Modification of the coupled integral equations method for calculation of the accelerating structure characteristics
In this paper we present modification of coupled integral equations method (CIEM) for calculating the characteristics of the accelerating structures. In earlier developed CIEM schemes the coupled integral equations are derived for the unknown electrical fields at interfaces that divide the adjacent...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2022
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| Zitieren: | Modification of the coupled integral equations method for calculation of the accelerating structure characteristics / M.I. Ayzatsky // Problems of Atomic Science and Technology. — 2022. — № 3. — С. 56-61. — Бібліогр.: 10 назв. — англ. |
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| author | Ayzatsky, M.I. |
| author_facet | Ayzatsky, M.I. |
| citation_txt | Modification of the coupled integral equations method for calculation of the accelerating structure characteristics / M.I. Ayzatsky // Problems of Atomic Science and Technology. — 2022. — № 3. — С. 56-61. — Бібліогр.: 10 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | In this paper we present modification of coupled integral equations method (CIEM) for calculating the characteristics of the accelerating structures. In earlier developed CIEM schemes the coupled integral equations are derived for the unknown electrical fields at interfaces that divide the adjacent volumes. In addition to the standard division of the structured waveguide by interfaces between the adjacent cells, we propose to introduce new interfaces in places where electric field has the simplest transverse structure. Moreover, the system of coupled integral equations is formulated for longitudinal electrical fields in contrast to the standard approach where the transverse electrical fields are unknowns. The final vector equations contain expansion coefficients of the longitudinal electric field at these additional interfaces. This modification makes it possible to deal with a physical quantity that plays an important role in the acceleration of particles (a longitudinal electric field), and to obtain approximate equations for the case of a slow change in the waveguide parameters.
Представлено модифікацію методу зв’язаних інтегральних рівнянь для розрахунку характеристик прискорювальних структур. У раніше розроблених схемах зв’язані інтегральні рівняння формулюються для невідомих електричних полів на поверхнях розділу, що ділять суміжні об’єми. На додаток до стандартного поділу структурованого хвилеводу на межі розділу між сусідніми комірками пропонуємо ввести нові інтерфейси в місцях, де електричне поле має найпростішу поперечну структуру. Крім того, система зв’язаних інтегральних рівнянь сформульована для поздовжніх електричних полів на відміну від стандартного підходу, де поперечні електричні поля невідомі. Кінцеві векторні рівняння містять коефіцієнти розкладання поздовжнього електричного поля на цих додаткових поверхнях розділу. Ця модифікація дає змогу мати справу з фізичною величиною, яка відіграє важливу роль у прискоренні частинок (поздовжнє електричне поле), та отримати наближені рівняння для випадку повільної зміни параметрів хвилеводу.
Представлена модификация метода связанных интегральных уравнений для расчета характеристик ускоряющих структур. В разработанных ранее схемах связанные интегральные уравнения формулируются для неизвестных электрических полей на границах раздела, разделяющих соседние объемы. В дополнение к стандартному разделению структурированного волновода границами раздела между соседними ячейками предлагается ввести новые границы раздела в местах, где электрическое поле имеет простейшую поперечную структуру. Кроме того, система связанных интегральных уравнений формулируется для продольных электрических полей в отличие от стандартного подхода, когда поперечные электрические поля неизвестны. Окончательные векторные уравнения содержат коэффициенты разложения продольного электрического поля на этих дополнительных границах раздела. Эта модификация позволяет оперировать с физической величиной, играющей важную роль в ускорении частиц (продольным электрическим полем), и получить приближенные уравнения для случая медленного изменения параметров волновода.
|
| first_indexed | 2025-11-24T11:31:21Z |
| format | Article |
| fulltext |
56 ISSN 1562-6016. ВАНТ. 2022. №3(139)
THEORY AND TECHNOLOGY OF PARTICLE ACCELERATION
https://doi.org/10.46813/2022-139-056
MODIFICATION OF THE COUPLED INTEGRAL EQUATIONS
METHOD FOR CALCULATION OF THE ACCELERATING
STRUCTURE CHARACTERISTICS
M.I. Ayzatsky
National Science Center “Kharkov Institute of Physics and Technology”, Kharkiv, Ukraine
E-mail: aizatsky@kipt.kharkov.ua
In this paper we present modification of coupled integral equations method (CIEM) for calculating the characte-
ristics of the accelerating structures. In earlier developed CIEM schemes the coupled integral equations are derived
for the unknown electrical fields at interfaces that divide the adjacent volumes. In addition to the standard division
of the structured waveguide by interfaces between the adjacent cells, we propose to introduce new interfaces in
places where electric field has the simplest transverse structure. Moreover, the system of coupled integral equations
is formulated for longitudinal electrical fields in contrast to the standard approach where the transverse electrical
fields are unknowns. The final vector equations contain expansion coefficients of the longitudinal electric field at
these additional interfaces. This modification makes it possible to deal with a physical quantity that plays an impor-
tant role in the acceleration of particles (a longitudinal electric field), and to obtain approximate equations for the
case of a slow change in the waveguide parameters.
