Numerical modelling of multibeam accelerating structures
The three-dimensional multibeam numerical simulation has been carried out on the basis of integral equations. Dispersion equations and expressions for shunt impedance and Q-factor have been obtained for E and H oscillations in cylindrical cavities. The tensor Green function and filament-like repre...
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irk-123456789-793282015-04-01T03:01:53Z Numerical modelling of multibeam accelerating structures Gavrilov, N.M. Komarov, D.A. Bogachenkov, D.A. Strukov, J.N. Элементы ускорителей The three-dimensional multibeam numerical simulation has been carried out on the basis of integral equations. Dispersion equations and expressions for shunt impedance and Q-factor have been obtained for E and H oscillations in cylindrical cavities. The tensor Green function and filament-like representation of the beam were used. На основі методу інтегральних рівнянь проведене тривимірне чисельне моделювання багатопроменевих прискорюючи структур. Для Е- и Н- резонаторів при використанні нитковидного зображення струму джерела і тензорної форми функції Гріна для циліндричного резонатора отримані дисперсійні рівняння, вирази для шунтового імпедансу і добротності. На основе метода интегральных уравнений проведено трехмерное численное моделирование многолучевых ускоряющих структур. Для Е- и Н- резонаторов при использовании нитевидного представления тока источника и тензорной формы функции Грина для цилиндрического резонатора получены дисперсионные уравнения, выражения для шунтового импеданса и добротности. 2004 Article Numerical modelling of multibeam accelerating structures / N.M. Gavrilov, D.A. Komarov, D.A. Bogachenkov, J.N. Strukov // Вопросы атомной науки и техники. — 2004. — № 2. — С. 53-56. — Бібліогр.: 3 назв. — англ. 1562-6016 PACS: 29.17.+w http://dspace.nbuv.gov.ua/handle/123456789/79328 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Элементы ускорителей Элементы ускорителей |
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Элементы ускорителей Элементы ускорителей Gavrilov, N.M. Komarov, D.A. Bogachenkov, D.A. Strukov, J.N. Numerical modelling of multibeam accelerating structures Вопросы атомной науки и техники |
| description |
The three-dimensional multibeam numerical simulation has been carried out on the basis of integral equations.
Dispersion equations and expressions for shunt impedance and Q-factor have been obtained for E and H oscillations
in cylindrical cavities. The tensor Green function and filament-like representation of the beam were used. |
| format |
Article |
| author |
Gavrilov, N.M. Komarov, D.A. Bogachenkov, D.A. Strukov, J.N. |
| author_facet |
Gavrilov, N.M. Komarov, D.A. Bogachenkov, D.A. Strukov, J.N. |
| author_sort |
Gavrilov, N.M. |
| title |
Numerical modelling of multibeam accelerating structures |
| title_short |
Numerical modelling of multibeam accelerating structures |
| title_full |
Numerical modelling of multibeam accelerating structures |
| title_fullStr |
Numerical modelling of multibeam accelerating structures |
| title_full_unstemmed |
Numerical modelling of multibeam accelerating structures |
| title_sort |
numerical modelling of multibeam accelerating structures |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2004 |
| topic_facet |
Элементы ускорителей |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/79328 |
| citation_txt |
Numerical modelling of multibeam accelerating structures / N.M. Gavrilov, D.A. Komarov, D.A. Bogachenkov, J.N. Strukov // Вопросы атомной науки и техники. — 2004. — № 2. — С. 53-56. — Бібліогр.: 3 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT gavrilovnm numericalmodellingofmultibeamacceleratingstructures AT komarovda numericalmodellingofmultibeamacceleratingstructures AT bogachenkovda numericalmodellingofmultibeamacceleratingstructures AT strukovjn numericalmodellingofmultibeamacceleratingstructures |
| first_indexed |
2025-07-06T03:24:34Z |
| last_indexed |
2025-07-06T03:24:34Z |
| _version_ |
1836866364789227520 |
| fulltext |
NUMERICAL MODELLING OF MULTIBEAM ACCELERATING
STRUCTURES
N.M. Gavrilov, D.A. Komarov, D.A. Bogachenkov, J.N. Strukov
The Moscow Engineering - Physical Institute (State University)
The three-dimensional multibeam numerical simulation has been carried out on the basis of integral equations.
Dispersion equations and expressions for shunt impedance and Q-factor have been obtained for E and H oscillations
in cylindrical cavities. The tensor Green function and filament-like representation of the beam were used.
PACS: 29.17.+w
With the purpose of creation of industrial linear
accelerators of ions with a current of a bunch more than
100 mА, works on studying multibeam accelerating
structures working on Е and Н kinds of fluctuations, and
also resonators with the electrodes of spiral type
overlapping a range of lengths of waves 2...15 m
recently are actively conducted.
