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In the article, based on solving the equations of vacuum technology, an iterative algorithm for calculating vacuum conductivity and the geometric parameters of a curvilinear channel for transporting a short-focus electron beam is proposed and studied. For such a type of channel, the dependence of it...
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| author | Melnyk, Igor Pochynok, Alina Skrypka, Mykhailo |
| author_facet | Melnyk, Igor Pochynok, Alina Skrypka, Mykhailo |
| author_institution_txt_mv | [
{
"author": "Igor Melnyk",
"institution": "National Technical University of Ukraine \"Igor Sikorsky Kyiv Polytechnic Institute\", Kyiv"
},
{
"author": "Alina Pochynok",
"institution": "Research Institute of Electronics and Microsystem Technology of the National Technical University of Ukraine \"Igor Sikorsky Kyiv Polytechnic Institute\", Kyiv"
},
{
"author": "Mykhailo Skrypka",
"institution": "National Technical University of Ukraine \"Igor Sikorsky Kyiv Polytechnic Institute\", Kyiv"
}
] |
| author_sort | Melnyk, Igor |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2025-05-20T17:56:07Z |
| description | In the article, based on solving the equations of vacuum technology, an iterative algorithm for calculating vacuum conductivity and the geometric parameters of a curvilinear channel for transporting a short-focus electron beam is proposed and studied. For such a type of channel, the dependence of its radius on the longitudinal coordinate is described by a power function. The proposed algorithm is based on the numerical solution of a set of nonlinear equations using the Steffensen method. The results of the test calculations are presented. The provided tests confirm the stability of the proposed algorithm’s convergence for correct pressure and pumping speed values in electron-beam technological equipment. Such curved transport channels can be used in electron beam equipment based on high-voltage glow discharge electron guns intended for welding, melting metals, and the deposition of thin films. The criterion for the optimal geometry of a nonlinear channel is the minimum power loss of the electron beam during its transportation while ensuring the required pressure drop between the discharge and technological chambers. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2025.1.05 |
| first_indexed | 2025-07-17T10:28:43Z |
| format | Article |
| fulltext |
Publisher IASA at the Igor Sikorsky Kyiv Polytechnic Institute, 2025
Системні дослідження та інформаційні технології, 2025, № 1 53
UDC 004.942:537.525:621.325
DOI: 10.20535/SRIT.2308-8893.2025.1.05
NUMERICAL ALGORITHM FOR CALCULATION
OF THE VACUUM CONDUCTIVITY OF A NON-LINEAR
CHANNEL FOR TRANSPORTING A SHORT-FOCUS ELECTRON
BEAM IN THE TECHNOLOGICAL EQUIPMENT
I. MELNYK, A. POCHYNOK, M. SKRYPKA
Abstract. In the article, based on solving the equations of vacuum technology, an it-
erative algorithm for calculating vacuum conductivity and the geometric parameters
of a curvilinear channel for transporting a short-focus electron beam is proposed and
studied. For such a type of channel, the dependence of its radius on the longitudinal
coordinate is described by a power function. The proposed algorithm is based on the
numerical solution of a set of nonlinear equations using the Steffensen method. The
results of the test calculations are presented. The provided tests confirm the stability
of the proposed algorithm’s convergence for correct pressure and pumping speed
values in electron-beam technological equipment. Such curved transport channels
can be used in electron beam equipment based on high-voltage glow discharge elec-
tron guns intended for welding, melting metals, and the deposition of thin films. The
criterion for the optimal geometry of a nonlinear channel is the minimum power loss
of the electron beam during its transportation while ensuring the required pressure
drop between the discharge and technological chambers.
Keywords: electron beam, electron beam technologies, electron beam transporta-
tion, nonlinear electron beam transportation channel, vacuum conductivity of the
transportation channel, high-voltage glow discharge electron gun, vacuum technol-
ogy equation, set of nonlinear equations, Steffensen method.
INTRODUCTION
The development of electron beam technologies is very important today, and such
advanced technologies are widely applied in different branches of industry, in-
cluding metallurgy, mechanical engineering, energetic industry, electronics, in-
strument-making industry, automotive industry, as well as aircraft and space in-
dustry [1–44]. Generally, it is caused by the many important advantages of the
electron beam as a technological instrument, which are as follows [1–10].
1. The total power and power density of the electron beam are extremely
high. Naturally, in industrial technological electron guns, the total power can
reach hundreds of kW, and the power density can be up to 109 W/m2.
2. Ease of control and changing of the geometric and energy focal parame-
ters of the electron beam using electric and magnetic fields.
3. Carrying out a technological operation under conditions of medium and
high vacuum, which ensures the repeatability of technological process parameters
during their control, the purity of the treated materials, and, as a result, the high
quality of products.
In particular, electron beam methods for purifying refractory metals and ce-
ramic materials are widely used today in metallurgy [36–41]. For example, with
I. Melnyk, A. Pochynok, M. Skrypka
ISSN 1681–6048 System Research & Information Technologies, 2025, № 1 54
the development of modern electronics, it is now very important to obtain pure
silicon for use in electronics for the production of effective and high-quality mi-
crochips [36]. Advanced technologies for refining refractory metals are very im-
portant for the production of reliable details for the automotive, aircraft, and space
industries [40; 41].
Recently, in the mechanical engineering, aviation, and space industries, elec-
tron beam technologies for three-dimensional metal printing have found wide ap-
plication and are gradually becoming quite cheap, highly efficient, and allow the
economical use of metal raw materials and electricity [21–25]. Such technologies
also make it possible to obtain high-strength and reliable parts for the chemical
industry, aviation, and space industries. Usually, the mechanical and chemical
properties of metals produced using three-dimensional printing are unique, and it
is really impossible to obtain metals of such quality using traditional metallurgy
methods [42–44].
