Тhe effect of nonreciprocity on the dynamics of coupled oscillators and coupled waves
The dynamics of nonreciprocally coupled oscillators and coupled waves is studied. Such coupling can lead to converting the energy of low-frequency (LF) oscillations to the energy of high-frequency (HF) oscillations. The influence of the resonant properties of the coupling elements on the condition...
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Buts, V.A. Vavriv, D.M. 2019-02-14T18:40:59Z 2019-02-14T18:40:59Z 2018 Тhe effect of nonreciprocity on the dynamics of coupled oscillators and coupled waves / V.A. Buts, D.M. Vavriv // Вопросы атомной науки и техники. — 2018. — № 4. — С. 213-216. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 04.30.Nk; 52.35.Mw; 52.35.Mw; 78.70.Gq https://nasplib.isofts.kiev.ua/handle/123456789/147438 The dynamics of nonreciprocally coupled oscillators and coupled waves is studied. Such coupling can lead to converting the energy of low-frequency (LF) oscillations to the energy of high-frequency (HF) oscillations. The influence of the resonant properties of the coupling elements on the conditions of the energy conversion is carried out. It is shown that this conversion can be realized when either the amplitude or the phase of the coupling coefficient is modulated at a low-frequency. By the example of waves in a rare magnetoactive plasma, it is shown for the first time that the discussed energy conversion takes also place in a system of interacting waves. Results of analytical and numerical studies illustrating the conditions for the excitation of high-frequency oscillations due to the energy conversion are presented. Досліджено динаміку зв'язаних осциляторів і хвиль при наявності невзаємного зв'язку між ними, який призводить до можливості перетворення енергії низькочастотних коливань в енергію високочастотних коливань. Проведено аналіз впливу резонансних властивостей елементів зв'язку на умови перетворення енергії. Показано, що перетворення енергії можна реалізувати як при низькочастотній амплітудній, так і при низькочастотній фазовій модуляції коефіцієнта зв'язку. На прикладі аналізу поширення хвиль у рідкій магнітоактивній плазмі вперше показана можливість перетворення енергії і при взаємодії хвиль. Наведено результати аналітичного і чисельного дослідження, що ілюструють умови збудження високочастотних коливань і їх властивості. Исследована динамика связанных осцилляторов и волн при наличии невзаимной связи между ними, которая приводит к возможности преобразования энергии низкочастотных колебаний в энергию высокочастотных колебаний. Проведен анализ влияния резонансных свойств элементов связи на условия преобразования энергии. Показано, что преобразование энергии можно реализовать как при низкочастотной амплитудной, так и при низкочастотной фазовой модуляции коэффициента связи. На примере анализа распространения волн в редкой магнитоактивной плазме впервые показана возможность преобразования энергии и при взаимодействии волн. Приведены результаты аналитического и численного исследований, иллюстрирующие условия возбуждения высокочастотных колебаний и их свойства. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Нелинейные процессы Тhe effect of nonreciprocity on the dynamics of coupled oscillators and coupled waves Роль невзаємності в теорії пов'язаних коливань і пов'язаних хвиль Роль невзаимности в теории связанных колебаний и связанных волн Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Тhe effect of nonreciprocity on the dynamics of coupled oscillators and coupled waves |
| spellingShingle |
Тhe effect of nonreciprocity on the dynamics of coupled oscillators and coupled waves Buts, V.A. Vavriv, D.M. Нелинейные процессы |
| title_short |
Тhe effect of nonreciprocity on the dynamics of coupled oscillators and coupled waves |
| title_full |
Тhe effect of nonreciprocity on the dynamics of coupled oscillators and coupled waves |
| title_fullStr |
Тhe effect of nonreciprocity on the dynamics of coupled oscillators and coupled waves |
| title_full_unstemmed |
Тhe effect of nonreciprocity on the dynamics of coupled oscillators and coupled waves |
| title_sort |
тhe effect of nonreciprocity on the dynamics of coupled oscillators and coupled waves |
| author |
Buts, V.A. Vavriv, D.M. |
| author_facet |
Buts, V.A. Vavriv, D.M. |
| topic |
Нелинейные процессы |
| topic_facet |
Нелинейные процессы |
| publishDate |
2018 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Роль невзаємності в теорії пов'язаних коливань і пов'язаних хвиль Роль невзаимности в теории связанных колебаний и связанных волн |
| description |
The dynamics of nonreciprocally coupled oscillators and coupled waves is studied. Such coupling can lead to
converting the energy of low-frequency (LF) oscillations to the energy of high-frequency (HF) oscillations. The
influence of the resonant properties of the coupling elements on the conditions of the energy conversion is carried
out. It is shown that this conversion can be realized when either the amplitude or the phase of the coupling coefficient is modulated at a low-frequency. By the example of waves in a rare magnetoactive plasma, it is shown for the
first time that the discussed energy conversion takes also place in a system of interacting waves. Results of analytical and numerical studies illustrating the conditions for the excitation of high-frequency oscillations due to the energy conversion are presented.
