Mathematical model of inhomogeneous cavity chain
Mathematical model of inhomogeneous chain of cylindrical cavities is developed. Coupling coefficients in the inhomogeneous cavity chain can be calculated with definite accuracy for the structure with arbitrary parameters. Influence of non-resonant fields and “long-range” couplings on the character...
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| Zitieren: | Mathematical model of inhomogeneous cavity chain / M.I. Ayzatskiy, K.Yu. Kramarenko // Вопросы атомной науки и техники. — 2004. — № 5. — С. 145-148. — Бібліогр.: 8 назв. — англ. |
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irk-123456789-806132015-04-20T03:01:57Z Mathematical model of inhomogeneous cavity chain Ayzatskiy, M.I. Kramarenko, K.Yu. Теория и техника ускорения частиц Mathematical model of inhomogeneous chain of cylindrical cavities is developed. Coupling coefficients in the inhomogeneous cavity chain can be calculated with definite accuracy for the structure with arbitrary parameters. Influence of non-resonant fields and “long-range” couplings on the characteristics of the structure is taken into account by calculation of the coupling coefficients. Розроблено математичну модель для опису неоднорідного ланцюжка зв’язаних циліндричних резонаторів. Коефіцієнти зв’язку у неоднорідному ланцюжку резонаторів можуть бути розраховані з певною точністю для структури з будь-якими параметрами. Вплив нерезонансних полів та “далеких” взаємодій на характеристики структури враховується при розрахуванні коефіцієнтів зв’язку. Разработана математическая модель для описания неоднородной цепочки связанных цилиндрических резонаторов. Коэффициенты связи в неоднородной цепочке резонаторов могут быть рассчитаны с заданной точностью для структуры с произвольными параметрами. Влияние нерезонансных полей и “дальних” взаимодействий на характеристики структуры учитывается при расчете коэффициентов связи. 2004 Article Mathematical model of inhomogeneous cavity chain / M.I. Ayzatskiy, K.Yu. Kramarenko // Вопросы атомной науки и техники. — 2004. — № 5. — С. 145-148. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 29.1 7.+w http://dspace.nbuv.gov.ua/handle/123456789/80613 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Теория и техника ускорения частиц Теория и техника ускорения частиц |
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Теория и техника ускорения частиц Теория и техника ускорения частиц Ayzatskiy, M.I. Kramarenko, K.Yu. Mathematical model of inhomogeneous cavity chain Вопросы атомной науки и техники |
| description |
Mathematical model of inhomogeneous chain of cylindrical cavities is developed. Coupling coefficients in the
inhomogeneous cavity chain can be calculated with definite accuracy for the structure with arbitrary parameters.
Influence of non-resonant fields and “long-range” couplings on the characteristics of the structure is taken into
account by calculation of the coupling coefficients. |
| format |
Article |
| author |
Ayzatskiy, M.I. Kramarenko, K.Yu. |
| author_facet |
Ayzatskiy, M.I. Kramarenko, K.Yu. |
| author_sort |
Ayzatskiy, M.I. |
| title |
Mathematical model of inhomogeneous cavity chain |
| title_short |
Mathematical model of inhomogeneous cavity chain |
| title_full |
Mathematical model of inhomogeneous cavity chain |
| title_fullStr |
Mathematical model of inhomogeneous cavity chain |
| title_full_unstemmed |
Mathematical model of inhomogeneous cavity chain |
| title_sort |
mathematical model of inhomogeneous cavity chain |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2004 |
| topic_facet |
Теория и техника ускорения частиц |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/80613 |
| citation_txt |
Mathematical model of inhomogeneous cavity chain / M.I. Ayzatskiy, K.Yu. Kramarenko // Вопросы атомной науки и техники. — 2004. — № 5. — С. 145-148. — Бібліогр.: 8 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT ayzatskiymi mathematicalmodelofinhomogeneouscavitychain AT kramarenkokyu mathematicalmodelofinhomogeneouscavitychain |
| first_indexed |
2025-07-06T04:36:53Z |
| last_indexed |
2025-07-06T04:36:53Z |
| _version_ |
1836870914338193408 |
| fulltext |
MATHEMATICAL MODEL OF INHOMOGENEOUS CAVITY CHAIN
M.I. Ayzatskiy, K.Yu. Kramarenko
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
e-mail: kramer@kipt.kharkov.ua
Mathematical model of inhomogeneous chain of cylindrical cavities is developed. Coupling coefficients in the
inhomogeneous cavity chain can be calculated with definite accuracy for the structure with arbitrary parameters.
