Simulation of Nonstationary Processes in Backward-Wave Tube with the Self-Modulation of Emission (Carcinotrode)

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ISSN 1064-2269, Journal of Communications Technology and Electronics, 2009, Vol. 54, No. 12, pp. 1403–1412. © Pleiades Publishing, Inc., 2009.
Original Russian Text © V.O. Melikhov, M.V. Nazarov, V.A. Solntsev, 2009, published in Radiotekhnika i Elektronika, 2009, Vol. 54, No. 12, pp. 1481–1490.
INTRODUCTION
Carcinotrode is the backward-wave tube (BWT) with
the self-modulation of emission that employs the working
principles of a conventional BWT and a klystrode [1]. In
the carcinotrode, the high-frequency (HF) output field of
the backward wave of the slow-wave structure (SWS) is
fed to the cathode to modulate the emission. Electron
bunches from the cathode are ejected to a strong HF field
at the beginning of the SWS, so that the efficiency of the
electron–field interaction increases in comparison with the
efficiency of the conventional BWT.
The analytical nonlinear theory of the carcinotrode
from [2–4] and the results of the numerical simulation
based on the equations of linear theory and simplified
equations of nonlinear theory indicate a possibility of a sig-
nificant increase in the efficiency. This work is devoted to
the carcinotrode simulation based on the nonstationary
nonlinear theory.
The amplification and generation of multifrequency
narrow-band signals in microwave devices with a relatively
long electron–wave interaction are analyzed in many
works. Note the resulting quasi-stationary methods, the
nonstationary theory of the waveguide excitation by the
current with a slowly varying amplitude, and the generali-
zation of the particle-in cell method on the nonstationary
processes. The interest in such study is due to the recently
observed regimes of the generation of chaotic signals in
BWTs and TWTs with the delayed feedback. The main
results can be found in [5]. A method and computer codes
for the calculation of the amplification of multifrequency
narrow-band signals in TWT were developed in [6, 7] with
the aid of the nonstationary excitation theory. Note that the
above analyses involve unmodulated electron fluxes at the
SWS entrance, whereas the modulation is important for the
carcinotrode operation. Therefore, we supplement the
known approaches to the simulation of nonlinear interac-
tion processes based on the equation for the waveguide
excitation by a nonstationary current with the features of
simulation related to the boundary conditions with respect
to the beam at the ends of the SWS segments. The results
on the carcinotrode simulation are presented.
1. WAVEGUIDE EXCITATION
BY NONSTATIONARY CURRENT
A nonstationary signal represents a harmonic oscilla-
tion with a slowly varying amplitude, frequency, or phase
(i.e., a slowly varying complex amplitude). The width of
the corresponding narrow spectrum Δω is significantly less
than oscillation frequency ω0. The equation of the nonsta-
tionary theory of the waveguide excitation establishes a
relation between the slowly varying complex amplitudes of
the field excited in the waveguide and the given nonstation-
ary current. Various variants of the derivation of this equa-
tion and the results of its application in the nonstationary
theory of BWTs and TWTs can be found in many works
(see [5]). Here, we present the simplest derivation of the
equation and various representations in terms of dimen-
sionless variables that are helpful in the study of the nonsta-
tionary processes in the carcinotrode.
Simulation of Nonstationary Processes in Backward-Wave Tube
with the Self-Modulation of Emission (Carcinotrode)
V. O. Melikhov, M. V. Nazarov, and V. A. Solntsev
Received July 3, 2009
Abstract—The equations that describe nonlinear nonstationary processes in carcinotrode (backward-
wave tube with the emission modulation in the presence of the field of the output signal fed to the
cathode via a feedback loop) are derived. An algorithm and the corresponding code are developed to
solve the equations with allowance for the modulation of emission using nonuniform (with respect to
time) large particles (electrons of equal charge) ejected from the cathode. The effect of the feedback
parameter on the intensity and shape of the carcinotrode oscillations is analyzed. It is demonstrated
that the carcinotrode efficiency can be increased to about 50% upon the generation of harmonic oscil-
lations. A more significant increase in the efficiency to 70% is possible in the regime of the weak self-
modulation of oscillations upon an increase in the feedback coefficient in the feedback loop involving
the slow-wave structure and the cathode and a decrease in the cathode–grid static field.
PACS numbers: 02.60.Cb, 84.40.Fe
DOI: 10.1134/S1064226909120109
MICROWAVE ELECTRONICS

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JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 54 No. 12 2009
MELIKHOV et al.
We represent nonstationary excitation current J(z, t) as
a Fourier integral or, approximately, as a sum with respect
to frequencies in its spectrum:
(1)
Here, J(z, ω) is the current spectral density and Jn(z) =
J(z, ωn)δω is the complex amplitude of current at fre-
quencies ωn resulting from the spectral discretization
with the step δω = Δω/M. The total number of discrete
frequencies in interval Δω is 2M + 1, and the frequen-
cies can be represented as
(2)
on the assumption that n = 0 corresponds to the center
of the spectrum.