Досліджено динаміку зв'язаних осциляторів і хвиль при наявності невзаємного зв'язку між ними, який
призводить до можливості перетворення енергії низькочастотних коливань в енергію високочастотних коливань. Проведено аналіз впливу резонансних властивостей елементів зв'язку на умови перетворення енергії. Показано, що перетворення енергії можна реалізувати як при низькочастотній амплітудній, так і при
низькочастотній фазовій модуляції коефіцієнта зв'язку. На прикладі аналізу поширення хвиль у рідкій магнітоактивній плазмі вперше показана можливість перетворення енергії і при взаємодії хвиль. Наведено результати аналітичного і чисельного дослідження, що ілюструють умови збудження високочастотних коливань і їх властивості.
Исследована динамика связанных осцилляторов и волн при наличии невзаимной связи между ними, которая приводит к возможности преобразования энергии низкочастотных колебаний в энергию высокочастотных колебаний. Проведен анализ влияния резонансных свойств элементов связи на условия преобразования энергии. Показано, что преобразование энергии можно реализовать как при низкочастотной амплитудной, так и при низкочастотной фазовой модуляции коэффициента связи. На примере анализа распространения волн в редкой магнитоактивной плазме впервые показана возможность преобразования энергии и
при взаимодействии волн. Приведены результаты аналитического и численного исследований, иллюстрирующие условия возбуждения высокочастотных колебаний и их свойства.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/147438 |
| citation_txt |
Тhe effect of nonreciprocity on the dynamics of coupled oscillators and coupled waves / V.A. Buts, D.M. Vavriv // Вопросы атомной науки и техники. — 2018. — № 4. — С. 213-216. — Бібліогр.: 5 назв. — англ. |
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2025-11-25T23:55:25Z |
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| fulltext |
ISSN 1562-6016. ВАНТ. 2018. №4(116) 213
THE EFFECT OF NONRECIPROCITY ON THE DYNAMICS
OF COUPLED OSCILLATORS AND COUPLED WAVES
V.A. Buts1,2, D.M. Vavriv3
1National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine;
2V.N. Karazin Kharkov National University, Kharkov, Ukraine;
3Institute of Radio Astronomy of NAS of Ukraine, Kharkov, Ukraine
E-mail: vbuts@kipt.kharkov.ua; vavriv@rian.kharkov.ua
The dynamics of nonreciprocally coupled oscillators and coupled waves is studied. Such coupling can lead to
converting the energy of low-frequency (LF) oscillations to the energy of high-frequency (HF) oscillations. The
influence of the resonant properties of the coupling elements on the conditions of the energy conversion is carried
out. It is shown that this conversion can be realized when either the amplitude or the phase of the coupling coeffi-
cient is modulated at a low-frequency. By the example of waves in a rare magnetoactive plasma, it is shown for the
first time that the discussed energy conversion takes also place in a system of interacting waves. Results of analyti-
cal and numerical studies illustrating the conditions for the excitation of high-frequency oscillations due to the ener-
gy conversion are presented.
PACS: 04.30.Nk; 52.35.Mw; 52.35.Mw; 78.70.Gq
INTRODUCTION
In the papers [1, 2], it was described that the intro-
duction of a non-reciprocal coupling between oscillating
systems opens a possibility of the energy converting
from low-frequency (LF) oscillations to high-frequency
(HF) oscillations. It was also noted that the most inter-
esting direction of using the discovered mechanism of
the energy conversion is related with the development
of novel types of sources of electromagnetic radiation,
including sources of terahertz radiation. It should be
noted that in this frequency range, the coupling ele-
ments themselves have usually resonant properties.
However, in [1, 2] these elements were considered as
frequency-independent elements. Therefore, it seems
important to investigate the influence of resonant prop-
erties of the coupling elements on the dynamics of cou-
pled oscillators. The corresponding analysis is presented
in the next section, where each of the coupling elements
is also considered as a separate oscillator coupled with
other oscillators.
