The nonlinear theory of the electromagnetic field excitation in orbitron
The nonlinear theory of electromagnetic waves generation in orbitron is developed. The set of the equations including the equations of field excitation and the equations of 2-dimential motion is constructed and numerically solved. It is shown, that mechanism of electron bunching and energy exchang...
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| Date: | 2009 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2009
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| Cite this: | The nonlinear theory of the electromagnetic field excitation in orbitron / Yu.V. Kirichenko, I.N. Onishchenko // Вопросы атомной науки и техники. — 2009. — № 5. — С. 110-117. — Бібліогр.: 9 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859703706659848192 |
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| author | Kirichenko, Yu.V. Onishchenko, I.N. |
| author_facet | Kirichenko, Yu.V. Onishchenko, I.N. |
| citation_txt | The nonlinear theory of the electromagnetic field excitation in orbitron / Yu.V. Kirichenko, I.N. Onishchenko // Вопросы атомной науки и техники. — 2009. — № 5. — С. 110-117. — Бібліогр.: 9 назв. — англ. |
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| description | The nonlinear theory of electromagnetic waves generation in orbitron is developed. The set of the equations including
the equations of field excitation and the equations of 2-dimential motion is constructed and numerically solved. It
is shown, that mechanism of electron bunching and energy exchange of electrons with the wave in orbitron and in
magnetron has much in common. For the fixed parameters of orbitron from the point of view of generated energy
and electronic efficiency there is some optimal value of electron density in the interaction region.
В роботi розвинуто нелiнiйну теорiю генерацiї електромагнiтних хвиль в орбiтронi. Побудовано i
чисельно розв’язано систему рiвнянь , що включає рiвняння збудження та руху. Показано, що механiзми групування та обмiну енергiєю електрона з хвилею в орбiтронi та магнетронi мають багато
спiльного. Для фiксованих параметрiв орбiтрона щiльнiсть електронiв у просторi взаємодiї має оптимальне значення з точки зору енергiї, що генерується, та коефiцiєнта корисної дiї. Достатньо точний
опис процесу збудження хвиль в орбiтронi можна отримати за допомогою основної власної гармонiки.
В работе развита нелинейная теория генерации электромагнитных волн в орбитроне. Построена и
численно решена система уравнений возбуждения и уравнений движения. Показано, что механизмы
группировки и обмена энергией электрона с волной в орбитроне и магнетроне имеют много общего.
Для фиксированных параметров орбитрона имеется некоторое оптимальное с точки зрения генерируемой энергии и электронного коэффициента полезного действия значение плотности электронов в
пространстве взаимодействия. Достаточно точное описание процесса возбуждения волн в орбитроне
можно получить, ограничиваясь основной собственной гармоникой.
|
| first_indexed | 2025-12-01T01:50:13Z |
| format | Article |
| fulltext |
THE NONLINEAR THEORY OF THE ELECTROMAGNETIC
FIELD EXCITATION IN ORBITRON
Yu.V. Kirichenko, I.N. Onishchenko∗
National Science Center ”Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
(Received July 6, 2009)
The nonlinear theory of electromagnetic waves generation in orbitron is developed. The set of the equations including
the equations of field excitation and the equations of 2-dimential motion is constructed and numerically solved. It
is shown, that mechanism of electron bunching and energy exchange of electrons with the wave in orbitron and in
magnetron has much in common. For the fixed parameters of orbitron from the point of view of generated energy
and electronic efficiency there is some optimal value of electron density in the interaction region.
PACS: 84.30.Jc, 85.10.Jz
1. INTRODUCTION
In [1] the original generator of millimeter waves - so
called ”orbitron” has been proposed, that represents
a coaxial structure, which inner cylinder is the thin
metal string. Advantage of such generator consists in
simplicity of a design, absence of slowing down system
and an external magnetic field. At applying to the
string (as an anode) of positive potential of several kV
electrons are pulled out from an inner surface of the
cylindrical resonator, being as a cathode. Multiply
spreading on molecules of residual gas, they get an
azimuthal component of the velocity Vϕ that allows
electrons to exchange energy with waves traveling in
an azimuthal direction. It has been shown [2], that
frequencies of rotating eigenwaves are determined by
the formula
ωmn ≈ πc
b
(
n +
|m|
2
+
1
4
)
, (1)
where c is speed of light, b is inner radius of an exter-
nal casing of coaxial structure, m = 0,±1,±2, ... and
n = 1, 2, ... - are azimuthal and radial numbers of
harmonics of eigenwave accordingly. As frequencies
of eigenwaves are discrete, we shall conventionally
name the considered coaxial system as a resonator.