Influence of non-resonant fields and “long-range” couplings on the characteristics of the structure is taken into
account by calculation of the coupling coefficients.
PACS: 29.17.+w
1. INTRODUCTION
The equivalent circuit analysis is widely used for
description of the behavior of accelerating structures
(see, for example, [1-3]). It proves to be useful at the
stage of primary study of electrodynamic properties of
the structure and its conceptual design. Using this
approach (equivalent circuit analysis) the technique of
consecutive cell-tuning in homogeneous and strongly
inhomogeneous structures is developed (see [4,5]).
Using the technique of consecutive cell-tuning the
comparative analysis of “random” and constant gradient,
quasi-constant gradient structures has been done in [5].
Next parameters were chosen for comparison: energy
gain, field gradient and damping. It was shown that
“random” structures are comparable with constant
gradient and quasi-constant gradient ones. However,
justification of using of equivalent circuit analysis for
description of chains of coupled cavities must be made
with the help of rigorous electrodynamic methods.
In [6] precise equations describing the RF-coupling
of two cavities is obtained. The method of partial cross-
over regions is used. On the base of these equations the
dependence of coupling coefficients versus frequency,
iris radius, etc. is calculated. The analytical solution of
these equations for various limited cases is presented
in [7]. It is shown that in the case of iris radius tending
to zero and infinitely small disk thickness (a → 0, t = 0)
the obtained equations agree with those in [2] obtained
on the base of quasi-static approach. In [8] the method
of partial cross-over regions is used to describe an
infinitely long chain of identical cylindrical cavities
coupled through irises. The dependence of the phase
shift per period versus frequency is calculated for
structures with various parameters.
Using of the method of partial cross-over regions for
description of inhomogeneous chain of cylindrical
cavities reduces to clumsy formulas and difficult
calculations. In this work we propose more simple, from
our point of view, method for description of
inhomogeneous structures. This method is based on
rigorous electrodynamic approach as well.
2. ELECTRODYNAMIC APPROACH
Inhomogeneous accelerating structure consists of an
array of ideally conducting co-axial cylindrical cavities
coupled through dividing irises with radii ai and
thickness t. The radii and lengths of the cavities we
denote by bi and di (see Fig.).
Longitudinal cross-section of inhomogeneous chain
of cylindrical cavities
The field functions in the region zi < z < zi+di are
represented in terms of cavity modes:
))(cos()( )()(
0
)()(
, i
i
p
ib
m
m p
i
mp
i
mpz zzkrkJAE −= ⊥∑ ∑ ,
))(sin()( )()(
1)(
)(
)()(
, i
i
p
ib
mib
m
i
p
m p
i
mp
i
mpr zzkrkJ
k
k
AE −= ⊥
⊥
∑ ∑ ,
=)(
,
i
mpHϕ (1)
))(cos()( )()(
1)(
)(
0)(
i
i
p
ib
mib
m
i
mp
m p
i
mp zzkrkJ
k
i
B −
−
= ⊥
⊥
∑ ∑
ωε
where )(i
mpA is the amplitude and )(i
mpω is the resonant
frequency of axially-symmetric Е0mp-mode in the i-th
cavity; )()()( i
mp
i
mp
i
mp AB ωω= .
The field functions in the region ti < z < ti+t (disk
aperture) can be expressed in terms of modes of uniform
waveguide of cylindrical symmetry:
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2004, № 5.