PACS: 02.10.Yn; 29.20.−c; 84.40.Az
INTRODUCTION
The main characteristic of the slow-wave accelerat-
ing structures is the distribution of the electric field in
both steady state and transient modes. This imposes
certain restrictions on the methods of calculating their
characteristics, manufacturing and tuning. The slow-
wave accelerating structures mainly belong to the class
of structured waveguides1 waveguides that consist of
similar, but not always identical, cells (disk-loaded wa-
veguides (DLW), chains of coupled resonators, etc.).
One of the effective approaches for calculating the
characteristics of structured waveguides is the coupled
integral equations method (CIEM) [1 - 5].
Based on a system of coupled integral equations, an
approximate method [6] is constructed for calculating
the characteristics of structured waveguides with slowly
varying dimensions [7]. It is the analog of classical Ei-
konal and WKB methods with taking into account not
only propagating waves, but also evanescent ones. The
advantage of this approach is the simple physical (but
not simple mathematical) interpretation of obtained eq-
uations and their solutions. This approximate method
was used to study the characteristics of the simplest case
of structured waveguide – a DLW with very thin diaph-
ragms [6, 7].
Analysis of the standard method of coupled integral
equations for studying the characteristics of DLWs with
real geometry showed that some modifications of the
standard approach can be useful.
In this paper we present such modification of
coupled integral equations method for calculating the
characteristics of the accelerating structures. In earlier
developed CIEM schemes the coupled integral equa-
tions are derived for the unknown electrical fields at
interfaces that divide the adjacent volumes. Usually
1 Accelerating structures on the base of waveguides
with dielectric can be smooth
these interfaces include geometrical singularities, such
as sharp edges. In this case it is needed to use special
basis functions.
In addition to the standard division of the structured
waveguide by interfaces between the adjacent cells, we
propose to introduce new interfaces in places where
electric field has the simplest transverse structure.
Moreover, the system of coupled integral equations is
formulated for longitudinal electrical fields in contrast
to the standard approach where the transverse electrical
fields are unknowns. The final vector equations contain
expansion coefficients of the longitudinal electric field
at these additional interfaces. This modification makes it
possible to deal with a physical quantity (longitudinal
electric field), which plays an important role in tuning
accelerator structures and particle acceleration, and to
obtain approximate equations for the case of a slow
change in the waveguide parameters
1. ACCELERATING STRUCTURE MODEL.
BASIC EQUATIONS
Consider a segment of DLW (circular corrugated
waveguide), the geometry of which is shown in Figure.
The right and left ends of segment are connected to
semi-infinite circular waveguides. All segment volumes
are filled with dielectric ( , 0i ). We
divide the DLW into subregions each of which is a cir-
cular waveguide. Unlike earlier works [1 - 3], we divide
each volume with large cross-section into two equal
subvolumes (in general, they can be different). Volumes
with large cross section will be numbered by the index
k (1 REZk N ), subvolumes – by 1k and 2k
( 1 2k k k ). A small cross-sectional volume placed
to the left of a large cross-sectional volume with an in-
dex k , will be numbered by the index k
( 1 1REZk N ).
ISSN 1562-6016. ВАНТ. 2022. №3(139) 57
We will consider only axially symmetric fields with
, ,z rE E H components (TM). Time dependence is
exp( )i t . Since we are interested in considering acce-
lerating structures, we must remember that it will be
necessary to take into account the beam loading. There-
fore, we will use initial expansions that are slightly dif-
ferent from the standard CIEM approach and give the
possibility to include current into consideration. In each
cylindrical volume (with index q ) we expand the elec-
tromagnetic field electromagnetic field in terms of the
complete orthogonal set of transverse functions
( ) ( )
, 0
( ) ( )
, 1
( ) ( )
. 1
, ,
, ,
, ,
q q m
z q z m
m q
q q m
r q r m
m q
q q m
q m
m q
E r z z E z J r
b
E r z z E z J r
b
H r z z H z J r
b
(1)
where 0 qz d , ( ) ( ) ( ) ( )Im 0, Re 0,q q q q
m m m m ,
0 ( ) 0mJ .