Three-dimensional modelling of similar systems
with the help of modern software packages of applied
electrodynamics such as MAFIA, ISFEEL 3D, HFSS
7.0 or ANSIS 7.0 is or expensive and designed on the
use of high-efficiency computers, or basically
unpromising, as in the case with spiral systems and
systems on basis Н-resonators in a range of meter
wavelengths. In this work the numerical algorithm of
calculation of similar systems is offered.
1. MODELLING MULTIBEAM SYSTEMS
ON BASIS Е AND Н RESONATORS
As is known [1] system of Maxwell equations which
describes electromagnetic fields in the given area, can
be shown to two wave equations concerning vector and
scalar sizes which name potentials of an
electromagnetic field. Electric and magnetic fields can
be determined from the following equations:
( ) ( )
( ) ( )
,
,
, ,
dA r t
E r t grad
dt
B r t rotA r t
φ= − −
=
r rr r
rr r r
, (1)
where ϕ, A
r
are defined from the d’Alambert equation:
0
2
2
2
02
2
2
1
1
ε
ρφφ
δµ
−=
∂
∂⋅−∆
⋅−=
∂
∂⋅−∆
tc
t
A
c
A
r
r
r
, (2)
In the right parts of the equations the vector of
density of a current and the volumetric density of a
charge stand respectively. In that specific case the time
dependence can be considered harmonious, and for the
solutions of equations (2) to search as late potentials:
∫
∫
→
→
→
→
⋅
=
⋅
=
V
m
V
m
dV
rrR
r
dV
rrR
r
A
,
),(4
)(
,
),(4
)(
0
00
0
00
rr
rr
rr
rr
r
π
ρεφ
π
δµ
(3)
where mA
r
, φm - complex amplitudes, 0→r
r
- vector
radius of a point source, rr - vector radius of an
observation point, R - distance from a source up to the
observer. For an external problem of electrodynamic a
unique condition, set in (2) is the regularity on infinity
or the condition of radiation which apparently is carried
out.
We shall consider now an internal problem of
electrodynamics in the following statement: it is
required to define an electromagnetic field in cylindrical
area with the limited conducting surface. Then the
system (2) remains constant, but the solution will be
searched in the following form:
( ) ( )0 0 0 0 0 0 0, , , , , , ,
V
A G r z r z r z dVµ φ φ δ φ= Ч Чт
r rt
, (4)
where ( )0 0 0, , , , ,G r z r zφ φ
t
is so-called tensor or a
Green dyad [2] taking into account influence of external
borders on electromagnetic process in volume. As
shown in [3] for the cylindrical system of coordinates
the dyad has an explicit expression in index
designations. As the table this tensor can be presented as
follows:
11 12
21 22
0
0
0 0
G G
G G G
G
ж ц
з ч= з ч
з ч
и ш
t
, (5)
where the factors of a matrix are the following:
),,(),(),(1
),(~),(~),(~
~
1
0
0
00
0
2
0
0
00
1 0 0
11
zzf
r
r
r
r
rrk
zzfrr
rrk
G
mn
mnmn
mn
mn
mnmn
m n mn
⋅
∂
∂⋅
∂
∂+
+⋅
∂
∂⋅
∂
∂⋅
⋅⋅
= ∑ ∑
∞
=
∞
=
φψφψ
φ
φψ
φ
φψ
),,(),(),(1
),(~),(~),(~
~
1
0
0
00
0
2
0
0
00
1 0
212
zzfr
r
r
rk
zzf
r
rr
rk
G
mn
mnmn
mn
mn
mnmn
m n mn
⋅
∂
∂⋅
∂
∂+
+⋅
∂
∂⋅
∂
∂⋅
⋅
= ∑ ∑
∞
=
∞
=
φ
φψφψ
φψ
φ
φψ
),,(
),(),(
~
1
),(~),(~),(~
~
1
0
0
00
2
0
0
00
1 0 0
221
zzf
r
rr
rk
zzf
r
r
r
rk
G
mn
mnmn
mn
mn
mnmn
m n mn
⋅
∂
∂
⋅
∂
∂
+
+⋅
∂
∂
⋅
∂
∂
⋅
⋅
= ∑ ∑
∞
=
∞
=
φψ
φ
φψ
φ
φψφψ
___________________________________________________________
PROBLEMS OF ATOMIC SIENCE AND TECHNOLOGY. 2004. № 2.