Series: Nuclear Physics Investigations (44), p. 145-148. 145
( ) [
] ,))(exp(
))(exp(
)()(
2
)()(
1
)(
0
)(
,
i
i
s
i
s
i
i
s
i
s
s
ia
s
i
sz
tzhC
tzhCrkJE
−+
+−−= ∑ ⊥
( ) [
] ,))(exp(
))(exp(
)(
)(
)()(
2
)()(
1
)(
1
)(
,
ia
s
i
s
i
i
s
i
s
i
i
s
i
s
s
ia
s
i
sr
k
h
tzhC
tzhCrkJE
⊥
⊥
−−
−−−= ∑
(2)
( ) [
] .))(exp(
))(exp(
)(
0)()(
2
)()(
1
)(
1
)(
,
ia
s
i
i
s
i
s
i
i
s
i
s
s
ia
s
i
s
k
itzhC
tzhCrkJH
⊥
⊥
−−+
+−−= ∑
ωε
ϕ
Tangential components of electric field at the left
and right boundaries of the i-th disk aperture are
expanded into the series with the complete set of the
first order Bessel functions:
( )
( )∑
∑
⊥−−
⊥++
=
=
s
ia
s
i
s
i
r
s
ia
s
i
s
i
r
rkJwE
rkJwE
)(
1
)()(
)(
1
)()( ,
(3)
where 0 < r < ai. Sign (+) refers to the right boundary of
the disk and sign (-) – to the left one, correspondingly.
As the tangential component of electric field is
continuous at the both boundaries of the i-th disk,
coefficients C and w are connected via the following
equations:
.
)(2
)exp(
,
)(2
)exp(
)(
)()()(
)(
)(
)(
2
)(
)()()(
)(
)(
)(
1
thsh
wthw
h
k
C
thsh
wthw
h
k
C
i
s
i
s
i
s
i
s
i
s
ia
si
s
i
s
i
s
i
s
i
s
i
s
ia
si
s
+−⊥
+−⊥
−−=
−=
(4)
Coefficients in expansion (1) are determined by the
tangential components of electric field )(i
rE + and )1( +
−
i
rE :
( ) ×=− )(
)(
2)(2)( 2
i
mp
i
mpi
mp
i
mp
iN
A
π ω
ωω (5)
−× ∫∫ ++
+
−
+
rdrzrHErdrtrHE
ii a
i
ii
r
a
i
ii
r
0
*)()(
0
1
*)()1( ),(),(
1
ϕϕ
where )(i
mpN is the norm of Е0mp-mode in the i-th cavity.
By matching tangential component of magnetic field
in the both planes of the i-th disk aperture (0 < r < ai),
one can write the equations for the coefficients )1( −i
mpA ,
)(i
mpA , )(i
sw − , )(i
sw + :
( )
( ) ( )
( ) ,
)1(
)(
)()()(
)(
)(
1
)1(
)1(
)1(
1
∑
∑ ∑
+−⊥
−
−
⊥
−
⊥
−=
=
−
s
i
s
i
s
i
s
i
s
i
s
ia
s
m
i
mp
p
ib
m
ib
m
p
thsh
wthchw
h
rkJ
A
k
rkJ
ω
ω
(6.1)
( )
( ) ( )
( ) .)(
)()()(
)(
)(
1
)(
)(
)(
1
∑
∑ ∑
+−⊥
⊥
⊥
−=
=
s
i
s
i
s
i
s
i
s
i
s
ia
s
m
i
mp
p
ib
m
ib
m
thsh
thchww
h
rkJ
A
k
rkJ
ω
ω
(6.2)
The left and right parts of Eqs.(6) are functions of
variable r. Since the functions are equal the expansion
coefficients of these functions with the complete set of
orthogonal functions ( )rkJ ia
n
)(
1 ⊥ are equal too:
( )
( )
( ) ( )
( ) ,
2
)1(
)(
)()()(
)(
1
)1(
2),1(2)1(
),1(
0
),1(
thsh
wthchw
h
J
A
k
J
i
n
i
n
i
n
i
n
i
n
n
m
i
mp
p
i
mn
ib
m
i
m
i
m
p
+−
−
+−−
⊥
+−+−
−=
=
−
−∑ ∑
λ
θλ
θθ
(7.1)
( )
( )
( ) ( )
( )thsh
thchww
h
J
A
k
J
i
n
i
n
i
n
i
n
i
n
n
m
i
mp
p
i
mn
ib
m
i
m
i
m
)(
)()()(
)(
1
)(
2),(2)(
),(
0
),(
2
+−
−
⊥
−−
−=
=
−
∑ ∑
λ
θλ
θθ
(7.2)
with i
ib
m
i
m ak )1(),1( −
⊥
+− =θ , i
ib
m
i
m ak )(),(
⊥
− =θ , n = 1, 2,… .