In the electronics industry, it is effective to use the technology of welding
with point-focus [14–19] and profile [20–22] electron beams for sealing the hous-
ings of electronic devices and welding metal and ceramic contacts. For example,
in papers [14; 15], the possibility of peer-reviewed welding of contacts of cryo-
genic electronic devices with a short-duration pulsed point-focus electron beam
has been considered [16–19]. Profile beam welding is a very low-cost technology
that provides high productivity and can be easily automated [20–22].
Another effective use of electron beam technologies in modern production is
the application of stoichiometric ceramic coatings, which contain active gases, in
particular oxides, carbides, sulfides, nitrides, etc. [23–32]. For example, multi-
layer coatings made of rare earth metal oxides are effective for forming insulating
coatings on electric vehicle contacts, as well as heat-protective coatings for inter-
nal combustion engines and jet engines [23–32]. Carbide and sulfide dielectric
films are used in microelectronics for the manufacture of high-quality capacitors
as well as for transmitting and receiving devices for communication microwave
electronic equipment [34; 35]. It was shown in the papers [11; 12; 23–32], that the
best way to obtain such coatings is electron beam evaporation in a vacuum with
stimulation of a chemical reaction between metal vapor and the residual gas by
igniting an auxiliary low-voltage discharge.
It is clear, that the main part of any electron beam technological equipment
is an electron gun, which ensures the generation of an electron beam with speci-
fied energy and geometric parameters for predetermined conditions of the techno-
logical process. Therefore, when one designing the electron beam technological
installation, an extremely important engineering aspect is always the coordination
of the physical operating conditions of the electron gun with the parameters of the
technological process that is being performed [1–10]. It is especially important to
ensure an appropriate pressure range in the area of electron beam formation and in
the area of technological operation.
In general, it should be noted that today the elaboration of electron guns for
technological use is carried out mainly in two directions: improving the designs of
traditionally used guns with heated cathodes [1–10] and the development of elec-
tron guns, the generation of beams in which is carried out in a fundamentally dif-
ferent way, for example, through field emission [1–10], photoemission [1–10] or
the ignition of various types of gas discharges [11–13; 45–47].
Numerical algorithm for calculation of the vacuum conductivity of a non-linear channel for …
Системні дослідження та інформаційні технології, 2025, № 1 55
HIGH-VOLTAGE GLOW DISCHARGE ELECTRON GUNS
AND THE PARTICULARITIES OF THEIR OPERATION IN INDUSTRIAL
TECHNOLOGICAL EQUIPMENT
In the papers [11–13; 23–32], it was pointed out that in the physical conditions of
low vacuum, on the order of 0.1–10 Pa, the High-Voltage Glow Discharge
(HVGD) electron guns usually operate stably and very reliably. The undeniable
advantages of such types of electron sources in comparison with traditionally used
electron guns with heated cathodes are the following [11–13; 23–32].
1. The ability of HVGD electron guns to operate in a soft and medium vac-
uum in the environment of various gases, in particular noble and active ones. This
makes it easy to coordinate the gun parameters with the required parameters of
the technological process. Typically, the pressure in the discharge chamber of the
gun lies in the range of 1–10 Pa. For welding and melting electron beam equip-
ment, it leads to a very important technical and economic effect of simplification
of technological installations [16; 17], and the technological process of deposition
of high-quality ceramic films in the medium of active gas is generally extremely
difficult to implement without the use of HVGD electron guns [23–32]. To coor-
dinate the operating parameters of HVGD electron guns and in the area where
technological operation is being performed, pressure decoupling is used through
an electron beam transport channel with a limited radius. This makes it possible to
maintain the required value of pressure in the HVGD combustion area and in the
technological chamber with a controlled injection of gas into the electron gun and
continuous pumping of the technological chamber [48; 49].
2. The simplicity of the design of HVGD electron guns and the possibility of
their assembly and disassembly in order to replace used components, in particular,
the HVGD cold cathode [11; 12]. Rough and precise estimates of the operating
physical conditions of the cold HVGD cathode and its surface temperature are
given in the papers [16; 17].
3. Ease of debugging HVGD electron guns and ensuring their operation as
part of technological electron beam installation [11; 12]. The only point related to
the complexity of assembling HVGD electron guns is ensuring the alignment of
the design parts [21; 22].
4. The relative simplicity of vacuum evacuation technological equipment,
since there is no need to ensure the operation of the HVGD electron guns under
high vacuum physical conditions [11; 12].
5. Ease of regulation of the electron beam power at a stable value of the ac-
celerating voltage. There are two ways existed for control the power of the elec-
tron beam: aerodynamic, through a controlled change of pressure in the gap of
HVGD lighting by regulating the gas flow [48; 49], and electrical, through the
ignition of an auxiliary discharge and changing the ion concentration in the anode
plasma [17–19]. A generalized description of the operating algorithm of the digi-
tal current control system of the HVGD electron gun based on the methods of dis-
crete mathematics and the theory of finite state machines is given in [50].