The longitudinal field of the matched wave is excited at
all of the spectral frequencies in the vicinity of frequency
ω0, and such a field can be represented using an expression
similar to expression (1):
(3)
The complex amplitude of the field at frequency En(z) =
E(z, ωn)δω is determined by the known equation of
excitation, and spectral density E(z, ω) satisfies the
same equation:
(4)
where h(ω) and R(ω) are the wave number and specific
coupling resistance of the matched wave at frequency ω
and the lower and upper signs correspond to the back-
ward-wave (BWT) and forward-wave (TWT) excita-
tion, respectively.
We represent the HF current of the beam and the corre-
sponding excited field as harmonic oscillations with carrier
frequency ω0 and slowly varying complex amplitudes.
Note that frequency ω0 can be arbitrarily chosen within
spectral width Δω.
Expressions (1) and (3) can be represented as
(5)
J z t,( ) Re
1
π
--- J z ω,( ) iωt–( )exp ωd
Δω
∫=
≈ Re Jn z( ) iωnt–( ).exp
n
∑
1
π
---
ωn ω0 n
Δω
M
--------, n+ 0 1 2 …M,,±,±,= =
Ez z t,( ) Re
1
π
--- E z ω,( ) iωt–( )exp ωd
Δω
∫=
≈ Re En z( ) iωnt–( ).exp
n
∑
1
π
---
dE z ω,( )
dz
--------------------- ih ω( )E z ω,( )–
R ω( )
2
-------------J z ω,( ),+−=
J z t,( ) ReJ˜ z t,( ) iω0t–( ),exp=
Ez z t,( ) ReE˜ z t,( ) iω0t–( ),exp=
where
(6)
are slowly varying complex amplitudes of the current
and field, since ω – ω0 < Δω Ӷ ω0. To derive the equa-
tion of excitation that establishes a relation between
these amplitudes, we employ a Taylor-series expansion
accurate to the terms of the first order:
(7)
Multiplying Eq. (4) by exp(–i(ω – ω0)t) and integrat-
ing with respect to all frequencies with allowance for
expressions (6) and (7), we obtain
It is seen that the integrals in expression (6) are deter-
mined by time derivatives of slowly varying amplitudes of
the field and current. Thus, we derive the following equa-
tion in terms of these amplitudes:
(8)
This equation for the waveguide excitation by the non-
stationary current is derived with disregard of the terms of
the second and higher orders in expansion (7) for h(ω),
which characterize the perturbation of the envelope upon
the propagation of waves in the waveguide. In this approx-
imation and with disregard of the derivative of the specific
coupling resistance with respect to frequency, the equation
was employed in many works devoted to the study of non-
J˜ z t,( )
1
π
--- J z ω,( ) i ω ω0–( )t–( )exp ωd
Δω
∫=
≈ Jn z( ) i ωn ω0–( )t–( ),exp
n
∑
E˜ z t,( )
1
π
--- E z ω,( ) i ω ω0–( )t–( )exp ωd
Δω
∫=
≈ En z( ) i ωn ω0–( )t–( )exp
n
∑
h ω( ) h ω0( )
dh
dω
-------
ω ω0=
ω ω0–( ),+=
R ω( ) R ω0( )
dh
dω
-------
ω ω0=
ω ω0–( ).+=
1
π
---
∂E˜ z t,( )
∂z
------------------- ih ω0( )E˜ z t,( )– i
dh
dω
-------
ω ω0=
1
π
--- ω ω0–( )
Δω
∫–
× E z ω,( ) i ω ω0–( )–( )exp dω
R ω0( )
2
---------------J˜ z t,( )±=
±
1
2
---
dR
dω
-------
ω ω0=
ω ω0–( )J z ω,( ) i ω ω0–( )t–( )exp dω.
Δω
∫
∂E˜ z t,( )
∂z
------------------- i
dh
dω
-------
ω ω0=
∂E˜ z t,( )
∂t
------------------- ih ω0( )E˜ z t,( )–+
=
R ω0( )
2
---------------J˜ z t,( )±
1
2
---
dR
dω
-------
ω ω0=
∂J˜ z t,( )
∂t
------------------.±

SIMULATION OF NONSTATIONARY PROCESSES IN BACKWARD-WAVE TUBE 1405
linear nonstationary and chaotic processes in BWTs (see
[5]) starting from [8].
We consider various representations of Eq. (8) in terms
of dimensionless variables. We introduce dimensionless
amplitudes of the field F(ζ, τ) and current I (ζ, τ), which
exhibit slow variations with respect to time and the coordi-
nate along the electron beam:
(9)
where
(10)
are dimensionless slowly varying coordinate and time
and v0, ω0, E0, and J0 are constant components of elec-
tron velocity, frequency, field, and current, respectively,
used for the normalization. We use the gain parameter
at frequency ω0 [9] as small quantity ε and choose the
normalization field
(11)
Then, the following equation is derived for slowly
varying amplitudes:
(12)
where vg(ω0) = 1/ (ω0) is the group velocity of
waves at frequency ω0 and
(13)
is the mismatch parameter involving the constant elec-
tron velocity and the phase velocity of wave at fre-
quency ω0 vp(ω0) = ω0/h(ω0).