Eqs. (4,5) let us to express all the coefficients Аmp,
except one, for example, А10, via ±sw . Then Eqs. (7)
turn into the following inhomogeneous set of algebraic
equations for ±sw :
[ ]∑
+−+ +
−
+
+
−
+
s
i
s
i
ns
i
s
i
ns
i
n
i
n wTwTfw )1(),()(),()()( ~~~2~
[ ] =
−+ −
+
−−
−
+− )1(),1()(),1()( ~~~~ i
s
i
ns
i
s
i
ns
i
n wTwTf (8.1)
( ) ( )
−
−
−
=
−
−
+−
+−−
2),(
1
2
,
10
)(
10)(
2),1(
1
2
,1
10
)1(
10)(~6
i
n
ii
i
ni
n
ii
i
n
JA
f
JA
f
θλ
θ
θλ
θ
π ,
[ ]( +−+ ∑ −
+
−−
−
+−
−
s
i
s
i
ns
i
s
i
ns
i
n
i
n wTwTfw )1(),1()(),1()()( ~~~2~
[ ] =
−+ +
−
+
+
− )1(),()(),()( ~~~~ i
s
i
ns
i
s
i
ns
i
n wTwTf (8.2)
( ) ( )
−
−
−
=
−
−
+−
+−−
2),(
1
2
,
10
)(
10)(
2),1(
1
2
,1
10
)1(
10)( ~6
i
n
ii
i
ni
n
ii
i
n
JA
f
JA
f
θλ
θ
θλ
θ
π
where
( )
( )( )∑ ±±
−±±±
+±
−−
=
m i
ms
i
mn
m
i
m
i
m
i
m
i
iii
ns
EJ
b
a
T
2),(22),(2
1),(),(2
0
3),(
,1),(
θλθλ
χθθπ
,
( ) ( )
( )( ) ,
~
~
2),(22),(2
),(),(
0
),(
0
),(2),(
,1
),(
∑ ±
±±
+
±
−−
=
=
m
i
ms
i
mnm
i
m
i
m
i
m
i
m
i
m
i
ii
i
ns
EJJ
b
a
T
θλθλχ
θθθθπ
)(/)( )()()()( thshthchahf i
n
i
ni
i
n
i
n = ,
)(/~ )()()( thshahf i
ni
i
n
i
n = ,
146
=∆−
Ω−
Ω−
≠
Ω−
Ω−
=
+
±
++
±
+
±
++
±
±
1,
/
1,
/
2
,1
2),(
,1
2
,1
2),(
2
,1
2),(
,1
2
,1
2),(
),(
m
adcth
m
adcth
E
ii
i
m
iiii
i
mi
ii
i
m
iiii
i
mi
i
m
θ
θ
θ
θ
,
=∆−
Ω−
Ω−
≠
Ω−
Ω−
=
+
±
++
±−
+
±
++
±−
±
1,
/
1,
/
~
2
,1
2),(
,1
2
,1
2),(1
2
,1
2),(
,1
2
,1
2),(1
),(
m
adsh
m
adsh
E
ii
i
m
iiii
i
mi
ii
i
m
iiii
i
mi
i
m
θ
θ
θ
θ
,
( )2
,1
2),(
1,1 / ii
i
iii da +
±
+ Ω−=∆ θ , ca iiii /,1,1 ω++ =Ω ,
2/)(2
1 πλλχ mmm J= , )(
1
)( )(3~ i
ss
i
s wJw ±± = λπ .
Since ±sw are the expansion coefficients of
tangential at the disk aperture cross-section component
of the electric field, Eqs. (8) are the interaction
equations for the fields defined in the circular regions.
