Calculation of non-uniform solenoids for focusing and acceleration of charged particles
In this work the more complex problem is solved by means of using Tikhonov′s regularization method: the sections parameters of a specific kind are calculated. These can be for example a winding thickness, length or a section internal radius etc., i.e. the values of magnetic field intensity are depen...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 1999 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
1999
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| Цитувати: | Calculation of non-uniform solenoids for focusing and acceleration of charged particles / V.I. Butenko, B.I. Ivanov // Вопросы атомной науки и техники. — 1999. — № 3. — С. 23-25. — Бібліогр.: 6 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860109540557586432 |
|---|---|
| author | Butenko, V.I. Ivanov, B.I. |
| author_facet | Butenko, V.I. Ivanov, B.I. |
| citation_txt | Calculation of non-uniform solenoids for focusing and acceleration of charged particles / V.I. Butenko, B.I. Ivanov // Вопросы атомной науки и техники. — 1999. — № 3. — С. 23-25. — Бібліогр.: 6 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | In this work the more complex problem is solved by means of using Tikhonov′s regularization method: the sections parameters of a specific kind are calculated. These can be for example a winding thickness, length or a section internal radius etc., i.e. the values of magnetic field intensity are depending, generally speaking, in the nonlinear way. The method described here is illustrated by solving the problem in case of defining the winding thickness for a solenoid with rectangular cross-sections.
|
| first_indexed | 2025-12-07T17:33:15Z |
| format | Article |
| fulltext |
CALCULATION OF NON-UNIFORM SOLENOIDS FOR FOCUSING AND
ACCELERATION OF CHARGED PARTICLES
V.I. Butenko, B.I. Ivanov
NSC KIPT, Kharkov, Ukraine
INTRODUCTION
For many problems of forming, acceleration
and focusing charged particles it is need to create non-
uniform axially symmetric stationary magnetic fields
with specified dependence on longitudinal coordinate
within the given accuracy. Usually such fields are
created by sectional solenoids consisted of coaxially
located sections (coils). The magnetic field intensity
created by each section depends on its parameters
generally nonlinearly. The parameters to be
manipulated are geometrical dimensions of the sections,
their mutual location (the section coordinates), current
density, etc.
In previous works related to the above-
mentioned problem, the sections’ current densities were
used as some unknown parameters. This is the simplest
way to solve the problem of parameters since current
density is a linear parameter of the equation for a
solenoid magnetic field intensity. The method proposed
in [1] is based on a functional minimization allowing to
determine the optimum (in sense of minimum standard
deviation of created field from required one) current
density distribution in all sections of the solenoid.
However, if the number of sections is large enough
(more then 20), the set of the linear equations – the
problem reduced – becomes mal-conditioned which
leads to a loss of accuracy in the calculations needed.
Hereupon, the load distribution among the sections
appears to be far from the optimum desired. Current
loading to the edge sections turns out to be many times
as large as that of all the others, resulting in thriftless
consumption of magnet wire and necessity of forced
cooling the outside sections. Power supply of the
solenoid is also complicated since setting up the current
density distribution among the sections requires either
to have an individual power source for each section or
to place a rheostat along with each section.
In this work the more complex problem is
solved by means of using Tikhonov′s regularization
method [2]: the sections parameters of a specific kind
are calculated. These can be for example a winding
thickness, length or a section internal radius etc., i.e. the
values of magnetic field intensity are depending,
generally speaking, in the nonlinear way. The method
described here is illustrated by solving the problem in
case of defining the winding thickness for a solenoid
with rectangular cross-sections.
THEORY
Let us consider a sectional solenoid of n
coaxial sections with an arbitrary cross-section. It is
necessary to create a magnetic field at the solenoid axis
in an interval [a, b] within the given accuracy δ, when
the field intensity is prescribed by a function f(z).