From Maxwell equations we obtain
2 ( ) ( )
, ,( )2 ( ) ( )2 ( )
, ,2
0 0
( )
,( ) ( )
, , 0( )2
( )
,( ) ( )
, ,( )2 2 ( )2
0
1 1
,
1
,
1
,
q q
r m z mq q q q m
m r m m r m
k
q
r mq qm
m z mq
km
q
r mq qm
z m z mq q
k m m
d E dI
E I
i i b dzdz
dE
H I i
b dz
dE i
E I
b dz c
(2)
where
2
( )
, 1( )
0 0
2
( )
, 0( )
0 0
( ) 2 2
1
1
, , ,
1
, , ,
k
k
b
k m
r m r kk
km
b
k m
z m z kk
km
k
m k m
I z j r z z J r rdrd
bW
I z j r z z J r rdrd
bW
W b J
(3)
The system of equations (2) is basic for the study
electromagnetic fields in accelerating sections.
In the semi-infinite waveguides the electromagnetic
field can be expanded in terms of the TM eigenmodes
( , ) ( , ),w p w p
s s
of a circular waveguide ( 1, 2p )
( , ) ( ) ( , ) ( ) ( , )w p p w p k w p
s s s s
s
H G G
, (4)
( , ) ( ) ( , ) ( ) ( , )w p p w p p w p
s s s s
s
E G G
. (5)
Chain of pieces of cylindrical waveguides that is connected with semi-infinite cylindrical waveguides
On the introduced interfaces we represent the elec-
tric fields as series of basis functions
1
2
( )( ) ( )
( 1 ) ( ) ( )
1
, /
, 0 /
kk r
r k s s k
s
k k r
r s s k
s
E r d C r b
E r C r b
(6)
1 2( ) ( ) ( ) ( ), / 2 , 0 /k k k z
z k z s s k
s
E r d E r Q r b . (7)
The boundary conditions for electric fields at the
junctions are written as
1
1
1
( )( ) ( )
, 1
( ) ( )
( )
, 1
/ , 0 ,
/ , 0 ,
0
0, ,
kk rm
r m k s s k k
m sk
k r
s s k k
k m s
r m
m k
k k
E d J r C r b r b
b
C r b r b
E J r
b b r b
(8)
1 2
2
2
2
( ) ( )
, 0 , 0
( ) ( )
( ) ( )
1 1
( )
, 1
1
( )( 1) ( )
, 1
1
/ 2 0
/ ,0 ,
/ , 0 ,
/ 2
0, ,
0 /
k km m
z m k z m
m mk k
k z
s s k k
s
k r
s s k k
k m s
r m k
m k
k k
kk rm
r m s s
m k
E d J r E J r
b b
Q r b r b
C r b r b
E d J r
b b r b
E J r C r b
b
1 1, 0 .k k
s
r b
(9)
Using the completeness and orthogonality of Bessel
functions 0
mJ r
b
and 1
mJ r
b
, it is easy to find
from (8),(9) coefficients of the left series. It should be
noted that that the boundary conditions (9) contain also
the longitudinal electric fields.
In the standard CIEM approach, the second group of
boundary conditions contains, as a rule, the continuity
of the tangential components of the magnetic field.
1
2
( )( )
, 1 , 1
( ) ( 1)
, 1 , 1 1
1
0 , 0 ,
/ 2 0 , 0 ,
kk m m
m k m k
m mk k
k km m
m k m k
m mk k
H d J r H J r r b
b b
H d J r H J r r b
b b
(10)
Multiplying the right and left sides of this relations
by a testing function /s kr b and integrating with
respect to r from 0 to kb , we get such equations
1
2
( )( ) ( , ) ( , )
, , , ,
( ) ( 1, ) ( 1) ( 1, 1)
, , , ,
0
/ 2 0
kk k k k k
m k s m m s m
m m
k k k k k k
m k s m m s m
m m
H d R H R
H d R H R
(11)
In our case, it is necessary to add additional condi-
tions for the continuity of the tangential components of
,
,
.
.
,
.
58 ISSN 1562-6016. ВАНТ. 2022. №3(139)
the electric field at the interfaces in the middle of vo-
lumes of large cross section
1
2 1 2
( )
, 1
( ) ( ) ( )
, 1 , ,
/ 2
0 / 2 0
k m
r m k
m k
k k km
r m r m k r m
m k
E d J r
b
E J r E d E
b
(12)
We will consider the case when the dimensions of
two semi-infinite waveguides are chosen such that only
the dominant mode 01TM propagates, and the higher-
order modes are all evanescent We will suppose that
there is an incident wave that travels from z with
amplitude (1)
1 1G ( (1) 0, 2sG s ).