There are several interesting results that may be
obtained from Eqs. (8). One is that the fields of only
four circular regions interact directly: at the left and
right boundaries of the i-th disk aperture )(~ i
sw − and )(~ i
sw + ,
at the right boundary of the (i–1)-th disk aperture )1(~ −
+
i
sw
and at the left boundary of the (i+1)-th disk aperture
)1(~ +
−
i
sw . It follows from the fact that i–th cavity contacts
directly only with two neighboring cavities: (i–1)-th and
(i+1)-th. Another important result coming from Eqs. (8)
is that the interaction of fields in adjacent apertures is
described by the terms which contain factors
),1(),( ~,~ −−+ i
ns
i
ns TT . It can be shown that
0~,~ ),1(),( →−−+ i
ns
i
ns TT when ai → 0 and t = 0. At the
same time factors ),( −i
nsT and ),1( +−i
nsT in terms, which
describe fields interaction at the left and right
boundaries of single aperture, tend to constant values
independent of ai when ai → 0 and t = 0.
The solution to the set of linear algebraic equations
(8) can be written in the following form:
∑
−=
++
±± =
N
Nn
ninij
s
j
s Aw )(
10
),()(~ ζ . (9)
The choice of the value of N depends on the fact
how many couplings of the i-th cavity with another ones
we want to take into account. Index j takes the values i–
N+1, i–N+2, … , i–1, i, i+1, … , i+ N, correspondingly.
For paired couplings (each cavity is coupled only with
adjacent ones) N = 1. Substituting expression (9) in
Eqs. (8) one can obtain inhomogeneous set of linear
algebraic equations for ),( nij
s
+
±ζ :
[ ](∑ +−+ ++
−
++
+
−+
+
s
nij
s
i
ns
nij
s
i
ns
i
n
nij
n TTf ),1(),(),(),()(),( ~2 ζζζ [ ] =
− +−
+
−−+
−
+− ),1(),1(),(),1()( ~~ nij
s
i
ns
nij
s
i
ns
i
n TTf ζζ
( ) ( )
−
−
−
= −
−
+
+−
+−
+−
2),(
1
2
,
10,)(
2),1(
1
2
,1
10,1)(~6 i
n
i
niii
ni
n
i
niii
n
J
f
J
f
θλ
θδ
θλ
θδ
π , (10.1)
[ ](∑ +−
+
−−+
−
+−+
− −+
s
nij
s
i
ns
nij
s
i
ns
i
n
nij
n TTf ),1(),1(),(),1()(),( ~2 ζζζ [ ] =
−+ ++
−
++
+
− ),1(),(),(),()( ~~ nij
s
i
ns
nij
s
i
ns
i
n TTf ζζ
( ) ( )
−
−
−
=
−
−
+
+−
+−
+−
2),(
1
2
,
10,)(
2),1(
1
2
,1
10,1)( ~6
i
n
i
niii
ni
n
i
niii
n
J
f
J
f
θλ
θδ
θλ
θδ
π (10.2)
where n = -N … 0 … N; j = i–N+1, i–N+2, … , i–1, i,
i+1, … , i+ N.
From Eq. (9) one can deduce that the electric field
tangential component in the circular regions, through
which i-th cavity is connected with other elements of the
structure under consideration, are only determined via
the Е010-mode amplitudes in the cavities. Eq. (5) for Е010-
mode amplitude in the i-th cavity will have the form:
( )
( )),()(),1()(
)(
10
2)(
10
2)(
10
2)(
10
nii
i
inii
i
i
N
Nn
niiii AA
+
−−
++
++
−=
+
Λ−Λ
×=− ∑
εε
ωωω
(11)
where )(i
±ε are the well known coupling coefficients
derived on the basis of quasi-static approximation (see,
for example, [2]) which are given by
ii
iii
db
a
J 2
3
,1
1
2
1
)(
)(3
2 +
± =
λπ
ε . (12)
The coupling coefficients Λ have frequency
dependence. They are given by
( )∑ +
++
−+++
+
−
=Λ
s i
s
nii
sinii
i J
2),(
1
2
),1(
),(
10
),1(
θλ
ζ
θ , (13.1)
( )∑ −
+
+−+
−
−
=Λ
s i
s
nii
sinii
i J
2),(
1
2
),(
),(
10
),(
θλ
ζ
θ . (13.2)
It is necessary to choice the number of terms in sum
on n in Eq. (11) equal the number of equations (10).