Series: Nuclear Physics Investigations (43), p.53-56. 53
),,(),(),(
),,(),(),(
~
1
),(~),(~
~
1
0
1 0
0033
0
0
00
0
2
0
1 0
222
zzgrrG
zzf
r
r
r
r
rrk
zzf
r
r
k
G
mn
m n
mnmn
mn
mnmn
mn
mn
mn
m n mn
∑ ∑
∑ ∑
∞
=
∞
=
∞
=
∞
=
⋅⋅=
⋅
∂
∂⋅
∂
∂+
+⋅
∂
∂⋅=
φψφψ
φψφψ
φψ
where ψmn, fmn, gmn are the own functions of the
Helmholtz equations with boundary conditions such as
an electric wall, and the functions mnmn f~,~ψ define
solutions with boundary conditions such as magnetic
walls for the given area. Definition of the given
functions and calculation of the component Green
tensor allows to obtain the explicit solution.
Representing accelerating systems as a closed area
with the currents distributed inside (current-carrying
elements are: a tube of drift for the Е-resonator, pin
holders for the Н-resonator) it is possible to obtain the
integrated equations describing the electromagnetic
process in the resonator. So for Е - the resonator relative
to the z-component the current density on a drift tube is
presented as Fourier series:
⋅⋅
⋅
−⋅−−⋅⋅
+
=⋅∫
zchk
Lshk
lLshkz
l
p
l
pk
c
dzzzzg
z
z
z
p
z
p
z
zmn
)()1(cos
)(),(
2
2
000
π
π
δ
(6)
For Н - the resonator the integrated equation can be
written down as:
( ) ( ) 0000
2
1
=+ ∫ drrrIKrI
r
r
, (7)
where ( )21 11
1 10.5K a r G G
r r r ϕ
м ь∂ ∂= −Чн э∂ ∂о ю
.
The decision of these equations can be carried by
Galerkin method, and for (1) it is more preferable to
choose Fourier series as basis, and for (2) to use sewing
together on points. Results of calculation are shown on
fig.1, fig.2, fig.3.
Fig.1. Dependence of wave number
of period length (Е - mode)
Fig.2. Dependence of shunt resistance from length
period (Е - mode)
Fig.3. Dependence of wavelength from length of the
resonator (Н - mode)
Testing of the program was carried out by
comparison of results of calculation by an offered
method and calculation with the help of package
ISFEEL 3D. Comparisons (for Е-modes) have shown,
that the error of definition of frequency is lesser than
3% at a time of processing 10...20 seconds while
computer modelling for four drift tubes on an azimuth
has taken 75 minutes for one point. By the shunt
impedance the error lays within the limits of 10%.
2. MODELLING MULTIBEAM SYSTEMS
ON BASIS OF SPIRAL ELECTRODES
Of greatest interest for acceleration of heavy ions are
structures with electrodes of a spiral type. In [4] one of
possible variants of the similar accelerator is presented.
Actually the resonator works as follows: the spiral
electrode being a resonant element is activated on its
own lowest frequency. The currents reaching the spiral
form the distribution of superficial charges at surfaces
of conducting ring (CR) with drift tubes (ТD) and then
between ТD the accelerating potential difference is
54
N=4
N=8
N=16
0 50 100 150
2
2.5
1.9
L,mm
R
s
,
MОрm/m
1.3
12
L=8см
L=10,
sm
L=20см
11 12.5 1
3.8
3.4
h,m
,m
3
11.5
formed. Thus, from the experiment it is known, that the
size of the external resonator does not influence the
frequency of a spiral, as well as accommodation of an
electrode. At spatial arrangement of spirals with
quadrupoles the characteristic frequency of a working
mode of fluctuations ceases to depend on the number of
electrodes, that also is the experimental fact.
All above-mentioned listed features allow us to
formulate the following mathematical consequences: in
the solution of the Helmholtz equation, there is no
necessity to use the Green tensor function as the
influence of the screen on the oscillatory process is
insignificant. The taking into account of several
electrodes is also unessential. The problem of the
analysis as a matter of fact is reduced to a problem
about a field created by a spiral, fixed on the conducting
plane. Thus, for the solution we shall use a method of
secondary sources: conducting surface we shall replace
by the mirror image of a spiral, and for a current we
shall write down, using threadlike model:
( ) ( )0dV I s s dsδ τ=
r r
, (8)
where ds is the element of spiral length ( )sτr is the
vector of a tangent to a spiral.
The vector potential will look like:
( ) ( )0
4
jkR jkR
s
e eA I s s ds
R R
µ τ
π
+ −− −
+ −
м ь
= −н э
о ю
т
r r
, (9)
where R+, R- the distances between an observation point
of a source and a point on the mirror image respectively.