Using the standard CIEM technique, we obtain such
system of vector equations
2 1
2 1
(1,1 ) (1 ) (2,1 ) ( ) ( ) ( ) ( ) ( )
( 1) ( )(1, ) (2, ) (2, , ) (1, , ) ( ) ( )
( ) ( 1)(1, , ) ( ) (2, 1) (2, 1, ) (1, 1) ( )
(( 1, )
,
2,...,
,
,
L L L L L
R
k kk k k k k k k k
k kk k k k k k k k
r k k
T C T C T C R Z
k N
T C T T C T Q Z
T Q T T C T C Z
T C
2 1
2
) ( )( , ) ( ) ( ) ( )
( )(2, 1 ) (1, 1 )( ) ( ) ( ) ( )
,
,RR R
k kr k k z k k k
NN NR R R R
T C T Q Z
T C T C T C Z
(13)
where ( )L
sC and ( )R
sC are the expansion coefficients of
the electric field tangential components at the left and
right interfaces between the DLW and the semi-infinite
waveguides. sZ (with different superscripts) are “cur-
rent” integrals that equal zero if current is absent. ,s sT
(with different superscripts) are such matrices
(1, ) ( , ) , , ,
, , ,( ) ( ) 2
1
( )
(2, ) ( , ) , , ,
, , ,( ) ( ) 2
1
2 ( )
(2, , )
, ( )
2
,
2
,
2 / 2
k k k r k kk
s s s m m sk k
mk m k m k m
k
m kk k k r k kk
s s s m m sk k
mk m k m k m
k
m kk k k
s s k
k m
b
T R R
b b sh d J
ch db
T R R
b b sh d J
sh db
T
b
( , ) , ( , )
, ,( ) 2
1
(1, , ) ( , ) ,
, , ,( ) 2
1
2
( , ) , ( , )
, ,2( ) ( )
( ) ,
, ,
,
/ 2
2 1
,
/ 2
,
2 / 2
,
k k r k k
s m m sk
m k m k m
k k k k z
s s s m m sk
m m m k m
r k k r k km k
m s m sk k
kk m m k
z k z
m s m s
R R
b ch d J
T R R
ch d J
b
T R
bb sh d
T R
(14)
where
1
, ( , ) ( )
, 1
0
/r k k r
m s s k m kR x J b x b xdx
,
1
, ( )
, 0
0
z z
m s s mR x J x xdx ,
1
( , )
, 1
0
/k k
s s s k s kR x J b x b xdx
.
Amplitudes of the eigen waves in the semi-infinite
waveguides are determined by the expansion coeffi-
cients ( )L
sC and ( )R
sC
1
1 1
1
1 1
2
2 2
2
(1) , ( )1 1
1 1,( )2 2
1 1 1
2
(1) , ( )1
,( )2 2
1
2
1(2) , ( )
,( )2 2
1
1 2 ,
2 , 2,3,...
2 , 1,2,...R
w L L
s sw
sw w
w L Ls
s s s sw
ss w s w
sN w R R
s s s sw
ss w s w
b
G R C
J b b
b
G R C s
J b b
b
G R C s
J b b
(15)
where
2 2
( , )2
2 2
,
w p s
s
w pb c
,
1
1
, ( )
, 1 1
0
/w L r
m s s m wR x J b x b xdx ,
2
1
, ( )
, 1 1
0
/
R
w R r
m s s m wN
R x J b x b xdx
.
For the numerical solution of system (13), it is ne-
cessary to limit the number of basis and testing func-
tions ( ) ( ), ,r z
s s s . We will suppose that
( ) 0, 0,r
s s rr r s N , ( ) 0,z
s zr s N .
Then we will have such sizes of defined matrices:
(1, ) (2, ) (2, , ), ,k k k kT T T are r rN N matrices, (1, , )k kT
are
r zN N matrices, ( , )r k kT
are a z rN N matrices, ( )
,
z k
m sT are
z zN N matrices.
2. INFINITIVE UNIFORM DISK LOADED
WAVEGUIDE
To demonstrate the difference between the standard
and the proposed approaches, consider an infinite ho-
mogeneous disk-loaded waveguide without current
( , , ,k k k kb a d t b b d d ).
If we omit the presence of boundaries for the uni-
form segment, we obtain from (13) the equations that
describe such waveguide. These difference equations in
the matrix form are written as
1 2
2 1
2 1
( ) ( 1)(2) (2) (1) (1) ( )
( ) ( 1)(2) (2) (1) (1) ( )
( ) ( ) ( ) 0
k k k
k k k
k kr r z k
T T C T C T Q
T T C T C T Q
T C T C T Q
, (16)
where T (with different superscripts) are complex ma-
trices, ( ) ( ),R zN Nk kC Q complex vectors.