THE STATE OF DEVELOPMENT OF TECHNOLOGICAL EQUIPMENT
WITH HIGH-VOLTAGE GLOW DISCHARGE ELECTRON GUNS AND THE
CONSIDERED PROBLEM OF SIMULATION OF TRANSPORT CHANNEL
However, despite the technical and economic advantages of HVGD electron guns
described above, there are also certain technical difficulties associated with their
I. Melnyk, A. Pochynok, M. Skrypka
ISSN 1681–6048 System Research & Information Technologies, 2025, № 1 56
industrial application. They are primarily related to the coordination of the physi-
cal operating conditions of HVGD guns with the pressure parameters of the tech-
nological process [11; 12]. For welding electron guns and guns intended for melt-
ing metals and ceramic materials for the purpose of cleaning them, it is very
important to separate the HVGD combustion zone from the zone of product proc-
essing [16; 17; 36–41], and for the process of deposition ceramic coatings, it is
important to ensure the required pressure in the HVGD combustion zone and in
the area of metal’s vapor interaction with the operation gas [23–32]. Therefore,
usually, to ensure stable operation of HVGD electron guns as part of technologi-
cal equipment, a pressure decoupling is used between the HVGD combustion area
and the area of the technological operation. For this purpose, electron beam trans-
port channels with a limited cross-sectional radius are usually used. Then the
pressure difference between the HVGD region and technological chamber is en-
sured through a controlled injection of gas into the electron gun and continuous
pumping of the technological chamber. The corresponding block diagram of the
electron beam technological installation is given in [48; 49], in the simplified
form it is presented at Fig. 1 [48; 49].
Typically, transport channels of cylindrical and conical cross-sections are
used to guide a short-focus electron beam. Corresponded relations of vacuum
technology for calculation the conductivity of such channels relative to the differ-
ence of pressure and gas flow are given in the manual books [51–57].
The aim of this paper is to analyze the possibilities of using transport chan-
nels with a nonlinear cross-section. An assessment of the vacuum conductivity of
a nonlinear channel with power dependence of its radius r on longitudinal coordi-
nate z under the physical conditions of carrying out technological operations for the
deposition of ceramic films and coatings has been provided. For electron beam
12 Control system
5
6
8
Technological
gas
Gas inlet system
7
pa
1
3
High Voltage
Power Source pg
4
Evacuation system
2
9
Outlet
pch
10 11
pm
Outlet flow
Fig. 1. Block diagram of the pumping and power supply system for an electron beam
installation with a HVGD gun: 1 — HVGD electron gun; 2 — channel for guiding an
electron beam into the technological chamber; 3 — gun current sensor; 4 — high voltage
power supply; 5 — electromagnetic valve for inputting gas into the HVGD electron gun;
9 — technological chamber; 10 — pumping channel; 11 — vacuum pump; 12 — elec-
tronic or microcontroller system for automatically changing the gun current; pa — at-
mospheric pressure; pch — pressure in the technological chamber; pg — pressure in the
HVGD electron gun; pm — minimum pressure in the technological chamber that can be
provided by applied pumping means
Numerical algorithm for calculation of the vacuum conductivity of a non-linear channel for …
Системні дослідження та інформаційні технології, 2025, № 1 57
welding and melting equipment, similar estimates can also be used [1–10; 51–57].
For providing generalized analyze of considered nonlinear function the standard
mathematical approaches of function theory and methods for calculating integrals
of power functions has been applied [58; 59]. To determine the conductivity of
the vacuum channel, well-known equations of vacuum technology were used, and
the required length of the channel was determined by numerically solving the
complex nonlinear equation using the Steffensen method [60–64].
BASIC ANALYTICAL RELATIONS AND FORMALIZING THE
OPTIMIZATION TASK FOR CHOOSING THE GEOMETRY PARAMETERS OF
A NON-LINEAR TRANSPORT CHANNEL
In the general case, to determine the pressure distribution along the length of an
axisymmetric transportation channel under the condition of the molecular regime
of gas flow in it, the basic equation of vacuum technology and the Knudsen equa-
tion are used [47–53]. The corresponding system of algebraic and integral equa-
tions is written in general form as follows [47–53]:
,
π
8
;
3
4
= ; 0
0
2
M
TR
v
S
Hdl
v
U
SU
Up
p
chl
cr
ch
pch
chg
ch
(1)
where chU is the vacuum conductivity of the channel; H is the perimeter of the
transportation channel in cross section; crS is its area; 0R is the universal gas
constant; T is the gas temperature; M is its molecular weight of gas atoms; gp
and chp are the pressure in the gun volume and in the technological chamber,
according to Fig. 1; chU is the vacuum conductivity of the transportation channel;
pS is the speed of the pumping system; v is the average thermal velocity of
movement of gas molecules; chl is the length of the channel.
At a relatively high pressure in the discharge chamber of the HVGD gun, an
intermediate gas flow regime is observed in the electron beam transport channel.
In this case, to calculate the channel conductivity, the corresponding correction
factor J is introduced [47–53]:
, ;
)(2361
))((2653)(2021
outin
2
outinoutin
mi JUU
RR
RRpRR
J
(2)
Using relations (1), it is possible to determine analytically the vacuum conductiv-
ity of where the index m corresponds to the molecular gas flow regime one and
the index i for the intermediate one; inR — input radius of beam transporting
channel; outR — it output radius correspondently; p is the average pressure in
the transportation channel; J is a semi-empirical coefficient for listing the con-
ductivity value.
a nonlinear channel for transporting an electron beam with an exponent of
/1 . The cross section of such a channel, depending on the longitudinal coordi-
nate z , is determined as follows:
I. Melnyk, A. Pochynok, M. Skrypka
ISSN 1681–6048 System Research & Information Technologies, 2025, № 1 58
,)()( )/1(
0
zzAzr (3)
where
,
)/1(
α
in
α
out
RR
l
A ch
,
)( α
in
α
out
α
in
α
in
0
chl
RRR
A
R
z
(4)
or
.