Owing to a certain arbitrariness of normalization veloc-
ity v0 and frequency ω0 (v0 ≈ ve is the velocity of unmod-
ulated electron beam and ω0 ≈ ω is the input-signal
frequency in TWT or the oscillation frequency in
BWT), we may assume that ξ(ω0) = 0 when v0 =
vp(ω0). With allowance for the fact that vg(ω0) > 0 in
TWT and vg(ω0) = –|vg(ω0)| < 0 in BWT, equation of
excitation (12) can be represented as
(14)
where ζ is defined above and τ = ε ω0t.
E˜ z t,( ) E0F ζ τ,( ) i
ω0
v0
------z
⎝ ⎠
⎛ ⎞ ,exp=
J˜ z t,( ) J0I ζ τ,( ) i
ω0
v0
------z
⎝ ⎠
⎛ ⎞ ,exp=
ζ ε
ω0
v0
------z, τ εω0t= =
ε
3 eR ω0( )J0
2mω0
2
-----------------------, E0
mω0v0
e
-----------------ε
2 R ω0( )J0v0
2ω0ε
---------------------------.= = =
∂F
∂ζ
------
v0
vg ω0( )
-----------------
∂F
∂τ
------ iξ ω0( )F–+ I,+−=
∂h
∂ω
-------
ξ ω0( )
v0 vp ω0( )–
εvp ω0( )
-----------------------------=
∂F
∂ζ
------
∂F
∂τ
------± I,+−=
vg ω0( )
v0
--------------------
We employ SWS geometrical length l to introduce
dimensionless coordinates
(15)
where L is the total dimensionless length.
In terms of these variables, the equation of excitation is
represented as
(16)
Alternative representations are derived when the initial
time
(17)
is used instead of dimensionless time τ given by for-
mula (10) and dimensionless coordinate ζ from formula
(10) is employed (Ä is the normalization constant).
Thus, we have
(18)
Substituting these derivatives in expression (12), we
derive the following equation of excitation:
(19)
Using the Nesta program for the calculation of nonsta-
tionary processes in TWT [7], we assume that A = 1, ω0 =
ω is the frequency of the input signal, and v0 = ve is the
constant velocity of electrons at the SWS entrance.
The SWS backward wave in BWT exhibits the anoma-
lous negative dispersion vg(ω0) = –|vg(ω0)| < 0. Therefore,
it is expedient to assume that Ä = 1 +
and to choose frequency ω0 from the electron–wave
matching condition vp(ω0) = v0, so that ξ(ω0) = 0.Then the
equation for the BWT excitation is written as
(20)
Following the above approach, we can select single
parameter L using relative coordinates
(21)
Thus, we have
(22)
L ε
ω0
v0
------l, ζ
z
l
--, τ t
vg ω0( )
l
--------------------,= = =
∂F
∂ζ
------
∂F
∂τ
------± LI.+−=
τ0 εω0 A t
z
v0
------–
⎝ ⎠
⎛ ⎞ A τ ζ–( )= =
∂F
∂ζ
------
τ
∂F
∂ζ
------
τ0
A
∂F
∂τ0
--------
ζ
,
∂F
∂τ
------
ζ
– A
∂F
∂τ0
--------
ζ
.= =
∂F
∂ζ
------ A
v0
vg ω0( )
----------------- 1–
⎝ ⎠
⎛ ⎞ ∂F
∂τ0
-------- iξ ω0( )F–+ I.+−=
v0
vg ω0( )
--------------------
⎝
⎛ 1
⎠
⎞ ,
∂F
∂ζ
------
∂F
∂τ0
--------– I.=
ζ
z
l
--, τ0
t z/v0–
l
1
vg
---------
1
v0
------+
⎝ ⎠
⎛ ⎞
-----------------------------.= =
∂F
∂ζ
------
∂F
∂τ0
--------– LI.=

1406
MELIKHOV et al.
For TWT, we can also choose frequency ω0 using the
exact matching condition vp(ω0) = v0. Then, we have
ξ(ω0) = 0 and the following scenarios are considered for
Eq. (19).
For the normal (positive) dispersion of waves in the
SWS, we have v0 ≈ vp(ω0) > vg(ω0) and obtain the follow-
ing equation of excitation assuming that Ä =
1 –
(23)
In the case of the anomalous positive dispersion, we
have v0 ≈ vp(ω0) < vg(ω0) ≤ c, assume that Ä = 1 –
and obtain
(24)
2. MATHEMATICAL MODEL
OF NONSTATIONARY PROCESSES
IN CARCINOTRODE
For the numerical simulation of the carcinotrode based
on the particle-in-cell method (below, we use electrons as
particles), we employ equation of excitation (22) and omit
subscript “0” of dimensionless time (21). The equation of
the field excitation must be supplemented with edge and
initial conditions for F. We consider the generation regime
of the carcinotrode and represent the edge condition at the
collector end of the tube as in the case of the conventional
BWT:
(25)
Output field F(0, τ) results from the calculations.