Excluding 2( )kC and ( )kQ from (16), we get the
standard matrix difference equation [4,5]
1 1 1( ) ( 1) ( 1)k k kTC T C T C . (17)
We supposed that all matrices are invertible. The
size of matrices , , R RN NT T T is defined by the
number of basis functions ( ) /r
s kr b in the rE expan-
sion (6).
The difference equation (17) is not symmetric
( T T ) as it includes only vectors that describe the
fields on the left side of the volumes with large cross
section. These fields have a different “interaction” with
right and left neighbors. The absence of symmetry
.
,
,
ISSN 1562-6016. ВАНТ. 2022. №3(139) 59
makes it more difficult2 to apply a transformation [8, 9],
which gives simple method of finding Floquet coeffi-
cients and possibility to use the WKB approach [6, 7].
Eliminating 1( )kC and 2( )kC we can transform (16)
into a symmetric difference equation ( k )
( ) ( 1) ( 1)k k kTQ Q Q , (18)
where
111(2) (2) (1) (2) (2) (1)
1(1, (2) (2 (1)
11(2) (2) (1) (2) (2) (1) (1)2 .
r
z r
T T T T T T T
T
T T T T
T T T T T T T T T
(19)
The size of matrix z zN NT is defined by the
number of basis functions ( ) /z
s kr b in the zE expan-
sion (7). The rE expansion (6) contains RN basis func-
tions ( ) /r
s kr b . Such approach gives possibility to
improve the accuracy of rE representation (to increase
RN ) without increasing the size of matrix T ( zN ). It
should also be noted that matrix T is not Hermitian.
Using the transformation [6, 8]
( ) ( ,1) ( ,2)
( 1) (1) ( ,1) (2) ( ,2)
,
,
k k k
k k k
Q Q Q
Q M Q M Q
(20)
where
( ) ( )2 0i iTM M I , (21)
we get ( 1,2i )
( 1, ) ( ) ( , )k i i k iQ M Q . (22)
It can be shown that in our case3 the matrix T is
non-defective, and can be decomposed as
1,T U U (23)
where U is the matrix of eigen vectors sU and
1 2( , ,...)diag , s eigen values.
Then the solutions of quadratic matrix equations
(21) are ( 1,2i )
( ) ( ) 1i iM U U , (24)
where ( ) ( ) ( )
1 2( , ,...)i i idiag and ( )i
s are the solu-
tions of the characteristic equations
( )2 ( )
2(1)
2(2)
1 0,
/ 2 / 2 1,
/ 2 / 2 1.
i i
s s s
s s s
s s s
(25)
The matrices ( )iM have the same eigen vectors,
therefore they are commutative. As (1) (2) 1s s , the
matrices ( )iM satisfy the condition (1) (2)M M I . We
will suppose that (1)Re 1s ( (2)Re 1s ).
2 Matrix equations, whose solutions are necessary to construct the
WKB equations, become more complicated.
3 The infinitive uniform disk-loaded waveguide has 2 zN differ-
ent independent solutions (waves).
Representing the vector ( )kQ as the sum of two new
vectors ( ,1)kQ and ( ,2)kQ we did not assume that they are
individually solutions to the difference equation (18).
Let us show that when ( )iM are chosen as solutions to
Eqs. (20), the vectors ( ,1)kQ and ( ,2)kQ are independent
solutions to the equation (18).
If we know the radial distribution of longitudinal
components of electric fields in two consecutive sec-
tions of the waveguide ( (0) (1),Q Q ) then we can find
vectors (0,1) (0,2),Q Q
1(0,1) (2) (1) (2) (0) (1)
1(0,2) (2) (1) (1) (0) (1)
,
.
Q M M M Q Q
Q M M M Q Q
(26)
To find the solutions of equations (22) with condi-
tions (26) and the conditions at the infinity for all values
of k we have to consider the equations (22) for 0k
and 0k separately.
Then the solutions of the difference matrix equations
(22) with taking into account the conditions at the in-
finity are
( ,1) (1) (0,1)
( ,2) (2) (0,2)
, 0,
, 1.
k k
k k
Q M Q k
Q M Q k
(27)
Vectors (0)Q and (1)Q we can represent as a sum of
eigen vectors ( 0,1i )
( ) ( )i i
s s
s
Q A U . (28)
The matrix T is not Hermitian and the vectors sU
are not orthogonal. In this case
( ) 1 ( )
,
i i
s ss s
s
A U Q
. (29)
Substitution (29) into (26) gives
1(0,1) (2) (1) (2) (0) (1)
(2) (0) (1)
(2) (1)
1(0,2) (2) (1) (1) (0) (1)
(1) (0) (1)
(2) (1)
,
.
s s s s
s
s s s
s
s s s
s s s s
s
s s s
s
s s s
Q M M A A U
A A
U
Q M M A A U
A A
U
(30)
Then the solution of the equation (18) takes the form
(2) (1) (0) (1)
(2) (1)
(1) (2) (0) (1)
(2) (1)
( )
(2) (1) (0) (1)
(2) (1)
(1) (2) (0) (1)
(2) (1)
, 1.