So far, the case was considered [1, 2] when the non-
reciprocity of the coupling and the LF modulation of the
coupling coefficient were provided by introducing an
amplitude modulation of the coupling coefficients. It
should be borne in mind that the amplitude modulation
is not always practical, therefore, we also consider in
this paper a possibility of applying a LF phase modula-
tion to realize the conversion of LF oscillations into HF
ones. The results of this study are presented in the third
section of the paper.
In our previous works [1, 2], we considered systems
where HF oscillations are excited as a result of three-
frequency interaction under the following resonance
conditions
1 2 ,n nω ω ω− = ±
where 1nω and 2nω are normal frequencies of a system
of two coupled oscillators, ω is the frequency of LF
modulation, which is much smaller than 1nω , 2nω . The
case was mainly considered when the partial frequen-
cies 1pω , 2pω of the oscillators are equal. In the fourth
section of this paper it is shown that, with an appropri-
ate choice of the coupling between the oscillators, the
excitation of HF oscillations is possible even if the
following condition is satisfied
1 2p pω ω ω− = ±
that is, with unequal partial frequencies of the interact-
ing oscillators.
In the fifth section, it is shown that the mechanism
of the energy conversion found in a system of interact-
ing oscillators can be also realized when waves interact.
This result is demonstrated by analyzing the propaga-
tion of two high-frequency transverse electromagnetic
waves in a rare magnetoactive plasma.
The last section summarizes the results presented in
the paper.
1. INFLUENCE OF RESONANT
PROPERTIES OF COUPLING ELEMENTS
As mentioned above, at high frequencies coupling
elements can have resonant properties and should also
be regarded as oscillatory systems. In this case, the
simplest model of coupled oscillators can be represent-
ed as a ring of four oscillators, as shown in Fig. 1.
Fig. 1. Scheme of the oscillatory system.
Arrows indicate the direction of connection
Numbers 1 and 3 show high-frequency oscillation
systems (resonators). The natural frequencies of these
resonators are equal to 0ω . Numbers 2 and 4 denote
nonreciprocal coupling elements. The eigenfrequencies
of these elements coincide and are equal to 1ω . Arrows
indicate the direction of the wave propagation in this
ring. A set of equations that describes the dynamics of
such a system can be represented in the form:
mailto:vbuts@kipt.kharkov.ua
ISSN 1562-6016. ВАНТ. 2018. №4(116) 214
2
1 0 1 4x x xω µ+ = ;
2
2 1 2 1x x xω µ+ = ; (1)
2
3 0 3 2x x xω µ+ = ;
2
4 1 4 3x x xω µ+ = .
It is easy to determine that this oscillatory system
has the following normal frequencies:
( )2 2 2
0 0 1 0/ 2N ω µ ω ω ωΩ = ± − .
Assuming that the coupling coefficients are small
( 2
0,1µ ω<< ), from (1), one can find the following aver-
aged equations for determining the complex amplitudes
of the coupled oscillators:
1 4
02
a a
i
µ
ω
= ;
2 1
12
a a
i
µ
ω
= ; (2)
3 2
02
a a
i
µ
ω
= ;
4 3
1
.
2
a a
i
µ
ω
=
Note that this set of equations is not changed if some
coupling coefficients are slow varying functions of
time. Let, for example, the coupling coefficients in the
third and fourth equations of (2) are such functions of
time 1( )tµ µ= . Then from (2), we find the following
equation describing the dynamics of the amplitudes of
the first and third oscillators (resonators):
[ ] [ ]
2
2
1 3 1 32 0d a a a a
dt
+ +Ω + = , (3)
where ( )22
1 0 1(t) / 4µ ω ωΩ = .
Let us assume that the function ( )21(t)µ has the fol-
lowing form ( ) ( )( )2 4
1 2(t) 1 cos 2µ ω ε t= + , where
2
2 0 1/ 2tt ω ω ω= ⋅ and 2
2 0 1ω ω ω<< . Then (3) can be
reduced to Mathieu equation:
[ ] ( )[ ]
2
1 3 1 32 1 cos2 0d a a a a
d
ε t
t
+ + + + = . (4)
From this equation, it follows that the presence of
the resonant properties of the coupling elements does
not prevent the energy transformation from LF oscilla-
tions into the energy of HF oscillations. One should
only take into account that the eigenfrequencies of the
coupling elements are essential parameters of the entire
oscillatory system, and their parameters should be ap-
propriately selected for the realization of the considered
energy transformation.