The initial condition
(26)
follows from the linear theory or is represented as an
arbitrary relatively small function F0(ζ) < 1 at F0(1) = 0.
In contrast to the study of the conventional BWT, the
analysis must involve the self-modulation of the cathode
emission, which is implemented using the nonuniform dis-
tribution of large particles (electrons) on the assumption of
the equality of charges. In addition, we take into account
the phase shift of the initial phases of electrons that corre-
sponds to the initial matching with the field, the presence of
delay, and the feedback parameters.
v0
vg ω0( )
-----------------
⎝
⎛ 1
⎠
⎞ :
∂F
∂ζ
------
∂F
∂τ0
--------+ I– .=
1
⎝
⎛
v0
vg ω0( )
-----------------
⎠
⎞ ,
∂F
∂ζ
------
∂F
∂τ0
--------– I– .=
F ζ 1=
0,=
F ζ τ,( ) τ 0=
F
0
ζ( )=
In the analysis of the modulation of emission, we fol-
low the approach from [4] and consider hot cathodes using
the 3/2 power law:
(27)
where J(0, t0) is the emission current, P is the per-
veance, and U is the cathode–grid (first anode) voltage.
For the carcinotrode, we have
U = U0 + U1cosωt0 = Umax(1 – μ + μcosωt0), (28)
where μ = U1/Umax is the coefficient of modulation with
respect to the maximum voltage Umax= U0 + U1.
In section 1, we analyze the excitation of the matched-
field wave by the current with frequency ω0 and slowly
varying amplitude (z, t) given by expressions (1) and (5).
In the device, current J(z, t) represents a sum of harmonics
at frequencies kω0 with slowly varying amplitudes (z, t):
With allowance for the slowness of amplitude varia-
tions and using the law of conservation of charge, we
obtain
As in the stationary regime, the harmonics of current
can be represented as integrals
(29)
where δ is the Kronecker delta (δmk = 1 at k = m and δmk = 0
at k ≠ m).
The harmonics of current at the cathode (0, t) corre-
spond to t(0, t0) = t0 and the following formula is derived
with regard to expression (27)
(30)
Note that θ = π at μ ≤ 0.5 and cosθ = 1 – 1/μ at μ > 0.5.
We find Jmax from expression (27) at U = Umax. In the non-
stationary regime, the amplitude of the HF voltage U1 and
quantities Jmax, μ, and αk slow variations with time (on the
scale 2π/ω0). Figure 1 demonstrates the typical depen-
J 0 t0,( ) PU
3/2
,=
J˜
J˜k
J z t,( ) J˜ t( ) Re J˜k z t,( ) ikω0t–( ).exp
k 1=
∞
∑+=
J z t,( ) dt J 0 t0,( )dt0.=
J˜k z t,( )
1
π 1 δ0k+( )
------------------------ J z t,( ) ikω0t( )exp ω0t( )d
0
2π
∫=
=
1
π 1 δ0k+( )
------------------------ J 0 t0,( ) ikω0t z t0,( )( )exp ω0t( ),d
0
2π
∫
J˜k
J˜k 0 t0,( ) αk Jmax,=
αk
1
π 1 δ0k+( )
------------------------=
× 1 μ– μ ω0t0cos+( )
3/2
kω0t0( )cos ω0t0( ).d
θ–
θ
∫

dences of the harmonics of current normalized by the max-
imum current α0 = /Jmax and α1 = /Jmax on coeffi-
cient of modulation μ. For real optoelectronic systems with
hot cathodes, the anode–grid characteristics may differ
from the characteristics given by expression (27), so that
the corresponding dependences are determined from
experiments or numerical simulation of optoelectronic sys-
tems.
Consider normalization coefficients (11) and parameter
L in expression (15) for the nonstationary model of the car-
cinotrode.
Mean J0 and maximum Jmax currents of electron beam
depend on modulation, so that the ratio of these quantities
decreases with increasing coefficient of modulation μ upon
an increase in the HF field and the amplitude of modulation
voltage U1 in the transient process. Therefore, we use cur-
rent J0 corresponding to the absence of the modulation of
emission in expressions (11) for the normalization coeffi-
cients and for L and obtain ε = ε|μ = 0 and L = L|μ = 0. Such an
approach simplifies the comparison of the results obtained
with the results corresponding to a conventional BWT.
However, we must additionally consider the excitation of
emission by the HF field (μ = 1) in the absence of static
voltage U0.
Fortheabovenormalization,theequationofthenonsta-
tionary model for the carcinotrode is similar to the equation
foraconventionalBWT[5].Noteonlythedifferenceofthe
expression for the normalized first harmonic of currentI. In
addition, we take into account a possibility of a significant
variation in the electron velocity, assume that ε ≠ 0, and
represent the equations in terms of independent variables
(21):
(31)
(32)
(33)
where u = u(u0, ζ) = ω0(t – z/v0) and the dimensionless
first harmonic of current I is introduced with the aid of
expression
The coefficient of modulation is written as
(34)
where feedback parameter G depends on the ratio of the
feedback coefficient to the static field.