)(
)(
)/1(
α
in
α
out
α
in
α
in
α
out
ch
cn
l
RRR
z
RR
l
zr (5)
Taking into account the well-known rules of integration of exponential func-
tion, solving of integral equation of system (1) giving the following result:
,3=α,
))(ln)((lnπ3
8
;3α,
)(π3
α
3α
8
=)α,,,(
00
α
3α
0
α
3α
0outin
zzlA
v
zzlA
v
lRRU
ch
chchch
or in the form of arithmetic-logic relation [65]:
))(ln)((ln
1
3=α
π3
8
=α,,,
00
outin zzlA
v
lRRU
ch
chch
.
)(
α
3α
))3>α(|)3<α((
α
3α
0
α
3α
0
zzlA ch
(6)
Taking into account relations (3), (4), as well as first relation of set of equa-
tions (1), one can obtain the following nonlinear relation for the value chl rela-
tively to transversal variable z :
,3=α,
))ln()(ln(π3
)(8
;3α,
)(π3
)(
α
3α
8
=
00
α
3α
0
α
3α
0
zzlSAJp
ppv
zzlSAJp
ppv
zl
chpch
gch
chpch
gch
ch
or, in the form of arithmetic-logic relation [65]:
Numerical algorithm for calculation of the vacuum conductivity of a non-linear channel for …
Системні дослідження та інформаційні технології, 2025, № 1 59
α
3α
0
α
3α
0 )(
α
3α
)3>(α|)3<(α
π3
)(8
=(z)
zzlA
SJp
ppv
l
ch
pch
gch
ch
.
)ln()ln(
1
)3=α(
00
zzlA ch
(7)
In further considerations, let us assume that 3 α . Therefore, the special
case 3 α in arithmetic-logic relation (7) is out of consideration. To solve the
nonlinear equation systems (5), (6) with respect to the parameter chl , let we intro-
duce the corresponding auxiliary variables:
. ; α
in
α
out RbRa (8)
With this substitution the analytical relations (4), (5) are rewritten as follows:
.0)))((()))((( αα2 chzlazrzrabb (9)
Clear, that (9) is the quadratic equation relatively to parameter b and for
fixed value of a and knowing set of (r, z) coordinate it can be solved analytically.
Let formulate the task of finding nonlinear channel parameters as the task of
optimization [61; 62]. Assume, that optimization is provided by four geometry
parameters of channel, namely: inR , outR , α and chl . In such conditions for four
basic points ), ( 111 zrP , ), ( 222 zrP , ), ( 333 zrP and ), ( 444 zrP the relations (8), (9)
are rewritten as the following set of nonlinear equations, which can be solved
numerically:
.0)))((()))(((
;0)))((()))(((
;0)))((()))(((
;0)))((()))(((
4
α
44out
α
44
α
out
α
in
2α
in
3
α
33out
α
33
α
out
α
in
2α
in
2
α
22out
α
22
α
out
α
in
2α
in
1
α
11out
α
11
α
out
α
in
2α
in
ch
ch
ch
ch
lzzrRzrRRR
lzzrRzrRRR
lzzrRzrRRR
lzzrRzrRRR
(10)
The criterium of optimization for the task of beam transporting in nonlinear
guiding channel is minimum current loses of electron beam in the case of provid-
ing required difference of pressure in the electron gun and in the technological
chamber chg pp , correspond to Fig. 1 [1–10]. For defining current losses in the
transporting channel, the boundary trajectory of electron beam has to be calcu-
lated. In such case beam losses on the small elementary range of longitudinal co-
ordinate dz are defined as follows [1–10]:
;
)(β
)(
exp
)(π
)(
;
)(π2
)(
)(β
2
02
0
n
nr
j
nr
nI
nj
nI
n
b
b
b
bb
b
22
0
2
2
)(
exp
2
)(β)(
exp)()(πβ)(
nrnnr
njnndI chch
bb , (11)
I. Melnyk, A. Pochynok, M. Skrypka
ISSN 1681–6048 System Research & Information Technologies, 2025, № 1 60
),()()1( ndInInI bbb ),()( ndzrnr chch ),)1(()1( dznrnr chch ,
it
ch
N
l
dz
where bI — beam current; bβ — beam parameter, which characterized the dissi-
pation of electrons by the velocity; n — current number of iteration, itN — total
number of iterations [1–10]. Generally, algorithm of calculation of boundary tra-
jectories of electron beam in the case of its’ propagation in the free space with the
constant pressure have been described in the papers [66–72]. Also, in this papers
an advanced method of interpolation and approximation of boundary trajectories
of electron beam, based on root-polynomial function and giving very small error,
has been proposed and analyzed. Using such approach, the optimization task in
this case is formulated as follows [61; 62]:
.)(min
α
0
out
in
zdI
l
R
R
chl
z
b
ch
(12)
Since the channel radius, defined from relations (4), (5), is depend on chan-
nel geometry parameter, the coordinates of basic points 41 PP – , as a result of
solving set of equations (10), are correspond to criterium (12). Usually, it is
enough to define 4 point on the beam boundary trajectory, for which beam radius
have the maximal value and choose the channel radius by the simple relation
[1–10; 45–48]:
)(β 3 )( zrzr bb . (13)
Relation (13), from the point of view of the physics of HVGD [12; 45–47],
is explained by the fact that with a large number of collisions of beam electrons
with residual gas atoms, the longitudinal distribution of current density j(r) corre-
sponds to the Gaussian law with a very high probability, more than 99% [1–4].
Generally, analyzing beam current losses using relations (11)–(13) is the
separate sophisticated problem for future research, which consideration is not the
subject of this paper. However, in any case, numerical solving of the set of equa-
tions (10) with the known set of basic points 41 PP – coordinates is a separate
complex problem because this set of equations has been formed by using the stiff
power function (5) [63; 64]. The corresponding proposed algorithm for solving
the set of equations (10), based on the Steffensen method [63; 64], will be de-
scribed in the next part of the article.