, 0,1,
, 1.
k
s s s s
s
s s s
k
s s s s
s sk
sk
s s s s s
s s
k
s s s s
s
s s s
A A
U k
A A
Q U k
A A
A A
U k
(31)
For the case when (0)
mQ U and (1) (1)
m mQ U we
have (0)
,s s mA , (1) (1)
,s m s mA and
60 ISSN 1562-6016. ВАНТ. 2022. №3(139)
( )
(1)
0, 0
, 0
fw k
k
m m
k
Q
U k
(32)
For the case (1) (2)
m mQ U
(2)
( ) , 1,
0, 1.
k
bw k m mU k
Q
k
(33)
Therefore, the vector sequences ( )i k
s sU can be con-
sidered as forward ( 1i ) or backward ( 2i ) eigen
solutions of the equation (18).
It was shown [6], that the vector equation (22) can
be transformed into a difference equations for any com-
ponent of the vector ( , )k iQ . For a homogeneous wave-
guide these equations have the same form. Therefore, if
we choose basis function that fulfill a condi-
tion ( ) 0 1z
s (for example, 0
sJ r
b
), we can write
a difference equation of the 2 zN -order that connects the
values of the electric field ( ) ( ,1) ( ,2)k k k
z s s
s
E Q Q at
different points of the axis 0, ( ) / 2kr z k d t d
1 1,2 1,
( )2,1 2
,1 ,2
...
... ...
det 0
... ... ... ...
...
z
z z z
N
k
z
N N N
L T T
T L
E
T T L
, (34)
where the operator det
is defined on the base of rules of
common determinants
1 1,2
1 2 1,2 2,1
2,1 2
det ,
L T
L L T T
T L
(35)
,i i iL T
, ( ) ( 1)k kb b
and
( ) ( 1)k kb b
are shift operators. It was shown [6]
that equation (34) does not have spurious solutions as it
was for the equation based on a coupled cavities model
[10].
3. MODIFIED VECTOR EQUATIONS
The system of vector equations (13) can be trans-
formed to a system with only unknowns ( )kQ
1 2( ) ( )(1) (2) (1)
( ) ( ) ( ) ( 1) ( ) ( 1) ( )
1 ( 1) ( ) ( )
,
2,..., 1
,
,NREZ NREZREZ REZ REZ
Q Q Q
REZ
k k k k k k Q k
Q QN N Q N
T Q T Q Z
k N
T Q T Q T Q Z
T Q T Q Z
(36)
where the sizes of all T matrices are z zN N .
There are additional equations relating ( )(1) , zNQ Q ,
( ) ( ),L R
s sC C , from which we can calculate the reflection
and transmission coefficients (see (15)). Based on sys-
tem (36), a computer code has been developed. The
results of studying the characteristics of inhomogeneous
DLWs will be presented in subsequent papers.
System (36) is similar to that analyzed in [6] and,
therefore, can be the basis for deriving the WKB equa-
tions.
CONCLUSIONS
The presented approach to the description of inho-
mogeneous disk-loaded waveguides can be a useful tool
in studying the properties of slow wave system. Pro-
posed modification of the coupled integral equations
method makes it possible to deal directly with a longi-
tudinal electric field and to obtain approximate equa-
tions for the case of a slow change in the waveguide
parameters.
REFERENCES
1. J. Esteban, J.M. Rebollar. Characterization of corru-
gated waveguides by modal analysis // IEEE Trans-
actions on Microwave Theory and Techniques.
1991, v. 39(6), p. 937-943
2. S. Amari, J. Bornemann, and R. Vahldieck. Accurate
analysis of scattering from multiple waveguide dis-
continuities using the coupled integral equation
technique // J. Electromag. Waves Applicat. 1996,
v. 10, p. 1623-1644.
3. J. Bornemann, S. Amari, and R. Vahldieck. Analysis
of waveguide discontinuities by the coupled-integral
equations technique // Recent Res. Devel. Micro-
wave Theory and Techniques. 1999, v. 1, p. 25-35.
4. S. Amari, R. Vahldieck, J. Bornemann. Analysis of
propagation in periodically loaded circular wave-
guides // IEE Proceedings Microwaves Antennas
and Propagation 1999, v. 146, № 1, p. 50-54.
5. S. Amari, R. Vahldieck, J. Bornemann, and
P. Leuchtmann. Spectrum of corrugated and periodi-
cally loaded waveguides from classical matrix ei-
genvalues // IEEE Transactions on Microwave
Theory and Techniques. 2000, v. 48, № 3, p. 453-
459.