J˜0 J˜1
∂
2
u
∂ζ
2
-------- L
2
1
ε
L
---
∂u
∂ζ
------+
⎝ ⎠
⎛ ⎞
3
ReF iu–( ),exp–=
∂F
∂ζ
------
∂F
∂τ
------– LI
1
1 μ–( )
3/2
----------------------,=
I
1
π
--- 1 μ– μ u0cos+( )
3/2
iu( )exp u0,d
0
2π
∫=
J˜1 z t,( ) JmaxI i
ω
v0
------z
⎝ ⎠
⎛ ⎞ .exp=
μ
G F 0 τ,( )
1 G F 0 τ,( )+
----------------------------------,=
We calculate the efficiency using the ratio of the HF
power to the mean power of electron beam, which exhibits
variations in the course of the transient process owing to a
variation in the mean current of the beam:
Using the normalized quantities, we obtain
(35)
3. MAIN EQUATIONS FOR SIMULATION
For the numerical modulation of the electron flux in the
carcinotrode, we consider large particles with equal
charges. The particles are nonuniformly distributed with
respect to t0 at z = 0 due to the modulation of emission. The
distribution results from expression (30) at k = 0 for the
constant component of current:
The condition for the equality of charges of the particles
Q ≡ J(0, t0j)Δt0j corresponds to the equality of all of the
terms in the last sum, so that
A transition from the integral to the sum with respect to
particlesyieldsthefollowingexpressionforthenormalized
first harmonic of current I with allowance for formula (33):
η
PHF
P0
--------
E˜ 0( )
2
2RJ˜0 0 t0,( )U0
----------------------------------.= =
η ε
F 0 τ,( )
2
2
----------------------
1 μ–( )
3/2
α0 μ( )
----------------------.=
J˜ 0 t0,( )
1
2π
------ J 0 t0 j,( )Δ ω0 t0 j,( )
j 1=
N
∑ Jmaxα0= =
= Jmax
1
2π
------ 1 μ u0cos–( )
3/2
Δu0 j.
j 1=
N
∑
1 μ– μ u0 jcos+( )
3/2
Δu0 j
2πα0
N
------------.≡
0.2
0.2
0.4 0.6 0.80
0.4
0.6
0.8
α0,
α1
1
2
μ
Fig. 1. Plots of the harmonics of current (1) α0 = J0/Jmax and
(2) α1 = J1/Jmax normalized by maximum current Jmax vs.
coefficient of modulation μ for the hot cathode.

1408
MELIKHOV et al.
(36)
Taking into account expression (36) and using a transi-
tion from Eqs. (31)–(33) to equations for large particles
with the phases uj(ζ) = u(ζ, t0j), we obtain
(37)
(38)
(39)
where
(40)
and expression (35) for the efficiency is represented as
(41)
These equations must be supplemented with the
expressions for coefficient of modulation μ and α0, which
can be found from feedback parameter G with the aid of
expressions (34) and (30).
Mathematical model (37)–(40) does not take into
account the space charge, the SWS loss, and several addi-
tional effects.
We assume that independent variables τ and ζ for time
and length along the tube range in the intervals
0 ≤ τ ≤ tEnd, 0 ≤ ζ ≤ 1.
I
2α0
N
--------- iuj( ).exp
j 1=
N
∑=
d
2
uj
dζ
2
---------- L
2
1
ε
L
---
duj
dζ
--------+
⎝ ⎠
⎛ ⎞
3
Re F iuj–( )exp( ),–=
j 1 2 … N,, , ,=
∂F
∂ζ
------
∂F
∂τ
------– LαI˜,=
I˜ 2
N
---- iuj( ),exp
j 1=
N
∑=
α μ( )
α0 μ( )
1 μ–( )
3/2
----------------------,=
η ε
F 0 τ,( )
2
2α μ( )
----------------------.=
The initial and boundary conditions for field F can be
found in section 1. At the collector of the tube, we have
the condition F(τ, 1) = 0. We choose the initial condi-
tion as an arbitrary small function, for example
F(0, ζ) = F0cos(0.5πζ),
where F0 is the maximum field at τ = 0 and ζ = 0. Note
that the determination of field F(τ, 0) at the left-hand
boundary is the main purpose of the simulation.
In the solution of Eq. (37), we assume that the initial
velocities of electrons are zeros: du/dζ = 0 at ζ = 0.
For the phases of electrons in expression (37), we
choose the initial conditions at ζ = 0 in accordance with the
distribution of electrons at the beginning of the interaction
space, which is assumed to be identical to the distribution
of particles at the cathode.We also assume that the charges
of electrons are identical and that the phase difference
between the electrons is inversely proportional to the cath-
ode current. Such a scenario corresponds to the nonuni-
form distribution of particles with respect to phases of u0 in
expression (37) [4]. In the analysis, we use the hot cathode
with the 3/2 power law for the current. Figure 2 shows
function 1/α(μ), which determines the right-hand side of
Eq. (38) and efficiency (41) and exhibits minor variations
at μ < 0.7. Evidently, the right-hand side in expression (38)
tends to infinity at μ 1 and the system of equations
under study needs to be renormalized with respect to F.