2. MODULATION OF THE PHASE
OF THE COUPLING COEFFICIENTS
In the papers [1, 2], an amplitude modulation of the
coupling coefficients was considered. For a number of
practical applications, it is more convenient to use a
phase modulation of the coupling coefficient. To de-
scribe the dynamics of a system of two coupled identi-
cal oscillators, in which the phase of the coupling ele-
ments is a function of time, we use the following system
of equations:
[ ]1 1 2( ) exp( ( )x x t i t xµ ϕ+ = ;
2 2 0 1x x xµ+ = . (5)
Here ( ) ( )t and tµ ϕ are real slow varying func-
tions of time. We look for a solution of (5) in the fol-
lowing form:
( ) ( )1 1 2exp expx A it A it= ⋅ + ⋅ −
( ) ( )2 1 2exp expx B it B it= ⋅ + ⋅ − . (6)
Here the amplitudes kA and kB (k=1, 2) are slow
varying functions of time. Applying the averaging tech-
nique to (6), we obtain to the following set of equations:
[ ]1 1( ) exp( ( )iA t i t Bµ ϕ= ,
1 0 1iB Aµ= , (7)
[ ]2 2( ) exp( ( )iA t i t Bµ ϕ− = ,
2 0 2.iB Aµ− =
Here the amplitudes kA and kB are complex func-
tions: k k kA A iA′ ′′= + ; k k kB B iB′ ′′= + . To find the real and
imaginary components, we come to the following sys-
tem of equations:
[ ] [ ]1 0 1 0 1( )cos ( )sinB t B t Bµ µ ϕ µ µ ϕ′ ′ ′′+ = ,
[ ] [ ]1 0 1 0 1( )cos ( )sinB t B t Bµ µ ϕ µ µ ϕ′′ ′′ ′+ = − . (8)
An analogous system can also be obtained for the
function k k kA A iA′ ′′= + . The sets of equations (5) and
(8) were solved numerically. In Fig. 2 typical build-up
of the amplitude of high-frequency oscillations is
shown. This example illustrate that the energy conver-
sion can be also realized when the phase of the coupling
coefficient is modulated at a low-frequency.
Fig. 2. The characteristic dependence of the amplitudes
of the oscillators on time ( )0nT tµ µ≡ ⋅ at sinϕ t= ; 1(0) 1x =
3. EXCITATION OF HF OSCILLATIONS
WHEN THE PARTIAL FREQUENCIES
DO NOT COINCIDE
In the works [1, 2], the partial frequencies of the in-
teracting oscillators were considered to be equal. Under
this condition, a three-frequency interaction and the
excitation of HF oscillations were realized when the
difference of the normal frequencies was approximately
equal to the frequency of the LF modulation.
In this section, we show that with a certain method
of oscillators coupling, HF oscillations are excited also
when the partial frequencies do not coincide. This case
is realized in the absence of constant in time coupling
between the oscillators. However, a nonreciprocity of
the coupling, as in the previous case, is needed. A set of
equations that describes such a coupled oscillatory sys-
tem can be represented as:
ISSN 1562-6016. ВАНТ. 2018. №4(116) 215
( )2
1 1 1 2 cosq q q tω µ ω+ ⋅ = ⋅
( )2
2 2 2 1 cosq q q tω µ ω+ ⋅ = − ⋅ . (9)
Here 2 1ω ω ω= − is the low frequency modulation
of the coupling between the high-frequency oscillators.
We look for the solution of (9) in the form:
( ) exp( ) ( ) exp( ).k k k k kq A t i t B t i tω ω= + −
To find equations for slowly varying amplitudes, we
at first come from (9) to the following system of equa-
tions:
[ ]
1 1 1 1
2 2 2 2
1
( )exp( ) ( )exp( )
exp( ) exp( ) cos( )
2
A t i t B t i t
A i t B i t t
i
ω ω
µ ω ω ω
ω
− − =
= + −
[ ]
2 2 2 2
1 1 1 1
2
( ) exp( ) ( ) exp( )
exp( ) exp( ) cos( ).
2
A t i t B t i t
A i t B i t t
i
ω ω
µ ω ω ω
ω
− − =
= − + −
(10)
From this equations, it is easy to determine the fol-
lowing relations between the complex amplitudes
1 2
14
A A
i
µ
ω
= ;
2 1
24
A A
i
µ
ω
= − ; (11)
2
1 1
1 2
0
16
A Aµ
ω ω
− = .