THE NUMERICAL ALGORITHM FOR DEFINING THE GEOMETRY
PARAMETERS OF A NON-LINEAR TRANSPORT CHANNEL
Having solved the quadratic equation (9) for the variable b and substituting, in-
stead of the common variables z and r , its values 1z and 1r , it is easy to obtain
following relation for the function )α,,( chb laf :
Numerical algorithm for calculation of the vacuum conductivity of a non-linear channel for …
Системні дослідження та інформаційні технології, 2025, № 1 61
α
11 ))((
2
1
)α,,( zralafb chb
.))((4))(())((2 α
111
α
11
α
11
2 zarlzzrzra ch (14)
Further, if one rewrites relation (9) with respect to the variable a and substi-
tutes, instead of the common variables z and r , their values 2z and 2r , then the
corresponding relation for the function )α,,( cha lbf will be written as follows:
.
))((2
))((
α),,(
α
22
2α
222
bzr
bzrblz
lbfa ch
cha
(15)
Similar, from relation (9) the function for calculation the channel length chl
by substitutes, instead of the common variables z and r , their values 3z and 3r ,
is written as follows:
.
)())((
α),,(
3
2α
33
z
abbbazr
bafl lch
(16)
And finally, the function for defining the exponent α in relations (3), (4),
through the known values a , b , chl , and substitutes, instead of the common vari-
ables z and r , their values 4z and 4r , is written as follows:
.
))(ln(
)ln()ln(
),,(α
44
2
4
α zr
baabblz
lbaf ch
ch
(17)
Taking into account relation (14)–(17), the values ,,, chlba and α , which are
satisfied for set of equations (10) for given coordinates of basic points 41 PP – ,
can be defined numerically using the Steffensen method of solving sets of nonlin-
ear equations [62–64]. Corresponding numerical relations for iterative calculation
of these values are written as follows:
nb
1
1 1 1 1
2
1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
( , ,α , , )
;
( ( ( , , , ), ,α , , ), ,α , , ) ( , ,α , , )
n
n n n n
b n ch n
b n a n b n ch ch n ch n b n ch n
f a l r z
f a f b f a l r z l r z l r z f a l r z
(18)
na
;
),,α,,(),,),,,α,),,,,(((
),,α,,(
22122221221
221
2
1111
1
zrlbfzrlzrlzrlbfafbf
zrlbf
nchnachnchchnanbna
nchna
nnnn
n
(19)
1
1 1 1
2
α 3 3
α 1 2 2 3 3 α 3 3
( , , , , )
α ;
(( ( , ,α , , )), , , , ) ( , , , , )
n
n n n
n n ch
n
n a n ch n n ch n n ch
f a b l r z
f a f b l r z b l r z f a b l r z
(20)
1
2
1 4 4
2 2 4 4 1 4 4
( , ,α , , )
,
(( ( , ,α , , )), ,α , , ) ( , ,α , , )n
n
l n n n
ch
l n a n ch n n n l n n n
f a b r z
l
f a f b l r z b r z f a b r z
(21)
where n is number of current iteration.
Iterative calculations using the given relations (14)–(21) have been carried
out until the vacuum conductivity of the transport channel reaches a value that
I. Melnyk, A. Pochynok, M. Skrypka
ISSN 1681–6048 System Research & Information Technologies, 2025, № 1 62
provides the required pressure difference chg pp – , corresponding to Fig. 1, while
the channel conductivity have been calculated from relation (6). Correspondently,
iterative calculations have been considered completed if the modulus of the dif-
ference in channel conductivities at the previous and current iterations did not
exceed ,
s
mPa
10
3
6 namely:
,
s
mPa
10δ
3
6
1
nn chch UU (22)
where δ is achieved accuracy of calculations on the current iteration n .
The corresponding flow chart of the described iterative algorithm of calcula-
tions, which have been carried out using relations (6; 7; 14–22), is presented
in Fig. 2.
The testing results of using proposed and described algorithm for solving the
task of designing electron-beam vacuum equipment will be presented in the next
part of the article. All numerical calculations have been provided using program-
ming manes, as well as numerical and graphic libraries of MATLAB software for
scientific ang technical calculations [73]. Among advanced programming means
of MATLAB structure and matrix approach have been implemented [65; 73].
4
Calculation
nchU using
relation (6)
5
End
7
Ні
6
?10δ 6
1
nn chch UU
4
7 n = n + 1
6
Calculation an, bn, ,
nchl αn
using relations (14)–(21)
4
Start
Reading input data for
starting approach: α, Rin,
Rout; and for further
calculations: z1, z2, z3, z4,
r1, r2, r3, r4, pg, pch, Sp
1
73 n = 1
Calculation lch for
starting approach
using relation (7)
2
5
Fig.2. Flow chart of described iterative algorithm for define geometry parameters of
nonlinear vacuum channel for transporting electron beam
Numerical algorithm for calculation of the vacuum conductivity of a non-linear channel for …
Системні дослідження та інформаційні технології, 2025, № 1 63
OBTAINED RESULTS OF NUMERICAL EXPERIMENTS AND ITS
DISCUSSION
Let’s considering and analyzing in this part of article the several task of defining
the geometry parameters of nonlinear transporting channel. In all tasks, which
will be considered below, on the start iteration such geometry parameters of
nonlinear tube have been taken: m 006.0 in R , m 015.0 out R , 5.3 α . All cal-
culations have been provided for such parameters of electron-beam equipment,
corresponding to Fig. 1: Pa, 5
gp Pa, 1.0 chp .
s
m
001.0
3
pS Starting value
for the length of channel has been calculated using relation (7). Therefore, only
the coordinates of basic points have been changed in considered tests.