From the formula (1) follows, that phase velocity of
eigenwave vph = ωmnr/m is less than speed of light
under condition of r ¿ b, where r is a distance from
the resonator axis. Thus, the wave appears slowed
down near the string. Just in this area electromag-
netic waves generation takes place that corresponds
to experimental data [1]. We shall emphasize, that
slowing down of a wave in orbitron occurs in absence
of the special slowing down system. The linear stage
of generation in orbitron has been investigated in [2-4]
where the conditions of instabilities originating have
been found and formulas for their increments have
been obtained. In [5] in of the given field approxima-
tion the nonlinear dynamics of nonrelativistic elec-
trons in orbitron has been considered at small ampli-
tudes of the wave. It is of interest to carry out more
general nonlinear consideration with refuse from the
assumptions made in [5], simplifying the picture of
wave generation in orbitron. In the present work the
nonlinear theory of electromagnetic waves excitation
in orbitron is developed, allowing to study the gener-
ation process starting from the field fluctuation am-
plification.
2. DERIVATION OF THE EQUATIONS OF
THE NONLINEAR THEORY
Let’s consider the high Q coaxial cylindrical res-
onator, unbounded along an axis z (the cylindri-
cal system of coordinates r, ϕ, z is used). Radius
of the charged string, which creates an electrosta-
tic field ~E = 2~eQ/r where Q is linear charge den-
sity of the string, is equal a << b . The follow-
ing two-dimentional non-stationary problem is being
solved that simulates the generation process in or-
bitron. At absence of electrons in orbitron there is
some fluctuation of an electromagnetic field having
components Hz, Eϕ, Er (H -wave). At the initial mo-
ment of time nonrelativistic electrons are uniformly
distributed along the circle of radius r0 . By virtue
of azimuthal symmetry they have equal initial speeds
Vr0, Vϕo . At the following moments of time electrons
start to move in plane r, ϕ in the electrostatic field of
the string and in the fluctuation field, giving up its
energy to the fluctuation. As a result of fluctuation
amplification the electromagnetic field is generated
in orbitron. We find the time-dependent field of the
wave in the form of expansion on eigen waves of the
∗Corresponding author E-mail address: onish@kipt.kharkov.ua
110 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2009, N5.
Series: Nuclear Physics Investigations (52), p.110-117.
resonator that forms a full set of functions
~E(r, ϕ, t) = Re
∞∑
m=−∞
∑
fmn~emn(r, ϕ),
~H(r, ϕ, t) = Re
∞∑
m=−∞
∑
gmn
~hmn(r, ϕ) ,
(2)
where ~emn(r, ϕ) ,~hmn(r, ϕ) is intensity electric and a
magnetic field of eigen harmonics of the wave
emn,r = −mRmn(kmnr)
kmnr
√
Nmn
exp(imϕ),
emn,ϕ = −i
m ´Rmn(kmnr)√
Nmn
exp(imϕ),
hmn,r =
Rmn(kmnr)√
Nmn
exp(imϕ),
(3)
where
Rmn = Jm(kmnr)N ′
m(kmna)− J ′m(kmna)Nm(kmnr),
R′mn = J ′m(kmnr)N ′
m(kmna)− J ′m(kmna)N ′
m(kmnr),
kmn/c, Jm(x) and Nm(x) are Bessel and Neumann
functions (the stroke means differentiation by argu-
ment x ),
Nmn = {[(kmnb)2 −m2]R2
mn(kmnb)−
[(kmna)2 −m2]R2
mn(kmna)}/(4k2
mn)
is a normalizing multiplier. In (2) the field of a spa-
tial charge is not taken into account. In the further
the harmonic with m = 0 will not be considered, as
it does not lead to bunching of electrons. Eigen func-
tions (3) are normalized by the following way:
∫ b
a
drr
∫ 2π
o
dϕ~hmn
~h∗m′n′ =
∫ b
a
drr
∫ 2π
o
dϕ~emn ~e∗m′n′ = 4πδmm′δnn′ .
(4)
Substituting relation (2) in Maxwell equations
rot ~H =
4π
c
~j +
1
c
∂ ~E
∂t
,
rot ~E = −1
c
∂ ~H
∂t
,
(5)
and using conditions of normalization (4), we shall
obtain the following equations of excitation for time-
dependent amplitudes of the expansion (2) :
dfmn(t)
dt
+ iωmngmn(t) = −2Kmn(t),
dgmn(t)
dt
+ iωmnfmn(t) = 0,
(6)
fmn|t=0 = f0mn = |f0mn|exp(iΦ0mn),
gmn|t=0 = g0mn = |g0mn|exp(iΦM
0mn),
(7)
where function of time
Kmn(t) =
∫ b
a
drr
∫ 2π
0
dϕ~j(r, ϕ, t)~e∗mn(r, ϕ) , (8)
makes sense a coefficient of coupling of eigen (cold)
wave with a flow of electrons. The current ~j in for-
mulas (5,8) is formed by electrons emitted from the
cathode. By means of (3) formula (8) can be repre-
sented in the form
Kmn(t) =
1√
Nmn
{K1mn(t)−K4mn(t)+
i(K2mn(t) + K3mn(t))},
(9)
where
K1mn =
∫ b
a
drr
∫ 2π
0
dϕjϕR′mn(kmnr)sin(mϕ),
K2mn =
∫ b
a
drr
∫ 2π
0
dϕjϕR′mn(kmnr)cos(mϕ),
(10)
K3mn =
∫ b
a
drr
∫ 2π
0
dϕjr
mRmn(kmnr)
kmnr
sin(mϕ),
K4mn =
∫ b
a
drr
∫ 2π
0
dϕjr
mRmn(kmnr)
kmnr
cos(mϕ).