From (11), it immediately follows that the excitation
of high-frequency oscillations can also occur in such a
system as illustrated in Fig. 3. This figure shows the
solution of the system of equations (9) at such parame-
ters: 0.2µ = ; 0.01ω = ; 1 1ω = ; 2 1.01ω = ;
1(0) 0.1q = .
Fig. 3. The excitation of oscillations of two coupled,
different high-frequency oscillators (see system (9)).
nT t≡
Fig. 4. Dispersion diagram of the waves
participating in the interaction
4. THE ROLE OF NONRECIPROCITY
IN THE DYNAMICS OF COUPLED WAVES
In the above sections, we discussed the existence in
coupled oscillators of LF- to HF energy transfer channel.
It can be expected that a similar channel can exist in
systems with interacting waves. Below, considering an
example of coupling of transverse high-frequency waves
to plasma waves, it is shown that such a channel does
exist. However, as before, it exists only in the presence of
a non-reciprocal coupling between interacting waves.
We consider the following problem. Suppose that
there are two high-frequency transverse electromagnetic
waves that propagate through rare magnetoactive plas-
ma ( ( )22 2~p Hω ω ω<< ). The frequencies of these waves
are large, and the difference of these frequencies is
close to the plasma frequency ( 2 1 pω ω ω− ≈ ). For sim-
plicity, we consider a one-dimensional motion, i.e. all
waves propagate and interact with each other only along
the axis z and in time. A diagram of a possible interac-
tion of the waves is shown in Fig. 4. It can be seen from
diagram 4 that a three-wave interaction occurs. In this
interaction, two transverse high-frequency waves and a
plasma wave are involved. The structure of these waves
and their dispersion characteristics are well known (see,
for example, [4, 5]). The plasma wave is longitudinal
with a large amplitude. We assume that this wave is
given. In this case, the plasma frequency can be repre-
sented only by its wave characteristic:
( )24 / . .p pe n m exp i z i t k cω p k ω = − + (12)
The equation for the electrical component of the
field of the transverse electromagnetic waves can be
obtained from the Maxwell equations:
2
2 2
1 (D) 0E
c t
∂
D − =
∂
. (13)
Here, ˆD Eε=
and ε̂ is the plasma permittivity tensor.
We look for the components of the electric field of
transverse waves in the form
, 2,1 2,1 2,1( , ) ( ) . .x yE z t A t exp ik z i t k cω = − +
(14)
Then it is convenient to represent equation (13) in
the form of the following set of equations with respect
to the amplitudes 1,2A :
2
2 22
2 2 1 12
A A i A
t
ω ε ω+
∂
+ = −
∂
;
2
2 21
1 1 2 22
A A i A
t
ω ε ω−
∂
+ =
∂
. (14)
Here ( )2 2 2/H p Hε ω ω ω ω ω± ± ±= − are off-diagonal
components of the permittivity tensor;
2 2 2 2 2 2
2 2 1 1/ , / .k c k cω ω= =
The upper sign in these expressions corresponds to a
wave propagating along the magnetic field 2ω ω+ = ; the
lower sign belongs to a wave propagating in the oppo-
site direction 1ω ω− = . When obtaining (14), the condi-
tion of spatial synchronism 2 1k kk = + has been used.
We note that the first wave 1 1, kω propagates in the
direction opposite to the direction of the external mag-
netic field.
ISSN 1562-6016. ВАНТ. 2018. №4(116) 216
For an effective interaction of the waves, it is neces-
sary that together with the spatial synchronism condition,
the time synchronism should be satisfied: 2 1 0pω ω ω− − = .
We look for a solution of (14) in the form
2,1 1,2 2,1( ) . .A a exp i t k cω= − + (15)
Substituting (15) into (14), we obtain the following
equations with respect to the slowly varying amplitudes 2,1a :
2
2,1
1 2 2,12 0
4
a
a
t
ε ε
ω ω+ −∂
− =
∂
. (16)
It can be seen from this equation that the amplitudes
of the transverse waves increase exponentially with the
increment:
( )2 2 2/ 2H p Hω ω ω ωΓ ≈ − . (17)
In this expression, it is taken into account that the
frequencies of the HF waves are close to each other
2 1 1pω ω ω ω− = << . High-frequency transverse elec-
tromagnetic waves receive energy from the LF Lang-
muir waves excited in the plasma.
CONCLUSIONS
We note the most important results presented in the
paper. There are two basic scientific results. The first is
that the conversion of the energy of LF oscillations to
the energy of HF frequency oscillations is a rather
"strong" effect, in the sense that it can be realized in
very different ways (see Sections 3 and 4), and also in
the presence of significant perturbations, like, for ex-
ample, additional resonances in the system (section 2).