Note that the calculations employ the spline interpolation
of function 1/α(μ).
4. THE MATHCAD SIMULATION
OF THE CARCINOTRODE
We search for a solution using Mathcad and the finite-
difference method at a mesh with the steps r = tEnd/M, and
h = 1/K, where M and K are the numbers of steps with
respect to τ and ζ, respectively. Indices m and k character-
ize the mesh points: m = 0, 1, 2, …, M, k = 0, 1, 2, …, K,
τ = mr, and ζ = kh.
For the difference scheme, we choose a five-point sam-
ple (Fig. 3). The first and second derivatives are calculated
using two and three neighboring points, respectively.
Points 1–3 are used to solve equation of motion (37), and
points 3–5 are used to solve equation of excitation (38).
The current is calculated at point 3, which corresponds to
indices m and k with respect to electron phases for this
point.Thefieldiscalculatedatpoint5usingthefieldvalues
at points 3 and 4 (i.e., from the values at the previous time
layer).
Each step r with respect to dimensionless time may cor-
respond to several HF periods with identical electron distri-
butions at each of them. For the calculation of current with
the aid of expression (39), we choose any period.
We obtain the matrix representation of the solution in
which each time layer with number m corresponds to N +
2 rows of the matrix (N is the number of electrons at one
period (normally, we have N = 12)). The first row from the
0.2 0.4 0.6 0.80 μ
0.5
1.0
1.5
1
α(μ)
Fig. 2. Function 1/α(μ) in the presence of the feedback
(μ = 0 in the absence of the feedback).

group of N + 2 rows contains the values of field at moment
τ, and the last row contains the values of current for this
moment calculated from the phases ofN electrons in N pre-
vious rows. Hence, the rows of matrix Y characterize vari-
ationsinthefield,phasesofelectrons,andcurrentalongthe
tube. The initial conditions for the field are recorded in the
zero row of the matrix.
When the row numbers of matrix Y range from p = 0 to
p = (2 + N)M, the rows for which the condition mod(p + 1,
N + 2) = 0 is satisfied correspond to the values of current
and the rows for which the condition mod(p, N + 2) = 0 is
satisfied correspond to the values of field. Standard func-
tion mod calculates a residue of division of integers. The
relationship of p and the mesh-point numbers (m, k) is rep-
resented as p = m(N + 2) + j, where j = 1, 2, …, N are elec-
tron numbers.
Elements of matrix Yp, k characterize the field, the phase
of one electron, or the current at the mesh point (m, k). The
main formula for the calculation is represented as
Yp, k = f(p, k, Y), (42)
where each function f for the matrix elements is algo-
rithmically determined with allowance for the discreti-
zation and the difference scheme.
Note that the dimension of solution matrix Y in expres-
sion (42) is (M + 1)(N + 2) × (K + 1). In the Mathcad cal-
culations, we can use up to several million elements. In the
course of solution, matrix Y is completely stored in mem-
ory. When the calculation using expression (42) is termi-
nated, we can analyze any part of matrix Y and can plot
curves using the rows and columns.
Note that the code allows the calculation of the initial
phases of electrons at ζ = 0 based on the nonuniform elec-
tron distribution. Prior to the solution of Eq. (42), we calcu-
late the total charge at the HF period Sq using the cathode
current and perform the spline interpolation of the corre-
sponding curve versus μ.
The distribution is performed at each time layer m. The
calculation consists of the following stages. Quantity Q =
Sq/N is the charge of one electron. For the first electron, we
have u0 = Q. For the subsequent electrons with numbers j =
2, 3, …, N/2, we find u0, j from the formulas
(43)
where is the normalized cathode current. For the case,
μ > 0.5, the calculation of the phases of electrons at
small currents i in the vicinity of the cutoff level is per-
formed with allowance for the interpolation and zero
current at cosu0 = 1/μ – 1. For the distribution of elec-
trons with the numbers j > N/2, we employ the symme-
try of the bunch.
When all of electrons are distributed, the phases of u0, j
at the cathode range from 0 to 2π, so that the bunch corre-
Q u0 j, u0 j 1–,–( )i˜,=
i˜ 1 μ– μ u0 j 1–,cos+( )
3/2
,=
i˜
sponds to the maximum of the HF field.At the entrance to
the interaction space, field F with the phase shift arg(F) is
exerted on the bunch. Note also an additional phase shift
ϕfb relative to the field maximum. This phase shift deter-
mines the phase difference between the field and current.
We assume that constant parameter ϕfb depends on the
feedback characteristics. Hence, the electrons that start at ζ
= 0 have the phase shift arg(F) +ϕfb relative to the maxi-
mum field F. Note that ϕfb is one of the three main param-
eters of the above mathematical model (L, G, and ϕfb) that
determines the characteristics of the generated oscillations
in the carcinotrode.
After the calculation of matrix Y in expression (42), we
analyze the steady-state oscillations of the field envelope
using zero column Yp, 0. The period is found from the cova-
riance of real parts of the field envelope Re(Fm, 0) at the last
two time intervals with equal lengths of no greater than tD.