(11)
As in absence of electrons there is no chosen direction
in azimuth, it is natural to assume, that initial fluctu-
ations of the field are standing waves in an azimuthal
direction. Therefore following relations between am-
plitudes and phases of the fluctuations should be ful-
filled
|g0mn| = |g0,−mn| = |f0mn| = |f0,−mn|,
Φ0mn = Φg
0mn = Φ0,−mn = Φg
0,−mn .
(12)
The solutions of the equations of excitation (6) with
initial conditions (7) are convenient to present in an
integrated view
fmn = f0mncos(ωmnt)− ig0mnsin(ωmnt)− exp(−iωmnt)
∫ t
0
dt′exp(iωmnt′)Kmn(t′)−
exp(iωmnt)
∫ t
0
dt′exp(−iωmnt′)Kmn(t′),
(13)
111
gmn = −if0mnsin(ωmnt) + g0mncos(ωmnt)− exp(−iωmnt)
∫ t
0
dt′exp(iωmnt′)Kmn(t′)+
exp(iωmnt)
∫ t
0
dt′exp(iωmnt′)Kmn(t′) .
(14)
Relations(10-14) allow to present any harmonic of the electric field of the wave (2) in the form
Emn,r(r, ϕ, t) = Re(fmn(t)emn,r(r, ϕ) + f−mn(t)e−mn,r(r, ϕ)) =
2mRmn(kmnr)
Nmnkmnr
{L1mn(t)cosΦ+
mn + L2mn(t)sinΦ+
mn + L3mn(t)cosΦ−mn + L4mn(t)sinΦ−mn−
√
Nmn
2
|f0mn|(cos(Φ+
mn + Φ0mn) + cos(Φ−mn − Φ0mn)} ,
(15)
Emn,ϕ(r, ϕ, t) = Re(fmn(t)emn,ϕ(r, ϕ) + f−mn(t)e−mn,ϕ(r, ϕ)) =
−2R′mn(kmnr)
Nmn
{L5mn(t)cosΦ+
mn + L6mn(t)sinΦ+
mn + L7mn(t)cosΦ−mn + L8mn(t)sinΦ−mn−
√
Nmn
2
|f0mn|(sin(Φ+
mn + Φ0mn)− sin(Φ−mn − Φ0mn)} ,
(16)
where
L1mn = −
∫ t
0
dt′{cos θ′K4mn(t′) + sinθ′K3mn(t′)}, L2mn =
∫ t
0
dt′{sin θ′K4mn(t′)− cosθ′K3mn(t′)}, (17)
L3mn =
∫ t
0
dt′{sin θ′K3mn(t′)− cosθ′K4mn(t′)}, L4mn = −
∫ t
0
dt′{sin θ′K4mn(t′) + cosθ′K3mn(t′)}, (18)
L5mn =
∫ t
0
dt′{cos θ′K2mn(t′) + sinθ′K1mn(t′)}, L6mn =
∫ t
0
dt′{cos θ′K1mn(t′)− sinθ′K2mn(t′)}, (19)
L7mn =
∫ t
0
dt′{cos θ′K2mn(t′)− sinθ′K1mn(t′)}, L8mn =
∫ t
0
dt′{cos θ′K1mn(t′) + sinθ′K2mn(t′)}, (20)
θ′ = ωmnt′, Φ+
mn = mϕ− ωmnt, Φ−mn = mϕ + ωmnt. (21)
From formulas (15,16,21) it is visible, that partial
waves of the field in orbitron represent waves travel-
ing in an azimuthal direction. To find electron cur-
rent~j(r, ϕ, t) , which determines functions (8-11), it is
necessary to solve nonrelativistic equations of motion
in cylindrical coordinates with corresponding initial
conditions
dVr
dt
=
V 2
ϕ
r
− V 2
Q
r
− e
me
Er(r, ϕ, t),
dVϕ
dt
= −VrVϕ
r
− e
me
Eϕ(r, ϕ, t),
dr
dt
= Vr,
dϕ
dt
=
Vϕ
r
, (22)
Vr|t=0 = Vr0, Vϕ|t=0 = Vϕ0, r|t=o = r0, ϕ|t=o = ϕ0. (23)
112
In the equations (22) −e < 0 and me are charge
and mass of electron, V 2
Q = 2eQ/me is square of a
certain scale velocity of electrons, and components of
the electric field of the wave are determined by for-
mulas (2). The set of equations (6,7,22) describes
self-consistently the process of electromagnetic waves
excitation in orbitron. For their solution we shall
present electron current in the form
~j(r, ϕ, t) = −e
Ñ∑
i
1
r
δ(r − ri(t))δ(ϕ− ϕi(t)), (24)
where ri(t), ϕi(t), ~Vi(t) are the solutions of the equa-
tions (22,23) for i electron, Ñ is full amount of elec-
trons in the interaction region. For application of
the method of macroparticles it is necessary to write
down the current (24) in the following view ,
~j(r, ϕ, t) = −eµe
N∑
j
1
r
δ(r − rj(t))δ(ϕ− ϕj(t)),
(25)
where j is number of macroparticle, N is full amount
of macroparticles, µe is mass of macroparticle, deter-
mined by the amount of electrons in macroparticle.