The second result is that the availability of nonreci-
procity can create a channel for converting the energy
of LF oscillations to HF oscillations not only in systems
with coupled oscillators, but also in systems with cou-
pled waves (see Section 5). In latter case, the presence
of a nonreciprocity leads to a qualitatively new dynam-
ics of the three-wave interaction. Indeed, it is well
known (see, for example, [4 - 5]) that if at the initial
instant of time the low-frequency wave has the largest
amplitude in a system with a three-wave interaction,
then practically no dynamics with energy exchange can
occur in such system. In the case considered above, the
presence of a nonreciprocity leads to the excitation of
HF waves, in spite of the fact that at the initial moment
only a low-frequency Langmuir wave exists.
In our paper, there are also several results of practi-
cal importance. At first, it is shown that when consider-
ing the excitation of high-frequency oscillations (for
example, in terahertz frequency range), it is necessary to
take into account the oscillatory properties of the cou-
pling elements (see Section 1). Secondly, it is proved
that a phase modulation of the coupling coefficient can
be used as well as an amplitude modulation to realize
the energy transfer. At third, it is shown that oscillatory
systems with different partial frequencies can be used
for the conversion of LF- to HF oscillations. The only
requirement in this case is that the modulation frequen-
cy of the coupling coefficients of these systems should
be approximately equal to the difference between the
partial frequencies.
REFERENCES
1. V.A. Buts, D.M. Vavriv, O.G. Nechayev, D.V. Tarasov.
A Simple Method for Generating Electromagnetic
Oscillations // IEEE Transactions on circuits and
systems II. Express Briefs. 2015, v. 62, № 1, p. 36-40.
2. V.A. Buts, D.M. Vavriv. Role of Non-Reciprocity in
the Theory of Oscillations // Radio Physics and Ra-
dio Astronomy. 2018, v. 23, № 1, p. 60-71.
3. A.I. Akhiezer, I.A. Akhiezer, et al. Plasma Electro-
dynamics. M.: “Nauka”. 1974, 719 p. (in Russian).
4. B.B. Kadomtsev. Collective Phenomena in Plasma.
M.: “Nauka”. 1976, 238 p. (in Russian).
5. H.A. Wilhelmsson, J. Weiland. Coherent Non-
Linear Interaction of Waves in Plasmas. M.: “Ener-
goizdat”, 1981, 224 p. (in Russian).
Article received 29.05.2018
РОЛЬ НЕВЗАИМНОСТИ В ТЕОРИИ СВЯЗАННЫХ КОЛЕБАНИЙ И СВЯЗАННЫХ ВОЛН
В.А. Буц, Д.М. Ваврив
Исследована динамика связанных осцилляторов и волн при наличии невзаимной связи между ними, ко-
торая приводит к возможности преобразования энергии низкочастотных колебаний в энергию высокоча-
стотных колебаний. Проведен анализ влияния резонансных свойств элементов связи на условия преобразо-
вания энергии. Показано, что преобразование энергии можно реализовать как при низкочастотной ампли-
тудной, так и при низкочастотной фазовой модуляции коэффициента связи. На примере анализа распро-
странения волн в редкой магнитоактивной плазме впервые показана возможность преобразования энергии и
при взаимодействии волн. Приведены результаты аналитического и численного исследований, иллюстри-
рующие условия возбуждения высокочастотных колебаний и их свойства.
РОЛЬ НЕВЗАЄМНОСТІ В ТЕОРІЇ ПОВ'ЯЗАНИХ КОЛИВАНЬ І ПОВ'ЯЗАНИХ ХВИЛЬ
В.О. Буц, Д.М. Ваврів
Досліджено динаміку зв'язаних осциляторів і хвиль при наявності невзаємного зв'язку між ними, який
призводить до можливості перетворення енергії низькочастотних коливань в енергію високочастотних ко-
ливань. Проведено аналіз впливу резонансних властивостей елементів зв'язку на умови перетворення енер-
гії. Показано, що перетворення енергії можна реалізувати як при низькочастотній амплітудній, так і при
низькочастотній фазовій модуляції коефіцієнта зв'язку. На прикладі аналізу поширення хвиль у рідкій маг-
нітоактивній плазмі вперше показана можливість перетворення енергії і при взаємодії хвиль. Наведено ре-
зультати аналітичного і чисельного дослідження, що ілюструють умови збудження високочастотних коли-
вань і їх властивості.
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