-Галатей Про стійкiсть левітуючих станів надпровідних міксин плазмових пасток-Галатей |
| title_full | About stability of levitating states of superconducting myxini of plasma traps-Galateas |
| title_fullStr | About stability of levitating states of superconducting myxini of plasma traps-Galateas |
| title_full_unstemmed | About stability of levitating states of superconducting myxini of plasma traps-Galateas |
| title_short | About stability of levitating states of superconducting myxini of plasma traps-Galateas |
| title_sort | about stability of levitating states of superconducting myxini of plasma traps-galateas |
| topic | Магнитное удержание |
| topic_facet | Магнитное удержание |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/82103 |
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