We note, that representation of the current in the
form (24,25) allows to pass simply from Euler coordi-
nates in (10,11) to Lagrange coordinates of macropar-
ticle, which are the solutions of the set of equations
(22,23). At that there are absent, from the comput-
ing point of view, laborious process of distribution
of a charge in cross-points of Euler grids of coordi-
nates and interpolation of force in points of particles
locations. Substituting (25) into formulas (10,11), we
shall obtain
K1mn = −eµe
N∑
j
Vjϕ(t)R′mn(kmnrj(t))sin(mϕj(t)),K2mn = −eµe
N∑
j
Vjϕ(t)R′mn(kmnrj(t))cos(mϕj(t)),
(26)
K3mn = −eµe
N∑
j
Vjr(t)
mRmn(kmnrj(t))
kmnrj(t)
sin(mϕj(t)),K4mn = −eµe
N∑
j
Vjr(t)
mRmn(kmnrj(t))
kmnrj(t)
cos(mϕj(t)).
(27)
For convenience of calculations the equation de-
duced above have been led to a dimensionless form.
For conciseness they are not presented. The formula
for energy of an excited field of the wave, correspond-
ing to unit of orbitron length, can be obtained from
expansion (2). It has a view ,
ε(t) =
∞∑
m=1
εmn(t), εmn(t) =
1
4
{|fmn(t)|2 + |f−mn(t)|2 + |gmn(t)|2 + |g−mn(t)|2}. (28)
The solution of the problem is determined by the
following main parameters: the ratio of radii b/a ;
amount of electrons, corresponding to unit of coax-
ial length l , i.e. Ñ/l ; voltage U in kV applied to
the string; coefficient of decreasing of particle energy
α = meV0/(4eQln(b/a)) < 1 , which characterizes of
its energy losses due to collisions with molecules of
residual gas and is equal to the ratio of its kinetic
energy in the point r0 to the maximal possible ki-
netic energy which the particle would gain, having
passed a way from the cathode up to the anode with-
out scattering; parameter of synchronism of particles
with a harmonic (m,n) of initial fluctuation of the
wave asmn = (Vϕ0−vphmn/vphmn) ; angle sc which is
formed by initial velocity of the particle ~V0 with ra-
dial direction, at that tgϕxc = −Vϕ0/Vr0. The values
βϕ0 = Vϕ0/c and ρ0 = r0/a are determined by para-
meters α , U , asmn , at that at their fixed values βϕ0
grows together with ρ0 . Parameters of initial fluctu-
ation |f0mn|,Φ0mn are chosen small enough that final
results did not depend on them. Accuracy of calcula-
tions is determined by amount of particles N and by
step of integration in time ∆t . Electron in orbitron
possesses not only kinetic Wk , but also potential Wp
energy which are given by the relations
Wk = Wkϕ + Wkr, Wkϕ =
meV
2
ϕ
2
, Wkr =
meV
2
r
2
, Wp = 2eQln
r
r′
, (29)
where r′ is reference point of potential. The elec-
tron efficiency for orbitron is determined as follows.
Let’s consider firstly the case when electrons fall down
only on the string. Let in the initial moment of time
their kinetic and potential energies are equal Wk0 and
Wp0 , and energy of initial fluctuation of an electro-
113
magnetic field is ε(0). After the termination of gener-
ation process when all particles will fall down on the
string, corresponding values are equal Wkf ,Wpf , εf .
From the law of energy conservation follows, that
Wk0 + Wp0 −Wpf = εf − ε(o) + Wk . (30)
Work of the external source creating a voltage be-
tween the anode and the cathode is spent on the ini-
tial kinetic energy of electrons and their potential en-
ergy relating to the string and consequently is equal
Wk0 + Wp0 − Wpf . From (30) it is visible, that at
the end of generation the work of the external source
transforms to the energy of the field and the energy
of the anode heating-up, which is equal Wkf . The
electron efficiency is equal
η =
εf − ε(0)
Wk0 + Wp0 −Wpf
= 1− Wkf )
Wk0 + Wp0 −Wpf
.
(31)
For magnetrons the relation Wk0 ≈ 0 is fulfilled. In
magnetron case formulas (31) transorm into efficiency
for magnetron [6-8]. Under certain conditions in orbi-
tron subsidence of some part of particles on the cath-
ode is possible. The similar phenomenon takes place
in magnetrons too [6-7]. In this case the part of work
of an external source is spent for increase in potential
energy of this part of particles which should be added
in the right part of equality (30) and in numerator of
the second formula (31). Subsidence of electrons on
the cathode leads to reduction of the value η .