6. M.I. Ayzatsky. Model of finite inhomogeneous cavi-
ty chain and approximate methods of its analysis//
Problems of Atomic Science and Technology. 2021,
№ 3, p. 28-37; Modelling of inhomogeneous disk-
loaded waveguides: matrix difference equations and
WKB approximation// https://arxiv.org/abs/2010.
10349, 2020.
7. M.I. Ayzatsky. Inhomogeneous travelling-wave ac-
celerating sections and WKB approach // Problems
of Atomic Science and Technology. 2021, № 4,
p. 43-48; http://arxiv.org/abs/2103.10664, 2021.
8. M.I. Ayzatsky. Transformation of the linear differ-
ence equation into a system of the first order differ-
ence equations // https: //arxiv.org/abs/1806.04378,
2018.
9. M.I. Ayzatsky. A note on the WKB solutions of dif-
ference equations // https://arxiv.org/abs/1806.02196,
2018.
10. M.I. Ayzatsky. Nonlocal Equations for the Electro-
magnetic Field in the Inhomogeneous Accelerating
Structures // Problems of Atomic Science and Tech-
nology. 2019, № 6, p. 71-76; Nonlocal Equations for
the Electromagnetic field in the Coupled Cavity
Model //https://arxiv.og/abs/1810.10382, 2018.
Article received 19.01.2022
.
,
ISSN 1562-6016. ВАНТ. 2022. №3(139) 61
МОДИФІКАЦІЯ МЕТОДУ ЗВ'ЯЗАНИХ ІНТЕГРАЛЬНИХ РІВНЯНЬ
ДЛЯ РОЗРАХУНКУ ХАРАКТЕРИСТИК ПРИСКОРЮВАЛЬНОЇ СТРУКТУРИ
М.І. Айзацький
Представлено модифікацію методу зв'язаних інтегральних рівнянь для розрахунку характеристик прис-
корювальних структур. У раніше розроблених схемах зв’язані інтегральні рівняння формулюються для неві-
домих електричних полів на поверхнях розділу, що ділять суміжні об’єми. На додаток до стандартного поді-
лу структурованого хвилеводу на межі розділу між сусідніми комірками пропонуємо ввести нові інтерфейси
в місцях, де електричне поле має найпростішу поперечну структуру. Крім того, система зв'язаних інтеграль-
них рівнянь сформульована для поздовжніх електричних полів на відміну від стандартного підходу, де по-
перечні електричні поля невідомі. Кінцеві векторні рівняння містять коефіцієнти розкладання поздовжнього
електричного поля на цих додаткових поверхнях розділу. Ця модифікація дає змогу мати справу з фізичною
величиною, яка відіграє важливу роль у прискоренні частинок (поздовжнє електричне поле), та отримати
наближені рівняння для випадку повільної зміни параметрів хвилеводу.
МОДИФИКАЦИЯ МЕТОДА СВЯЗАННЫХ ИНТЕГРАЛЬНЫХ УРАВНЕНИЙ
ДЛЯ РАСЧЕТА ХАРАКТЕРИСТИК УСКОРЯЮЩЕЙ СТРУКТУРЫ
Н.И. Айзацкий
Представлена модификация метода связанных интегральных уравнений для расчета характеристик уско-
ряющих структур. В разработанных ранее схемах связанные интегральные уравнения формулируются для
неизвестных электрических полей на границах раздела, разделяющих соседние объемы. В дополнение к
стандартному разделению структурированного волновода границами раздела между соседними ячейками
предлагается ввести новые границы раздела в местах, где электрическое поле имеет простейшую попереч-
ную структуру. Кроме того, система связанных интегральных уравнений формулируется для продольных
электрических полей в отличие от стандартного подхода, когда поперечные электрические поля неизвестны.
Окончательные векторные уравнения содержат коэффициенты разложения продольного электрического
поля на этих дополнительных границах раздела. Эта модификация позволяет оперировать с физической ве-
личиной, играющей важную роль в ускорении частиц (продольным электрическим полем), и получить при-
ближенные уравнения для случая медленного изменения параметров волновода.