Assume that field intensity, created by the i-th section
at the point , is described by the function Hi(Ni, z),
which in turn depends on a cross-section of the
geometrical configuration, its size and location towards
the measuring point. Then magnetic field created by the
solenoid on its axis can be formulated as
( ) ( )∑
=
=
n
i
ii zNHzB
1
, . We shall estimate deviation of
B(z) from f(z) in square metric, i.e. by the formula:
( ) ( ) ( )[ ]
21
2
−=ρ ∫
b
a
dzzfzBfB, . This problem pertains
to the class of incorrect ones in sense that within given
accuracy it has not single-valued solution; moreover, if
a sections’ number is large enough, its solution
becomes unstable towards small variations of initial
data. So, from a set of solutions matching ( ) δ≤ρ fB,
where δ is a given number, one has to pick up the only
solution optimized on a certain criteria (a solenoid
volume or power consumption can serve as such). The
problem is formulated as follows: to find an optimum
set of parameters (N1, N2,..., Nn) under which the
functional
( ) ( ) ( )
( )n
b
a
n
i
iin
NN
dzzfzNHNNF
,...,
,,,...,
1
2
1
1
Ωβ+
+
−=β ∫ ∑
= (1)
reaches its greatest lower bound [2]. Here Ω(N1,...Nn) is
stabilizing functional determined by the optimization
criteria, β is a parameter of regularization. The
conditions, under which
( ) 0,,...,1 =
∂
β∂
k
n
N
NNF
, lead us
to a set of nonlinear equations:
( ) ( ) 0,
1
=
∂
Ω∂β+
∂
∂
−∫ ∑
= k
b
a k
k
n
i
ii N
dz
N
H
zfzNH , (2)
that, in turn, can be solved by one of the gradient
methods (see [3 ]).
CALCULATIONS
Solution of the problem on solenoid
calculation includes following stages: 1) choice of
values determined by a device parameters, heat removal
conditions and other factors; 2) choice of initial
estimates for both 0
iN and the regularization parameter
β to be ensured that an iterative process converges; 3)
solving the set of equations (1) within given accuracy;
4) updating of the initial estimate (current value of a
desirable parameter is accepted as such) and that of the
parameter β (if the iterative process converges too
slowly, β would decrease); 5) repeated solving of (2)
until a satisfactory result has been found.
In this section the method offered has been
used in case of a homogeneous magnetic field to be
produced by a solenoid built of coils with rectangular
cross-section. As a desirable parameter Ni the winding
thickness di = Ri - ri has been accepted. In this case the
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 1999. №3.
Серия: Ядерно-физические исследования (34), с. 23-25.
23
approximating functions Hi(di, z) and their first
derivatives are:
( ) ( ) ( )
( )
( ) ( )
( )
ζ−−++
ζ−−++
ζ−−
−
ζ−++
ζ−++
ζ−
π
=
22
22
22
22
ln
ln
2
iiii
iiii
ii
iii
iii
i
i
i
azrr
azRR
az
zrr
zRR
z
c
J
zH
,
( ) ( )
ζ−−+
ζ−−
−
ζ−+
ζ−π
=
∂
∂
2222
2
iii
ii
ii
ii
i
i
azR
az
zR
z
c
J
d
H
where Ji is the current density [GS]; ζi is the section left
edge coordinate [cm]; ri is the section inner radius [cm];
Ri is its outer radius [cm]; ai is the length of the section
[cm]; c is the light velocity [cm/s].
As a stabilizing functional the square of the
solution Euclidean norm have been used:
( ) ( )∑
=
−=Ω
n
j
jjn dddd
1
20
1 ,..., , where 0
iN is an
initial value of the i-th section parameter to be found.
Such choice actually corresponds to the criteria of
minimum space of wire the solenoid is wound by. As a
pattern to be calculated we picked up the solenoid
design described in [1] , where the problem of current
density definition had been solved under β = 0. The
solenoid length is equal to 727 mm, its inner radius is
85 mm. Current density is the same through all sections
and equal to 2 A/mm2. A homogeneous magnetic field
of intensity f(z) = 1 kOe is created along the length of
436 mm (60% of the solenoid length). The set of
equations (2) was solved by the Newton method under
following initial values of parameters: the winding
thickness was 0
id = 0 for all sections, the parameter of
regularization β was altered from 105 to 0, absolute
accuracy of winding thickness calculated value was
ε = 10-4 cm (i.e. the iterative processes were to stop
when β = 0 and the next correction to di was less than ε
for all the sections taken into account). The
regularization parameter β was modified by means of
its dividing to a constant factor k = 2 in the course of its
gaining the accuracy prescribed, and it was made equal
to zero when getting the value below β = 10-10.
The Table 1 shows the meanings of the mean-
square deviation ρ(B, f), maximum of the relative
deviation ( ) ( )( ) ( )zfzfzB −=∆ max and volume
of the conductor with which the solenoid is reeled up
when being splitted into various number of sections m.