3. RESULTS OF CALCULATIONS
The numerical solution the equations obtained above
yielded following results. Values ε, η as functions of
the angle ϕsc have a maximum at φsc = π/2 . Devi-
ations of the angle ϕsc from this value leads to sharp
reduction of energy of the generated field and effi-
ciency of its excitation. It means, that only those
electrons effectively interact with the wave, which
trajectories are close to circular and which compo-
nents of velocity are subjected to the condition
Vϕ >> Vr. (32)
It means, that electron exchanges with the wave
only by an azimuthal part Wkϕ of its kinetic energy.
Power, transferred to the wave by an electron, by
virtue of (32) and relation Eϕ ≈ Er [5] is equal
P = −e(VϕEϕ + VrEr) ≈ −eVϕEϕ. (33)
Besides for interaction of electrons with a harmonic
(m,n) , similarly to magnetron [6-8], should be satis-
fied the condition of synchronism of angular velocity
of electrons ω0 and angular phase velocity ωphmn of
this harmonic
ω0 ≈ ωphmn, ω0 =
Vϕ
r
, ωphmn =
ωmn
m
. (34)
It is known [8], that in the field of the charged cylin-
der the frequency of radial fluctuations of electron
more than
√
2 times exceeds its angular velocity. The
incommensurability of frequencies of radial and az-
imuthal motion of electron leads to that under con-
dition (34) there is no synchronism of components of
the field Er with radial motion of an electron. There-
fore, and also by virtue of (33) it is possible to con-
sider, that the electron bunching and waves genera-
tion in orbitron slightly depend on Er and mainly are
determined by component Eϕ During some initial in-
terval of time electrons, being uniformly distributed
on phase, do not exchange energy with initial fluctu-
ation of the field (2,7). In the further under action of
azimuthal nonuniformity of the wave one part elec-
trons gets in decelerating phases, where Eϕ > 0 ,
and another part gets in accelerating phases, where
Eϕ < 0 . Electrons of the first part, being slowed
down, approach to the string as the balance of cen-
trifugal force and force of an attraction to the string
is broken. These electrons give to the wave a part of
their energy consisting from Wkϕ and Wp
W ′ = Wkϕ + Wp. (35)
Electrons the second part, being accelerated, gain a
part of energy from the wave. At that the part W´
of their energy is increased and they approach to the
cathode. Thus, electrons in orbitron exchange with
the wave not only by azimuthal part Wkϕ of their
kinetic energy, but also by potential energy. In this
respect orbitron reminds magnetron, in which how-
ever only potential energy of electrons is transferred
to the wave[6-8]. The bunching of electrons in or-
bitron is determined by dependence of their angular
speed on time. Using the formula for ω0 (34) and sec-
ond equation of (22), we obtain the following relation
dω0
dt
= −1
r
(
2VrVϕ
r
+
e
me
Eϕ) ≈ −2VrVϕ
r2
. (36)
In (36) it is used the fact, that in RF-devices of
small and moderate power the amplitude of an ex-
cited field Eϕ is less than electrostatic fields. From
(36) follows, that electrons, being in decelerating
phase where Eϕ > 0, Vr < 0 have positive angular
velocity. Meanwhile electrons, being in accelerating
phase where Eϕ < 0, Vr > 0 have negative angular
velocity. Therefore in an azimuthal direction elec-
trons move contrary to the force acting on them, and
being displaced on radius. Sometimes this phenom-
enon is named the effect of ”negative mass” [9]. As a
result electrons are bunching on an azimuth at tran-
sition from the phase of deceleration to the phase of
acceleration. Electrons trapped by the wave move to-
gether with the wave. For them the condition of syn-
chronism (34) is satisfied. If parameters of orbitron
are those, that electrons, being in decelerating phase
more than in accelerating phase the electromagnetic
waves will be generated in orbitron. Let’s consider
an electron of the bunch, being in decelerating phase.
Having given to the wave the part of its kinetic Wkϕ
and potential Wp energy, electron leaves synchronism
(34). At that it is decelerated in an azimuthal direc-
tion, but owing to the effect of ”negative mass” its
114
angular velocity increases. As a result electron again
gets in synchronism with the wave and gives to it the
next portion of the energy W ′ , gradually approaches
to the string. Electron, being in accelerating phase
gains a part of its energy from the wave and also
leaves synchronism (34). At that its Wkϕ and Wp
are increased. Acceleration in an azimuthal direction
results owing to the effect of ”negative mass” to that
angular velocity of electron decreases. Again it gets
in synchronism with the wave (34) and gains from
the wave the next portion of energy, gradually ap-
proaches to the cathode. Thus, during the process
of energy exchange with the wave in orbitron elec-
tron restores the angular velocity ω0 and does not
leave synchronism (34). In magnetron electrons also
continuously restore a condition of synchronism (34)
during interaction with the wave [6]. A condition for
increments dVϕ and dr , at which angular velocity is
restored after interaction of electron with the wave it
is possible to obtain from the formula (34) for ω0 . It
has the view
dVϕ
Vϕ
=
dr
r
. (37)
If by means of (29) to pass in the ratio (37) to vari-
ables Wp and Wkϕ , then the expression (37) takes
the following view:
dWp
dWkϕ
=
V 2
Q
V 2
ϕ
. (38)
The formula (38) gives the ratio of a part of poten-
tial energy to a part of kinetic energy of electron lost
or gained by electron during its interaction with the
wave. If there are too much electrons in the interac-
tion region, the intensive energy exchange of the wave
with electrons it can be occurred electron bunching.