Task 1. m, 01.01 z m, 02.02 z m, 03.03 z m; 04.04 z m, 017.01 r
m, 02.02 r m, 021.03 r m. 022.04 r Obtained solution: m, 0429.0 in R
m, 0825.0 out R m, 0691.0 chl .0496.5 α This solution has been obtained after
23 iterations. Obtained graphic dependence )(zrch for this task is presented at
Fig. 3 as straight line.
Task 2. m, 01.01 z m, 02.02 z m, 03.03 z m; 04.04 z m, 008.01 r
m, 009.02 r m, 01.03 r m. 012.04 r Obtained solution: m, 0701.0 in R
m, 1128.0 out R m, 0412.0 chl .7743.5 α This solution has been obtained af-
ter 28 iterations. Obtained graphic dependence )(zrch for this task is presented at
Fig. 3 as dash line.
Task 3. m, 01.01 z m, 02.02 z m, 03.03 z m; 04.04 z m, 0066.01 r
m, 0085.02 r m, 01.03 r m. 011.04 r Obtained solution: m, 0062.0 in R
m, 0153.0 out R m, 2234.0 chl .7278.2 α This solution has been obtained af-
ter 13 iterations. Obtained graphic dependence )(zrch for this task for all range of
changing of longitudinal coordinate z is presented at Fig. 4, a, and for the start
Fig. 3. Graphic dependences )(zrch , obtained using iterative relations (14)–(21) for data
sets of Task 1 and Task 2
I. Melnyk, A. Pochynok, M. Skrypka
ISSN 1681–6048 System Research & Information Technologies, 2025, № 1 64
range of changing z coordinate, correspondently, in Fig. 4, b. This solution is
reflected in Fig. 4 a, b, as straight line.
Task 4. m, 01.01 z m, 02.02 z m, 03.03 z m; 04.04 z m, 0093.01 r
m, 0121.02 r m, 0141.03 r m. 0157.04 r Obtained solution: m, 0059.0 in R
m, 0151.0 out R m, 32211.0 chl .64456.2 α This solution has been obtained
after 15 iterations. Obtained graphic dependence )(zrch for this task for all range
of changing of longitudinal coordinate z is presented at Fig. 4, a, and for the start
range of changing z coordinate, correspondently, in Fig. 4, b. This solution is re-
flected in Fig. 4 a, b, as dash line.
Task 5. m, 01.01 z m, 02.02 z m, 03.03 z m; 04.04 z m, 0067.01 r
m, 0087.02 r m, 0102.03 r m. 0113.04 r Obtained solution: m, 0061.0 in R
m, 0149.0 out R m, 2323.0 chl .6456.2 α This solution has been obtained af-
ter 17 iterations. Obtained graphic dependence )(zrch for this task for all range of
a
z, m
b
Fig. 4. Graphic dependences )(zrch , obtained using iterative relations (14)–(21) for data
sets of Tasks 3, 4, 5 and 6 in the all range (a) and in start range (b) of changing of longi-
tudinal coordinate z
Numerical algorithm for calculation of the vacuum conductivity of a non-linear channel for …
Системні дослідження та інформаційні технології, 2025, № 1 65
changing of longitudinal coordinate z is also presented at Fig. 4, a, and for the
start range of changing z coordinate, correspondently, in Fig. 4, b. This solution
is reflected in Fig. 4 a, b, as dot line.
Task 6. m, 01.01 z m, 02.02 z m, 03.03 z m; 04.04 z m, 0099.01 r
m, 013.02 r m, 0152.03 r m. 0171.04 r Obtained solution: m, 0063.0 in R
m, 0152.0 out R m, 352.0 chl .5546.2 α This solution has been obtained after
14 iterations. Obtained graphic dependence )(zrch for this task for all range of
changing of longitudinal coordinate z is also presented at Fig. 4, a, and for the
start range of changing z coordinate, correspondently, in Fig. 4, b. This solution
is reflected in Fig. 4 a, b, as dash-dot line.
From the obtained calculation results and graphical dependencies presented
in Fig. 3 and Fig. 4, it is clear that the iterative numerical algorithm based on rela-
tions (14)–(21), the flow chart of which is presented in Fig. 2, converges stably
for various sets of numerical data for base points . – 41 PP It should be noted that
this is a very important practical result, since the numerical solution of sets of
equations containing power and logarithmic functions with a high degree of rigid-
ity cannot always be implemented using standard numerical methods [62–64].
A good proof of the stable convergence of the proposed iterative method is
that for close values of the base point data sets, the solutions to the tasks posed are
almost identical. This is clearly visible in the obtained solutions for Task 3 and
Task 5. Indeed, as can be seen from the graphical dependencies shown in Fig. 4,
a, the solutions obtained for given sets of points with close coordinates practically
coincide.
It should also be noted that all the considered examples are of a practical na-
ture, and the numerical data sets given in problems 1–6 correspond to the actual
dimensions of the channels for transporting electron beams in industrial techno-
logical equipment. For example, with an input diaphragm radius of mm 8 and a
radius of a cylindrical beam transportation channel of mm 16 , a pressure drop
from Pa 5 to Pa 1.0 at a pumping speed of
s
m
001.0
3
is provided with a transpor-
tation channel length of m. 28.0 It is possible in the model task to consider such a
transporting channel as a nonlinear one described by the function (4) with pa-
rameter 20 α . The calculation results for such a model of the beam transport
channel give a result of m, 24.0 which is in good agreement with experimental
data, taking into account the complexity of the simulation of real vacuum systems
[51–57]. To further harmonize theoretical and experimental data, empirical coef-
ficients can be introduced into calculation formulas (14)–(17). In any case, the
stable convergence of iterative formulas (18)–(21) is an undeniable advantage of
the proposed algorithm for solving important practical problems of modern elec-
tron beam technologies.