The limiting value tD is the input parameter of about 1/2 or
1/3oftEnd. Period TF correspondstothefirstmaximumof
covariance. After finding the period, we perform the dis-
crete Fourier transform for the samples of the real part of
the field Re(F) at the last of the two periods. Finally, we
calculate the efficiency.
The above mathematical model is implemented using
the GAMS3 code. The code units are as follows: initial
data, calculation of the discretization steps, determination
of the initial and boundary conditions, spline interpolation
of the total cathode current Sq(μ) and function 1/α(μ),
determination of additional functions for the distribution of
particles, determination of function f in formula (42), calcu-
lation of the elements of matrix Yp, k using expression (42)
where second subscript k is the first subscript to be auto-
matically changed, determination of period TF of steady-
state oscillations, calculation of the spectrum of field
Re(F), calculation of the efficiency, data plotting for analy-
sis, and file recording of the main data.
We employ the following initial data (several parame-
ters are characterized above): L, G, ϕfb, ε, tEnd, M, K, N, L,
tD,andinitialfieldperturbationF0;itokzer—smallcurrents
at the cathode in the vicinity of the bunch edges which can
be neglected.
The simulation error depends on discretization parame-
ters K, M, and N. We choose parameters K and M based on
the stability condition, which is represented as r ≤ h in the
classical transport equation (r and h are the discretization
steps in time and space, respectively). A decrease in the
1 2 3 4
5
(m, k)
(m + 1, k)
(m, k + 1)(m, k – 2) * * * *
*
Fig. 3. Pattern of difference scheme.

1410
MELIKHOV et al.
steps (i.e., an increase in K and M) causes an increase in the
simulation time proportional to product KM.
Note that the computation time substantially increases
in the calculation of the chaotic regime owing to significant
instabilities. We must analyze such instabilities to perform
correct calculations. Note a significant effect of the discret-
ization steps. In particular, time stepr must be decreased by
a factor of approximately s, which characterizes an
increase in the number of harmonics in the spectrum of
Re(F).
5. RESULTS OF SIMULATION
Nonlinear equations of nonstationary processes in the
carcinotrode (Eqs. (31–(34) or (37)–(40)) contain three
main parameters: parameter L that characterizes the inter-
actionofelectronfluxwithbackwardwave,parameterGof
the cathode feedback, and feedback phase ϕfb. In addition,
gain parameter ε is needed for the calculations of the effi-
ciency. (The theory of conventional TWTs and BWTs
shows that this parameter affects the solution at an effi-
ciency of greater than 20% (ε ≥ 0.05–0.1).) The equations
of the nonstationary theory of a conventional BWT employ
onlyonesignificantparameterL,whoseeffectonthework-
ing regimes was analyzed in many works (see [5]). The
BWT self-excitation takes place at L = 1.98, and the har-
monic self-modulation of oscillations occurs at L ≈ 3.5.
When this parameter further increases, the self-modulation
becomes multifrequency and chaotic. Note the feedback
parameter G and backward-wave phase ϕfb substantially
affect the working regimes and the parameters of oscilla-
tions in the carcinotrode.We predominantly concentrate on
transient processes and characteristics of stationary oscilla-
tions in the ranges of the single-frequency regimes of the
carcinotrode, the effect of the feedback parameter on the
efficiency, the limits of the single-frequency range, etc.
Figure 4 demonstrates the curves of increasing field
|F(0, τ)| at L = 2 for several values of feedback parameter
G. The feedback phase (ϕfb = – ) corresponds to the
maximum efficiency in accordance with the estimations of
the approximate nonlinear theory [2, 4], and the gain
parameter is ε = 0.1. When G = 0, we have μ = 0 at the
entire time interval (as in the case of the conventional
BWT). When G increases, the oscillation rise time signifi-
cantly decreases, the regime remains single-frequency, and
the efficiency increases to η = 27% at G = 0.3. Coefficient
of modulation μ increases to 0.43, which corresponds to
the transition from a weakly modulated electron beam at
the beginning of the transient process to the deep modula-
tion in the absence of the current cutoff.
Figure 5 shows the dependences of the efficiency on G
in the stationary single-frequency regime at several L. The
curves of the efficiency versus feedback parameter G sub-
stantially depend on parameter L. In the case L > 1.98, the
efficiency gradually increases to η ~ 40% with increasing
G. When L = 2.5, we observe the self-modulation oscilla-
tions in the interval 0.38 < G < 0.42. In the case L < 1.98,
the efficiency may exhibit stepwise variations with an
increase in G and may reach the level η ≈ 45%. In both
cases, the efficiency increases with an increase in G and the
maximum corresponds to μ ≈ 0.8 (note a significant cutoff
of current at the cathode θ = 1.82. An increase in the feed-
back parameter G leads to a decrease in the start-up value
Lstart and the values of L that correspond to the transition
from the single-frequency regime to the self-modulation
regime (Fig. 6).