At that the amount of electrons, being in deceler-
ating phase and in accelerating phase are approxi-
mately equal. The decelerated electrons quickly ex-
cite the wave of very big amplitude and at once set-
tle on the string. Accelerated electrons, being in the
field of high amplitude, gain energy from the wave
and quickly settle on the cathode. Finally energy
of the field in orbitron appears close to zero. The
similar phenomenon takes place also in magnetron
[7]. According to experimental data [1] the density
of electrons in orbitron is rather low, as their plasma
frequency approximately is much less than the fre-
quency of generated waves. In [2] it is shown, that
increments of cold waves growth in orbitron quickly
decrease with growth of azimuthal number of a har-
monic |m| . It allows to consider, that the main con-
tribution to expansion (2) will be given by harmonics
with |m| = 1 and by several first numbers n of ra-
dial harmonics. Firstly the calculations for a single
wave have been carried out with |m| = 1 and n = 1 .
Energy of a wave ε everywhere is presented in terms
of erg /cm. Calculations which results are presented
below, are executed at the following parameters:
b
a
= 100, as11 = −0.1, ϕsc =
π
2
. (39)
In Fig. 1 it is shown how energy ε (erg /cm) of the
excited wave in orbitron changes in time at the fol-
lowing values parameters:
Ñ
l
= 1011, U = 1.5, α = 0.2, βϕ0 = 0.0343, ρ0 = 2.07.
(40)
it0
21
10.5
ε
7141 4234114281 2856121421 3570111
Fig.1. Dependence energy ε from temporal step it
Along the axis of abscissa the amount of steps in
time is marked. The value of the step in time ∆t is
equal
∆t = 0.05
a
Vϕ0
(41)
and amount of particles N = 800 . On time interval
I initial fluctuation (7,12) amplifies. On time interval
II bunching of electrons takes place. In the beginning
of this interval there is a small splash in energy of the
wave, which is explained by that approximately the
half of non bunched electrons, being in decelerated
phase, gives energy to the wave, and other half of
electrons, which is being in accelerating phase, gains
energy from the wave. At a stage III delays electrons
bunched in decelerating phase give energy to the wave
and settle on the string. Dependence of the amount of
particles which settle on the string on time is similar
to the dependence ε and for brevity is not presented.
In Fig. 2-5 it is shown how values η in % (continuous
lines) and ε in erg/cm (dashed lines) depend on ρ0 at
various values of parameters Ñ/l, U . First of all we
note, that the wave is effectively excited by electrons,
which initial radial coordinates ρ0 lay in a cylindrical
layer in immediate proximity from the string (anode).
1 1.5 2 2.5 3 3.5
10
12
14
16
18
20
22
24
N/l=10
11
, V=1.5 kV
η
ρ
0
η
(
%
)
ε
ε
(
e
rg
/c
m
)
0
5
10
15
20
25
30
35
~
Fig.2. Dependence ε, η from ρ0
115
3 3.5 4 4.5 5 5.5 6
8
10
12
14
16
18
20
22
24
26
28
N/l = 10
11
, V = 5 kV
η
ρ
0
η
(
%
)
ε
ε
(
e
rg
/c
m
)
20
40
60
80
100
120
140
~
Fig.3.Dependence ε, η from ρ0
1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
3
4
5
6
7
8
9
10
11
12
N/l = 10
12
, V = 1.5 kV
η
ρ
0
η
(
%
)
ε
ε
(
e
rg
/c
m
)
0
20
40
60
80
100
120
140
~
Fig.4. Dependence ε, η from ρ0
2.5 3 3.5 4 4.5 5 5.5 6
6
8
10
12
14
16
18
N/l = 10
12
, V = 5 kV
η
ρ
0
η
(
%
)
ε
ε
(
e
rg
/c
m
)
100
200
300
400
500
600
700
800
900
~
Fig.5. Dependence ε, η from ρ0
Thickness of this layer is small in comparison with
radius b of the external cylinder of the coaxial. It
is in the consent with experimental data [1], and as-
sumptions of work [2] according to which near to the
string the field generation in orbitron takes place.
Outside of this layer the values ε, η decrease. The
bottom border of the layer is determined by that elec-
trons, located near to it, have the small initial energy
W ′ and cannot excite significant energy of the field.
Besides because of deviations of orbits from circular
one these electrons can quickly settle on the string,
without exciting the wave. Electrons, located near
to the right border of the layer ρ0, have greater az-
imuthal velocity. Therefore significant part of them
gets on the cathode that reduces ε, η . Comparing
Fig. 2, 4 on the one hand and Fig. 3, 5 on the other
hand, it can see that, having increased a voltage U
at constant electron density, it is possible to increase
considerably the energy of the excited field without
reduction of efficiency of its excitation.