|
| id | nasplib_isofts_kiev_ua-123456789-195392 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-24T11:31:21Z |
| publishDate | 2022 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Ayzatsky, M.I. 2023-12-05T09:45:15Z 2023-12-05T09:45:15Z 2022 Modification of the coupled integral equations method for calculation of the accelerating structure characteristics / M.I. Ayzatsky // Problems of Atomic Science and Technology. — 2022. — № 3. — С. 56-61. — Бібліогр.: 10 назв. — англ. 1562-6016 PACS: 02.10.Yn; 29.20.−c; 84.40.Az https://nasplib.isofts.kiev.ua/handle/123456789/195392 In this paper we present modification of coupled integral equations method (CIEM) for calculating the characteristics of the accelerating structures. In earlier developed CIEM schemes the coupled integral equations are derived for the unknown electrical fields at interfaces that divide the adjacent volumes. In addition to the standard division of the structured waveguide by interfaces between the adjacent cells, we propose to introduce new interfaces in places where electric field has the simplest transverse structure. Moreover, the system of coupled integral equations is formulated for longitudinal electrical fields in contrast to the standard approach where the transverse electrical fields are unknowns. The final vector equations contain expansion coefficients of the longitudinal electric field at these additional interfaces. This modification makes it possible to deal with a physical quantity that plays an important role in the acceleration of particles (a longitudinal electric field), and to obtain approximate equations for the case of a slow change in the waveguide parameters. Представлено модифікацію методу зв’язаних інтегральних рівнянь для розрахунку характеристик прискорювальних структур. У раніше розроблених схемах зв’язані інтегральні рівняння формулюються для невідомих електричних полів на поверхнях розділу, що ділять суміжні об’єми. На додаток до стандартного поділу структурованого хвилеводу на межі розділу між сусідніми комірками пропонуємо ввести нові інтерфейси в місцях, де електричне поле має найпростішу поперечну структуру. Крім того, система зв’язаних інтегральних рівнянь сформульована для поздовжніх електричних полів на відміну від стандартного підходу, де поперечні електричні поля невідомі. Кінцеві векторні рівняння містять коефіцієнти розкладання поздовжнього електричного поля на цих додаткових поверхнях розділу. Ця модифікація дає змогу мати справу з фізичною величиною, яка відіграє важливу роль у прискоренні частинок (поздовжнє електричне поле), та отримати наближені рівняння для випадку повільної зміни параметрів хвилеводу. Представлена модификация метода связанных интегральных уравнений для расчета характеристик ускоряющих структур. В разработанных ранее схемах связанные интегральные уравнения формулируются для неизвестных электрических полей на границах раздела, разделяющих соседние объемы. В дополнение к стандартному разделению структурированного волновода границами раздела между соседними ячейками предлагается ввести новые границы раздела в местах, где электрическое поле имеет простейшую поперечную структуру. Кроме того, система связанных интегральных уравнений формулируется для продольных электрических полей в отличие от стандартного подхода, когда поперечные электрические поля неизвестны. Окончательные векторные уравнения содержат коэффициенты разложения продольного электрического поля на этих дополнительных границах раздела. Эта модификация позволяет оперировать с физической величиной, играющей важную роль в ускорении частиц (продольным электрическим полем), и получить приближенные уравнения для случая медленного изменения параметров волновода. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Theory and technology of particle acceleration Modification of the coupled integral equations method for calculation of the accelerating structure characteristics Модифікація методу зв’язаних інтегральних рівнянь для розрахунку характеристик прискорювальної структури Модификация метода связанных интегральных уравнений для расчета характеристик ускоряющей структуры Article published earlier |
| spellingShingle | Modification of the coupled integral equations method for calculation of the accelerating structure characteristics Ayzatsky, M.I. Theory and technology of particle acceleration |
| title | Modification of the coupled integral equations method for calculation of the accelerating structure characteristics |
| title_alt | Модифікація методу зв’язаних інтегральних рівнянь для розрахунку характеристик прискорювальної структури Модификация метода связанных интегральных уравнений для расчета характеристик ускоряющей структуры |
| title_full | Modification of the coupled integral equations method for calculation of the accelerating structure characteristics |
| title_fullStr | Modification of the coupled integral equations method for calculation of the accelerating structure characteristics |
| title_full_unstemmed | Modification of the coupled integral equations method for calculation of the accelerating structure characteristics |
| title_short | Modification of the coupled integral equations method for calculation of the accelerating structure characteristics |
| title_sort | modification of the coupled integral equations method for calculation of the accelerating structure characteristics |
| topic | Theory and technology of particle acceleration |
| topic_facet | Theory and technology of particle acceleration |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/195392 |
| work_keys_str_mv | AT ayzatskymi modificationofthecoupledintegralequationsmethodforcalculationoftheacceleratingstructurecharacteristics AT ayzatskymi modifíkacíâmetoduzvâzanihíntegralʹnihrívnânʹdlârozrahunkuharakteristikpriskorûvalʹnoístrukturi AT ayzatskymi modifikaciâmetodasvâzannyhintegralʹnyhuravneniidlârasčetaharakteristikuskorâûŝeistruktury |