To carry out further theoretical research in order to solve complex practical
and engineering problems of modern industrial electron beam technologies, it is
necessary to combine the numerical method described in this article for calculat-
ing the geometric parameters of the vacuum channel for transporting an electron
beam with the previously proposed modern methods of interpolation and ap-
proximation of the boundary trajectories of an electron beam propagating in soft
I. Melnyk, A. Pochynok, M. Skrypka
ISSN 1681–6048 System Research & Information Technologies, 2025, № 1 66
vacuum [66–72]. In this way, the important practical problem of optimizing beam
current losses in the guide channel, described by equations (11)–(13), can be solved.
All research work described in this paper has been provided in the Scientific and
Educational Laboratory of Electron Beam Technological Devices of the National
Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnical Institute”.
CONCLUSION
The test numerical experiments carried out in the research work showed that the
proposed algorithm for the numerical calculation of the geometric parameters of a
nonlinear vacuum channel for transporting a short-focus electron beam, specified
by iterative relations (14)–(21) and shown in the form of a flow chart in Fig. 2,
converges stably for the pumping speed of the vacuum chamber and the range of
pressures in the electron gun and in the technological chamber used in electron
beam technological installations. The obtained results of numerical calculations
are in good agreement with experimental data. The divergence of calculated and
experimental data for test tasks did not exceed 10%. The implementation of the
proposed algorithm together with the solution of problems of interpolation and
approximation of the boundary trajectory of an electron beam in a single software
package will allow, at the initial stage of designing electron beam technological
equipment, to estimate the energy losses of the electron beam in the transportation
channel and to study complexly the issue of the efficiency of applying of
nonlinear beam transportation channels in industrial installations, taking into
account the complexity of their production.
Currently, for the manufacture of beam transportation channels with
nonlinear geometry, electron beam technologies of three-dimensional metal
printing can be applied. In such cases, the manufacturing technology of
curvilinear tubes is significantly simplified, and the energy costs as well as the
consumption of the material used are also valuably reduced. The accuracy of
manufacturing a nonlinear channel using the electron beam three-dimensional
printing method is also much higher than when using traditional mechanical proc-
essing methods, and the mechanical and thermodynamic properties of the result-
ing metal are always much better [43; 44].
The results presented in the article may be of great practical interest to ex-
perts involved in the development of electron beam equipment and its applying in
modern industry. Experts in the field of computational mathematics and numeri-
cal methods may be interested in the proposed algorithm for the numerical solu-
tion of a stiff set of nonlinear equations (10). The distinguishing feature of this
algorithm stably converges for different values of the power function exponent 1/α.
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Received 23.01.2024
INFORMATION ON THE ARTICLE
Igor V. Melnyk, ORCID: 0000-0003-0220-0615, National Technical University of
Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: imel-
nik@phbme.kpi.ua
Alina V. Pochynok, ORCID: 0000-0001-9531-7593, Research Institute of Electronics
and Microsystem Technology of the National Technical University of Ukraine “Igor Si-
korsky Kyiv Polytechnic Institute”, Ukraine, e-mail: alina_pochynok@yahoo.com
Mykhailo Yu. Skrypka, ORCID: 0009-0006-7142-5569, National Technical Univer-
sity of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail:
scientetik@gmail.com
ЧИСЛОВИЙ АЛГОРИТМ РОЗРАХУНКУ ВАКУУМНОЇ ПРОВІДНОСТІ
НЕЛІНІЙНОГО КАНАЛУ ТРАНСПОРТУВАННЯ КОРОТКОФОКУСНОГО
ЕЛЕКТРОННОГО ПУЧКА У ТЕХНОЛОГІЧНОМУ ОБЛАДНАННІ / І.В. Мель-
ник, А.В. Починок, М.Ю. Скрипка
Анотація. На основі розв’язування рівнянь вакуумної техніки запропоновано і
досліджено ітераційний алгоритм розрахунку вакуумної провідності та геоме-
тричних параметрів криволінійного каналу транспортування короткофокусно-
го електронного пучка, для якого залежність радіуса каналу від поздовжньої
координати описують степеневою функцією. Запропонований алгоритм засно-
ваний на числовому розв’язуванні системи нелінійних рівнянь методом Стеф-
фенсена. Наведені результати тестових розрахунків підтверджують стійку збі-
жність запропонованого алгоритму для реальних значень тиску та швидкості
відкачування у технологічному обладнанні. Такі криволінійні канали транспо-
ртування можуть бути використані в електронно-променевому обладнанні на
основі гармат високовольтного тліючого розряду, призначеному для зварю-
вання, плавлення металів та для нанесення тонких плівок. Критерієм оптима-
льності геометрії нелінійного каналу є мінімальні втрати потужності електро-
нного пучка під час його транспортування за умови забезпечення необхідного
перепаду тиску між розрядною та технологічною камерами.