It is explained by that for existence of trajectories
of electrons, close to circular ones, with U growth Vϕ0
should be increased that leads to increase in energy
which electron can give to the wave. From compar-
ison Fg. 2, 3 with Fig. 4, 5 it is visible that the
increase in density of electrons at a constant volt-
age leads to reduction of efficiency of the field gen-
eration in orbitron. It is connected with that the
increase of amount of electrons in the interaction re-
gion leads to increase in the wave amplitude. At that
electrons, being in accelerating phase, settle on the
cathode that leads to reduction of η value. If one
takes electron density even greater then not only wave
amplitude increases, but increment of its growth in-
creases too. At that nonbunched electrons interact
with the wave. They do not excite the field, about
what it was spoken above. This case is illustrated
by Fig.6 where calculations with a set of parame-
ters (39-41), in which the density of electrons is in-
creased up to the value Ñ/l = 1013 , are presented.
it0
610
305
ε
7141 4234114281 2856121421 357011
Fig.6. Dependence energy ε from temporal step it
Thus, from the point of view of values η, ε, there
is some optimal value of electron density in the inter-
action region. Calculations under the formula (38)
showed, that in the initial moment of time for var-
ious parameters dWp/dWkϕ = V 2
Q/V 2
ϕ0 ≈ 0.5...1.0.
In process of transforming energy to the wave V 2
ϕ
decreases, and it leads to increase in a share of po-
tential energy in the energy V 2
ϕ , given to the wave.
The part W ′
p of potential energy of electrons trans-
forms to kinetic energy W ′
kr of their radial motion
which goes on a warming up of the anode. We note
that in magnetron the magnetic field turns trajec-
tories of electrons in such a way, that their radial
motion transforms into azimuthal one. As a result
Wkr transforms into energy Wkϕ which is given to
the wave. Therefore in magnetron the efficiency is
higher, than in orbitron. The analysis lead above
has shown, that mechanisms of electromagnetic waves
generation in orbitron and in magnetron have much
in common.The calculations with taking into account
higher radial harmonics were also carried out. At that
116
parameters (39-41) were used. Calculations with tak-
ing into account two and three radial harmonics gave
the following values of field energy and efficiency: at
n = 1 it is obtained ε =20.3, η =24%; at n = 1, 2
ε=17.5, η =20%; at n = 1, 2, 3 ε =18.2,η =21%. The
results obtained with two and three harmonics, differ
by several percents, therefore there is a saturation of
results at increase of n .
4. THE CONCLUSION
The nonlinear theory of electromagnetic waves gen-
eration in orbitron is developed. The set of the equa-
tions including the equations of field excitation and
the equations of 2-dimensional motion is constructed
and numerically solved. It is shown, that mechanism
of electron bunching and energy exchange of electrons
with the wave in orbitron and in magnetron has much
in common. For the fixed parameters of orbitron from
the point of view of generated energy and electronic
efficiency there is some optimal value of electron den-
sity in the interaction region.
References
1. I. Alexeff, F. Dyer. Millimeter microwave emis-
sion from a maser by use of plasma-produced
electrons orbiting a positively charged wire //
Phys. Rev. Lett. 1980, v.45, N5, p.351-354.
2. V.V. Dolgopolov, Yu.V. Kirichenko, Yu.F. Lonin,
I.F. Kharchenko. Generatiion of electromagnetic
waves in the cylindrical resonator by electrons
rotating in radial electrostatic field // J.T.Ph.
1998, v.68, N8, p.91-94 (in Russian).
3. V.V. Dolgopolov, Yu.V. Kirichenko,
I.F. Kharchenko. Generation of electromag-
netic waves by relativistic electrons rotating in
a radial electrostatic field // Izvestia Vuzov.
Radioelectronics. 1999, v.42, N2, p.33-40 (in
Russian).
4. Yu.V. Kirichenko. Generation of electromagnetic
waves by relativistic electrons in the resonator
with crossed radial electrostatic and axial mag-
netic fields in conditions of a plasma resonance //
J.T.Ph. 1999, v.69, N6, p.112-114 (in Russian).
5. Yu.V. Kirichenko. Nonlinear dynamics of elec-
trons in rotating electromagnetic field // Izvestia
Vuzov. Radioelectronics. 2005, v.48, N6, p.29-36
(in Russian).
6. I.V. Lebedev. Technique and devices of the mi-
crowave. V.2. M.: ”High school”, 1972, 376p. (in
Russian).
7. Electron microwave devices with the crossed
fields. Volume 1. Basic elements of devices.
M.: Ed. ”Foreign literature”., 1961, 556p. (in
Russian).
8. V.I. Gajduk, K.I. Palatov, D.M. Peters. Physical
base of microwave electronics. M.: ”Sov. Radio”,
1971, 600p. (in Russian).
9. A..A. Kolomenskij, A.N. Lebedev. Stability of
the charged beam in storage systems // Atom-
naya Energia. 1959, v.7, N6, p.549-550 (in
Russian).