Figure 7 demonstrates the effect of the feedback phase
ϕfb on the efficiency. Note the optimal feedback phaseϕfb =
– at which the efficiency is η ≈ 50% in the single-fre-
quency regime at L = 2 and G = 0.6 (Fig. 8). Thus, we can
π
2
---
3π
4
------,
5
3.0
2.5
2.0
1.5
1.0
0.5
0 10 15 20 25
τ
F
1
2
3
Fig. 4. The build-up of the field in the carcinotrode at the
feedback parameters G = (1) 0, (2) 0.1, and (3) 0.3 for
L = 2 and ϕfb = –1.57.
0.2 0.4 0.6 0.8
10
20
30
40
50
η, %
0 G
3
3
2
1
Fig. 5. Plots of the carcinotrode efficiency vs. feedback
parameter G at ϕfb = –1.57 and L = (1) 1.5, (2) 2, and (3) 2.5.

reach a carcinotrode efficiency of about 50% in the single-
frequency regime at the gain parameter ε ∼ 0.1.
A transition to the self-modulation regime with an
increase in G leads to the oscillations of the field amplitude
as in the case of the conventional BWT. Figure 9 shows the
characteristic regimes of the oscillation build-up. Note that,
at the optimal phase ϕfb = – , the amplitude oscillations
are relatively small, the side components are no greater
than –36 dB, and the efficiency is η ≈ 70%. The side com-
ponents increase when the phase differs from the optimal
value. In the self-modulation regime, even greater effi-
ciency results from a further decrease in L and an increase
in G (e.g., η = 80% at L = 1 and G = 1.15). In this case, the
3π
4
------
self-modulation coefficient is μ = 0.97 (the B regime of the
cathode–grid diode).
CONCLUSIONS
The proposed theory of the nonlinear nonstationary
processes in the carcinotrode makes it possible to derive
equations with three main parameters: parameter L that
characterizes the interaction of electron flux with the back-
ward-wave field and depends on the SWS length and the
beam current, parameters G that characterizes the feedback
between the SWS exit and the cathode, and feedback phase
ϕfb. To solve the equations, we developed an algorithm and
a Mathcad code that take into account the modulation of
emission at the cathode using the nonuniform time distri-
5 10 15 20 25
0.5
1.0
1.5
2.0
0
G
Lstart
Fig. 6. Effect of feedback parameter G on the start-up
regime of the carcinotrode determined by parameter Lstart at
ϕfb = –1.57.
–1.8 –1.6–2.0–2.2–2.4–2.6–2.8–3
35
30
40
45
50
55
η, %
ϕfb
Fig. 7. Plot of the efficiency vs. the feedback phase at
G = 0.6 and L = 2.
0.2 0.4 0.6 0.80 G
10
20
30
40
50
η, %
3
2
1
Fig. 8. Plots of the carcinotrode efficiency vs. feedback
parameter G for the feedback phases ϕfb = (1) –1.47,
(2) –1.57, and (3) –2.3 at L = 2.
5 10 15 20 250
τ
5
10
15
20
25
F
2
1
Fig. 9. The build-up of the carcinotrode oscillations at ϕfb =
(1) –1.57 and (2) –2.3.

1412
MELIKHOV et al.
bution of large particles (electrons) with equal charges and
the corresponding phase matching of the particles with
respect to the field with allowance for the feedback phase.
Thenumericalsimulationyieldsneweffects:adecrease
in the start-up length and the start-up current (and the cor-
responding parameter Lstart) with an increase in the feed-
back level, a decrease in parameter L that separates the
regimes of harmonic and modulated oscillations, a signifi-
cant (to 50%) increase in the efficiency of harmonic oscil-
lations at the gain parameter ε = 0.1 and even more signif-
icant increase in the efficiency to 80% in the self-modula-
tion regime. The results prove and develop the conclusions
from [2–4], drawn with the aid of an approximate analyti-
cal theory and the model of stationary nonlinear processes
inthecarcinotrode.Theproposedmethodsandcodesmake
it possible to study the nonlinear nonstationary processes
and parameters of the carcinotrode in wide ranges of
parameters that characterize the interaction of electron flux
with backward wave and the feedback between the SWS
exit and the cathode.
ACKNOWLEDGMENTS
This work was supported by the Russian Foundation
for Basic Research (project no. 07-02-00947).
REFERENCES
1. V. A. Solntsev, RF Patent No. 2121194, Byull. Izobret.,
No. 30 (1998).
2. V. A. Solntsev, Izv. Vyssh. Uchebn. Zaved., Prikl. Nelinein-
aya Din. 6 (1), 54 (1998).
3. V.A. Solntsev, J. Commun.Technol. Electron. 45 (Suppl. 1),
39 (2000).
4. V. A. Solntsev, R. P. Koltunov, and V. O. Melikhov,
Radiotekh. Elektron. (Moscow) 50, 483 (2005) [J. Com-
mun. Technol. Electron. 50, 448 (2005)].
5. D.I.TrubetskovandA.E.Khramov,LecturesonMicrowave
Electronics for Physicists (Fizmatlit, Moscow, 2003) [in
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6. V. A. Solntsev, “Particle-in-cell Method and Mathematical
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