The figures for the value, which is inversely
proportional to the conditioned number
1
1
1
cond −= AAA of a matrix A with components
ji
ij dd
Fa
∂∂
∂=
2
, where ∑
=≤≤
=
m
i
ijmj
a
111
maxA , related
to the last iteration are also given. As we can see from
the Table, upon improvement of the field
approximation the solenoid volume (and weight) tends
to grow.
Table 1.
m ρ, Oe⋅cm1/2 ∆ V, cm3 1/cond A
4 18.1 1.09⋅10-2 2.12⋅104 8.22⋅10-2
6 3.13 2.16⋅10-3 2.19⋅104 4.74⋅10-3
8 0.549 4.31⋅10-4 2.26⋅104 1.67⋅10-4
10 8.56⋅10-2 7.50⋅10-5 2.35⋅104 3.95⋅10-6
12 1.16⋅10-2 1.10⋅10-5 2.50⋅104 6.64⋅10-8
14 1.24⋅10-3 1.29⋅10-6 2.80⋅104 9.57⋅10-10
16 1.19⋅10-4 1.32⋅10-7 3.34⋅104 1.40⋅10-11
18 1.38⋅10-5 1.47⋅10-8 4.20⋅104 2.00⋅10-13
18 4.13⋅10-3 5.63⋅10-6 2.41⋅104 1.56⋅10-8
It should be noted that as far as the
regularization parameter final figure gets equal to zero,
these results correspond to the best magnetic field
approximation in sense of the lowest root-mean-square
deviation. However, the results like that cannot always
be taken as satisfactory. For example, the Fig. 1
pictures a magnetic field design for the solenoid with 18
coils, m = 18 (as the solenoid is symmetric, only a half
of it is shown). As we see from the picture, the edge
sections manufacturing would most likely be a toilsome
affair, and the magnetic field approximation of such
accuracy as 10-8 in most cases is hardly required.
0 10 20 30 40
0
10
20
30
40
r, cm
z, cm
Fig. 1.
In order to optimize the solution towards both
parameters – accuracy of a magnetic field
approximation and the solenoid volume – the iterative
process was terminated when a root-mean-square
discrepancy ρ(B, f) reached value of δ = 4⋅10-3. The
relative error of the field approximation thus amounted
to ∆ = 5.63⋅10-6, as the solenoid volume got to V = 2.41⋅
104 cm3. These results are shown in the bottom line of
Table 1, and the solenoid optimum configuration
corresponding to the solution being found is pictured in
Fig. 2. It goes without saying that the solution of ours is
not optimum in sense of the lowest root-mean-square
deviation, however it has allowed to reduce both
dimension of edge sections and almost twice – volume
of the solenoid as a whole.
0 10 20 30 40
0
5
10
15
20
r, cm
z, cm
Fig. 2.
Let us note that current density in all sections
was installed identical that excludes the edge sections
overloading. Due to this, magnet wire might be
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 1999. №3.
Серия: Ядерно-физические исследования (34), с. 23-25.
23
consumed more rationally and the power consumption
be reduced in comparison with solenoids of different
kind where a required magnetic field is provided by the
proper current density distribution over sections. Power
supply of the solenoid might also be simplified as, at
common current for all sections, they could be
connected in series and fed up by a common power
source. All this eventually determines the solenoid
dimensions, weight and cost.
APPLICATIONS
As an example, let us consider the results of
modeling a concrete solenoid with prescribed magnetic
field distribution on its length. The appropriate
calculations had been carried out within a work on two-
beam electron-ion accelerator design [5] and working
out of adiabatic plasma lenses designed for 5 MeV
proton beam focusing [6].
Two-beam electron-ion accelerator. To ensure
electron-cyclotron resonance happening inside a
resonator for a wave excited by a driving electron beam
with natural oscillations of a non-uniform slow-wave
structure, it is necessary to place the resonator in a non-
uniform magnetic field, the diagram of which is shown
in a Fig. 3. The relative accuracy of a field
approximation ∆ should be less than 1%.
0 20 40 60 80 100 120 140
450
500
550
600
650
H, Gs
z, cm
Fig. 3.
The configuration of solenoid creating a field
in need, is shown in Fig. 4. Its main parameters are as
follows: length- 200 cm, internal radius- 39 cm,
sections length -10 cm, number of sections- 15, average
current density over the winding cross-section –
2 A/mm2. The parameter to be calculated was thickness
of winding. The field approximation relative error has
not exceeded ∆ = 10-3 over length of 140 cm, i.e. 70%
of the solenoid length.