НЕЛИНЕЙНАЯ ТЕОРИЯ ВОЗБУЖДЕНИЯ ЭЛЕКТРОМАГНИТНОГО ПОЛЯ В
ОРБИТРОНЕ
Ю.В. Кириченко, И.Н. Онищенко
В работе развита нелинейная теория генерации электромагнитных волн в орбитроне. Построена и
численно решена система уравнений возбуждения и уравнений движения. Показано, что механизмы
группировки и обмена энергией электрона с волной в орбитроне и магнетроне имеют много общего.
Для фиксированных параметров орбитрона имеется некоторое оптимальное с точки зрения генери-
руемой энергии и электронного коэффициента полезного действия значение плотности электронов в
пространстве взаимодействия. Достаточно точное описание процесса возбуждения волн в орбитроне
можно получить, ограничиваясь основной собственной гармоникой.
НЕЛIНIЙНА ТЕОРIЯ ЗБУДЖЕННЯ ЕЛЕКТРОМАГНIТНОГО ПОЛЯ В ОРБIТРОНI
Ю.В. Кириченко, I.М. Онiщенко
В роботi розвинуто нелiнiйну теорiю генерацiї електромагнiтних хвиль в орбiтронi. Побудовано i
чисельно розв’язано систему рiвнянь , що включає рiвняння збудження та руху. Показано, що ме-
ханiзми групування та обмiну енергiєю електрона з хвилею в орбiтронi та магнетронi мають багато
спiльного. Для фiксованих параметрiв орбiтрона щiльнiсть електронiв у просторi взаємодiї має опти-
мальне значення з точки зору енергiї, що генерується, та коефiцiєнта корисної дiї. Достатньо точний
опис процесу збудження хвиль в орбiтронi можна отримати за допомогою основної власної гармонiки.
117
|
| id | nasplib_isofts_kiev_ua-123456789-96517 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-01T01:50:13Z |
| publishDate | 2009 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Kirichenko, Yu.V. Onishchenko, I.N. 2016-03-17T21:19:31Z 2016-03-17T21:19:31Z 2009 The nonlinear theory of the electromagnetic field excitation in orbitron / Yu.V. Kirichenko, I.N. Onishchenko // Вопросы атомной науки и техники. — 2009. — № 5. — С. 110-117. — Бібліогр.: 9 назв. — англ. 1562-6016 PACS: 84.30.Jc, 85.10.Jz https://nasplib.isofts.kiev.ua/handle/123456789/96517 The nonlinear theory of electromagnetic waves generation in orbitron is developed. The set of the equations including the equations of field excitation and the equations of 2-dimential motion is constructed and numerically solved. It is shown, that mechanism of electron bunching and energy exchange of electrons with the wave in orbitron and in magnetron has much in common. For the fixed parameters of orbitron from the point of view of generated energy and electronic efficiency there is some optimal value of electron density in the interaction region. В роботi розвинуто нелiнiйну теорiю генерацiї електромагнiтних хвиль в орбiтронi. Побудовано i чисельно розв’язано систему рiвнянь , що включає рiвняння збудження та руху. Показано, що механiзми групування та обмiну енергiєю електрона з хвилею в орбiтронi та магнетронi мають багато спiльного. Для фiксованих параметрiв орбiтрона щiльнiсть електронiв у просторi взаємодiї має оптимальне значення з точки зору енергiї, що генерується, та коефiцiєнта корисної дiї. Достатньо точний опис процесу збудження хвиль в орбiтронi можна отримати за допомогою основної власної гармонiки. В работе развита нелинейная теория генерации электромагнитных волн в орбитроне. Построена и численно решена система уравнений возбуждения и уравнений движения. Показано, что механизмы группировки и обмена энергией электрона с волной в орбитроне и магнетроне имеют много общего. Для фиксированных параметров орбитрона имеется некоторое оптимальное с точки зрения генерируемой энергии и электронного коэффициента полезного действия значение плотности электронов в пространстве взаимодействия. Достаточно точное описание процесса возбуждения волн в орбитроне можно получить, ограничиваясь основной собственной гармоникой. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Электродинамика The nonlinear theory of the electromagnetic field excitation in orbitron Нелiнiйна теорiя збудження електромагнiтного поля в орбiтронi Нелинейная теория возбуждения электромагнитного поля в орбитроне Article published earlier |
| spellingShingle | The nonlinear theory of the electromagnetic field excitation in orbitron Kirichenko, Yu.V. Onishchenko, I.N. Электродинамика |
| title | The nonlinear theory of the electromagnetic field excitation in orbitron |
| title_alt | Нелiнiйна теорiя збудження електромагнiтного поля в орбiтронi Нелинейная теория возбуждения электромагнитного поля в орбитроне |
| title_full | The nonlinear theory of the electromagnetic field excitation in orbitron |
| title_fullStr | The nonlinear theory of the electromagnetic field excitation in orbitron |
| title_full_unstemmed | The nonlinear theory of the electromagnetic field excitation in orbitron |
| title_short | The nonlinear theory of the electromagnetic field excitation in orbitron |
| title_sort | nonlinear theory of the electromagnetic field excitation in orbitron |
| topic | Электродинамика |
| topic_facet | Электродинамика |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/96517 |
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