Thus, the problem of coupled cavities has been
rigorously reduced to the problem of the coupling of
electric fields which are determined in circular regions.
147
Eqs. (11) are similar to the equations of equivalent
circuit analysis. Only one equation corresponds to each
cavity. Existence of infinite number of cavity modes
besides Е010-mode (non-resonant fields) affects on the
form of coupling coefficients Λ. Besides the term which
contains factor )(
10
iA the number of terms in the right-
hand side of Eqs. (11) depends on the fact how many
couplings of the considered cavity with another ones we
want to take into account.
3. CONCLUSIONS
Mathematical model of inhomogeneous chain of
cylindrical cavities is developed. This model combines
the explicitness of the model of equivalent coupled
circuit chain with the possibility to control rigorously
the influence of non-resonant fields and “long-range”
coupling of cavities on the structure characteristics.
Coupling coefficients can be calculated with definite
accuracy for the structure with arbitrary parameters.
Influence of non-resonant fields and “long-range”
coupling on the characteristics of the structure is taken
into account by calculation of the coupling coefficients.
REFERENCES
1. P.M. Lapostolle, A.L. Septier. Linear
accelerators. Amsterdam: “North-Holland
publishing company”, 1970, р. 86-89.
2. V.V. Vladimirsky. Coupling of hollow
electromagnetic resonators through small hole //
Journal of Technical Physics. 1947, v. 17, №11,
p. 1277-1282 (in Russian).
3. H.A. Bethe. Theory of diffraction by small
holes // Phys. Rev. 1944, v. 66, №7, p. 163-182.
4. M.I. Ayzatsky, E.Z. Biller. Development of
inhomogeneous disk-loaded accelerating
waveguides and RF-coupling. Proc. of the 18-th
ILAC, 1996, v. 1, p. 119-121.
5. M.I. Ayzatsky, K.Yu. Kramarenko.
Characteristics of inhomogeneous accelerating
structures // Problems of Atomic Science and
Technology. Series: Nuclear Physics
Investigations. 2004, №2(43), p. 69-71.
6. M.I. Ayzatsky. On two cavity coupling //
Journal of Technical Physics. 1996, v. 66, №9,
p. 137-147 (in Russian).
7. M.I. Ayzatsky. Analytical solutions in the
two-cavity coupling problem. Proc. of the 5-th
EPAC, 1996, v. 3, p. 2023-2025.
8. M.I. Ayzatsky. New mathematical model of
an infinite cavity chain. Proc. of the 5-th EPAC,
1996, v. 3, p. 2026-2028.
МАТЕМАТИЧЕСКАЯ МОДЕЛЬ НЕОДНОРОДНОЙ ЦЕПОЧКИ СВЯЗАННЫХ
ЦИЛИНДРИЧЕСКИХ РЕЗОНАТОРОВ
Н.И. Айзацкий, Е.Ю. Крамаренко
Разработана математическая модель для описания неоднородной цепочки связанных цилиндрических
резонаторов. Коэффициенты связи в неоднородной цепочке резонаторов могут быть рассчитаны с заданной
точностью для структуры с произвольными параметрами. Влияние нерезонансных полей и “дальних”
взаимодействий на характеристики структуры учитывается при расчете коэффициентов связи.
МАТЕМАТИЧНА МОДЕЛЬ НЕОДНОРІДНОГО ЛАНЦЮЖКА ЗВ’ЯЗАНИХ
ЦИЛІНДРИЧНИХ РЕЗОНАТОРІВ
М.І. Айзацький, К.Ю. Крамаренко
Розроблено математичну модель для опису неоднорідного ланцюжка зв’язаних циліндричних
резонаторів. Коефіцієнти зв’язку у неоднорідному ланцюжку резонаторів можуть бути розраховані з певною
точністю для структури з будь-якими параметрами. Вплив нерезонансних полів та “далеких” взаємодій на
характеристики структури враховується при розрахуванні коефіцієнтів зв’язку.
148
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
PACS: 29.17.+w
3. CONCLUSIONS
REFERENCES
Н.И. Айзацкий, Е.Ю. Крамаренко
М.І. Айзацький, К.Ю. Крамаренко
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