The equation of a spiral in this case will be written
down as follows:
[ ]
0
0
0 0
0
2 ,
0,
r a
zz ctg
b a
z l
πϕ ψ
=м
пп = =н
п
п Оо
, (10)
where r0, φ0, z0 are the coordinates on a spiral, а is the
radius of the coil, b is the step of a spiral, ψ0 is the angle
of winding. Distance up to observation points :
( ) ( )2 2 2
0 02 cosR z z r a ar ϕ ϕ± = + − − −m (11)
The length element is correct in the form
0
sin
dzds
ψ
= , (12)
where ( ) 22
sin
2
b
b a
ψ
π
=
+
.
Note the importance of Equation (12) as
parametrical forms of an length element, since in this
case the current acts as an explicit function of the
coordinate z that allows essentially to facilitate the
model.
Substituting (10) in expression for vector potential,
we shall receive the integrated equation being
mathematical model of a physical problem:
( ) { } ( )0 0
02sin
l
I z N rot G t I z dzδ τ
ψ
− Ч ґ Ч Ч Чт
r rr
. (13)
Making consistently all the vector operations and
entering explicit expressions for the Frene’s trihedron, it
is possible to write the kernel of the integrated equation
in the following final form:
∂
∂+
+
∂
∂−
∂
∂
×
×
+
+=
G
r
G
z
G
r
b
K
α
αψψ
ϕ
ψψψαδψ
cos
sincossin1
sincossinsincos 22
(14)
and to reduce the equation to the initial form:
( ) ( )0 0
0
0
l
I z K I z dzλ− =т , (15)
with
2sin
δλ
ψ
= .
The solution can be obtained also by the method of
sewing together on points. All given algorithms are
realized as program modules, the time of the account by
which is some tens seconds. Results are submitted on
fig. 4.
Fig.4. Dependence of resonant frequency of a spiral on
its length
Results of calculations compared with experimental
data showed that the error of definition of frequency
lays within the limits of 5%.
3. CONCLUSIONS
In this paper were presented the electrodynamic
models of multibeam systems based on Е and Н
resonators and structures with the spiral electrodes,
admitting numerical realization in MATCAD
environment.
REFERENCES
1. Dg. Stratton The theory of electromagnetism. M.:
State technical Publishing house, 1948, p.539.
___________________________________________________________
PROBLEMS OF ATOMIC SIENCE AND TECHNOLOGY. 2004. № 2.
Series: Nuclear Physics Investigations (43), p.53-56. 55
D=4,cm
D =5,cm
D = 6,cm
6 7.5 9 10.5 1
50
7
40
L,cm
F МHz
30
60
2. G.T.Markov, E.N.Vasilev Mathematical methods
of applied electrodynamics. M.: Sov. radio, 1970,
p.118 (in Russian).
3. А.L.Gulianhc, Е.N.Tuirin Тensores Green's
functions of round wave guides and resonators.
Accelerators. M: Energy Atom Publishing house,
1981, p.76-85.
ЧИСЛЕННОЕ МОДЕЛИРОВАНИЕ МНОГОЛУЧЕВЫХ
УСКОРЯЮЩИХ СТРУКТУP
Н.М. Гаврилов, Д.А. Богаченков, Д.А. Комаров, Ю.Н. Струков
На основе метода интегральных уравнений проведено трехмерное численное моделирование
многолучевых ускоряющих структур. Для Е- и Н- резонаторов при использовании нитевидного
представления тока источника и тензорной формы функции Грина для цилиндрического резонатора
получены дисперсионные уравнения, выражения для шунтового импеданса и добротности.
ЧИСЕЛЬНЕ МОДЕЛЮВАННЯ БАГАТОПРОМЕНЕВИХ
ПРИСКОРЮЮЧИХ СТРУКТУР
М.М. Гаврилов, Д.А. Богаченков, Д.А. Комаров, Ю.Н. Струков
На основі методу інтегральних рівнянь проведене тривимірне чисельне моделювання
багатопроменевих прискорюючи структур. Для Е- и Н- резонаторів при використанні нитковидного
зображення струму джерела і тензорної форми функції Гріна для циліндричного резонатора отримані
дисперсійні рівняння, вирази для шунтового імпедансу і добротності.
56
Numerical modelling of multibeam accelerating structures
References
Численное моделирование многолучевых
Н.М. Гаврилов, Д.А. Богаченков, Д.А. Комаров, Ю.Н. Струков
ЧИСЕЛЬНЕ МОДЕЛЮВАННЯ БАГАТОПРОМЕНЕВИХ
М.М. Гаврилов, Д.А. Богаченков, Д.А. Комаров, Ю.Н. Струков
|