0 50 100 150
0
10
20
30
40
50
z, cm
r, cm
Fig. 4.
Adiabatic plasma lens. In devices like this a
radius of the focusing current channel is determined by
an external magnetic field topography. For the greater
efficiency of a current lens a focused proton beam
should fill a focusing current channel as good as
possible. To this effect one has to create such external
magnetic field over the device length that the focused
protons trajectories would coincide with its magnetic
lines of force. The diagram of such field is shown in
Fig. , and configuration of the solenoid creating it is in
Fig. . Its main parameters are as follows: length –
114 cm, internal radius – 10 cm, length of each section
– 11.5 cm, number of the sections – 10, average current
density over cross-section of winding – 2 A/mm2. The
parameter to be calculated was thickness of winding.
The relative error of the field approximation has not
exceeded ∆ = 2.48⋅10-3 over the length of 80 cm, i.e.
70 % of the solenoid length.
REFERENCES
1. Lugansky L.B. Calculation of a Solenoid with a
Magnetic Field Given Distribution. JTP, 1985, V.
55, No.7, P.1263–1271.
2. Tikhonov A.N., Arsenin V.Ya. Methods of Incorrect
Problems Solution. Moscow. Nauka, 1986.
3. Korn G., Korn T. Guide to Mathematics for Research
Workers and Engineers. Moscow. Nauka, 1984.
831 P.
4. Butenko V.I. Calculation of a Solenoid Creating a
Magnetic Field with Given Dependence on
Longitudinal Coordinate. JTP, 1992, V. 62, No.7,
P.157–165.
5. V. Butenko, B. Ivanov, V. Ognivenko,
I. Onishchenko, V. Prishchepov, A. Yegorov.
Development of a Collective Linear Ion
Accelerator with RF Field Excitation Using
Doppler Effect. VANT, ser. Fizika plazmy, 1999,
No. 3,4(3,4), P. 212.
6. V. Belan, V. Butenko, B. Ivanov, V. Kiselev,
A. Kitsenko, A. Linnik, I. Onishchenko,
V. Prishchepov, A. Yegorov. Plasma Lens with a
Current Density Depended on External Magnetic
Field. VANT, ser. Fizika plazmy, 1999,
No. 3,4(3,4), P. 233.
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 1999. №3.
Серия: Ядерно-физические исследования (34), с. 23-25.
23
-20 0 20 40 60 80 100
0
5
10
15
20
r, cm
z, cm
Fig. 6
0 20 40 60 80
150
200
250
300
350
400
450
H, Gs
z, cm
Fig. 5
INTRODUCTION
THEORY
CALCULATIONS
APPLICATIONS
|
| id | nasplib_isofts_kiev_ua-123456789-81141 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T17:33:15Z |
| publishDate | 1999 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Butenko, V.I. Ivanov, B.I. 2015-05-11T17:46:32Z 2015-05-11T17:46:32Z 1999 Calculation of non-uniform solenoids for focusing and acceleration of charged particles / V.I. Butenko, B.I. Ivanov // Вопросы атомной науки и техники. — 1999. — № 3. — С. 23-25. — Бібліогр.: 6 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/81141 In this work the more complex problem is solved by means of using Tikhonov′s regularization method: the sections parameters of a specific kind are calculated. These can be for example a winding thickness, length or a section internal radius etc., i.e. the values of magnetic field intensity are depending, generally speaking, in the nonlinear way. The method described here is illustrated by solving the problem in case of defining the winding thickness for a solenoid with rectangular cross-sections. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Calculation of non-uniform solenoids for focusing and acceleration of charged particles Расчет неоднородных соленоидов для фокусировки и ускорения заряженных частиц Article published earlier |
| spellingShingle | Calculation of non-uniform solenoids for focusing and acceleration of charged particles Butenko, V.I. Ivanov, B.I. |
| title | Calculation of non-uniform solenoids for focusing and acceleration of charged particles |
| title_alt | Расчет неоднородных соленоидов для фокусировки и ускорения заряженных частиц |
| title_full | Calculation of non-uniform solenoids for focusing and acceleration of charged particles |
| title_fullStr | Calculation of non-uniform solenoids for focusing and acceleration of charged particles |
| title_full_unstemmed | Calculation of non-uniform solenoids for focusing and acceleration of charged particles |
| title_short | Calculation of non-uniform solenoids for focusing and acceleration of charged particles |
| title_sort | calculation of non-uniform solenoids for focusing and acceleration of charged particles |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/81141 |
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