Ключові слова: електронний пучок, електронно-променеві технології, транс-
портування електронного пучка, нелінійний канал транспортування електро-
нного пучка, вакуумна провідність каналу транспортування, електронна гар-
мата високовольтного тліючого розряду, рівняння вакуумної техніки, система
нелінійних рівнянь, метод Стеффенсена.
|
| id | journaliasakpiua-article-330021 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-09-17T09:26:02Z |
| publishDate | 2025 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/8c/5a538735d1cfbef29b50bc68fe4f478c.pdf |
| spelling | journaliasakpiua-article-3300212025-05-20T17:56:07Z Numerical algorithm for calculation of the vacuum conductivity of a non-linear channel for transporting a short-focus electron beam in the technological equipment Числовий алгоритм розрахунку вакуумної провідності нелінійного каналу транспортування короткофокусного електронного пучка у технологічному обладнанні Melnyk, Igor Pochynok, Alina Skrypka, Mykhailo focal parameters of the electron beam electron beam technologies electron beam transportation nonlinear electron beam transportation channel vacuum conductivity of the transportation channel high-voltage glow discharge electron gun vacuum technology equation set of nonlinear equations Steffensen method електронний пучок електронно-променеві технології транспортування електронного пучка нелінійний канал транспортування електронного пучка вакуумна провідність каналу транспортування електронна гармата високовольтного тліючого розряду рівняння вакуумної техніки система нелінійних рівнянь метод Стеффенсена In the article, based on solving the equations of vacuum technology, an iterative algorithm for calculating vacuum conductivity and the geometric parameters of a curvilinear channel for transporting a short-focus electron beam is proposed and studied. For such a type of channel, the dependence of its radius on the longitudinal coordinate is described by a power function. The proposed algorithm is based on the numerical solution of a set of nonlinear equations using the Steffensen method. The results of the test calculations are presented. The provided tests confirm the stability of the proposed algorithm’s convergence for correct pressure and pumping speed values in electron-beam technological equipment. Such curved transport channels can be used in electron beam equipment based on high-voltage glow discharge electron guns intended for welding, melting metals, and the deposition of thin films. The criterion for the optimal geometry of a nonlinear channel is the minimum power loss of the electron beam during its transportation while ensuring the required pressure drop between the discharge and technological chambers. На основі розв’язування рівнянь вакуумної техніки запропоновано і досліджено ітераційний алгоритм розрахунку вакуумної провідності та геометричних параметрів криволінійного каналу транспортування короткофокусного електронного пучка, для якого залежність радіуса каналу від поздовжньої координати описують степеневою функцією. Запропонований алгоритм заснований на числовому розв’язуванні системи нелінійних рівнянь методом Стеффенсена. Наведені результати тестових розрахунків підтверджують стійку збіжність запропонованого алгоритму для реальних значень тиску та швидкості відкачування у технологічному обладнанні. Такі криволінійні канали транспортування можуть бути використані в електронно-променевому обладнанні на основі гармат високовольтного тліючого розряду, призначеному для зварювання, плавлення металів та для нанесення тонких плівок. Критерієм оптимальності геометрії нелінійного каналу є мінімальні втрати потужності електронного пучка під час його транспортування за умови забезпечення необхідного перепаду тиску між розрядною та технологічною камерами. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2025-03-28 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/330021 10.20535/SRIT.2308-8893.2025.1.05 System research and information technologies; No. 1 (2025); 53-72 Системные исследования и информационные технологии; № 1 (2025); 53-72 Системні дослідження та інформаційні технології; № 1 (2025); 53-72 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/330021/319501 |
| spellingShingle | електронний пучок електронно-променеві технології транспортування електронного пучка нелінійний канал транспортування електронного пучка вакуумна провідність каналу транспортування електронна гармата високовольтного тліючого розряду рівняння вакуумної техніки система нелінійних рівнянь метод Стеффенсена Melnyk, Igor Pochynok, Alina Skrypka, Mykhailo Числовий алгоритм розрахунку вакуумної провідності нелінійного каналу транспортування короткофокусного електронного пучка у технологічному обладнанні |
| title | Числовий алгоритм розрахунку вакуумної провідності нелінійного каналу транспортування короткофокусного електронного пучка у технологічному обладнанні |
| title_alt | Numerical algorithm for calculation of the vacuum conductivity of a non-linear channel for transporting a short-focus electron beam in the technological equipment |
| title_full | Числовий алгоритм розрахунку вакуумної провідності нелінійного каналу транспортування короткофокусного електронного пучка у технологічному обладнанні |
| title_fullStr | Числовий алгоритм розрахунку вакуумної провідності нелінійного каналу транспортування короткофокусного електронного пучка у технологічному обладнанні |
| title_full_unstemmed | Числовий алгоритм розрахунку вакуумної провідності нелінійного каналу транспортування короткофокусного електронного пучка у технологічному обладнанні |
| title_short | Числовий алгоритм розрахунку вакуумної провідності нелінійного каналу транспортування короткофокусного електронного пучка у технологічному обладнанні |
| title_sort | числовий алгоритм розрахунку вакуумної провідності нелінійного каналу транспортування короткофокусного електронного пучка у технологічному обладнанні |
| topic | електронний пучок електронно-променеві технології транспортування електронного пучка нелінійний канал транспортування електронного пучка вакуумна провідність каналу транспортування електронна гармата високовольтного тліючого розряду рівняння вакуумної техніки система нелінійних рівнянь метод Стеффенсена |
| topic_facet | focal parameters of the electron beam electron beam technologies electron beam transportation nonlinear electron beam transportation channel vacuum conductivity of the transportation channel high-voltage glow discharge electron gun vacuum technology equation set of nonlinear equations Steffensen method електронний пучок електронно-променеві технології транспортування електронного пучка нелінійний канал транспортування електронного пучка вакуумна провідність каналу транспортування електронна гармата високовольтного тліючого розряду рівняння вакуумної техніки система нелінійних рівнянь метод Стеффенсена |
| url | https://journal.iasa.kpi.ua/article/view/330021 |
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