Numerical simulation of the magnetron operation with resonance load

Abstract

The results of numerical simulations and a comparison with experimental data obtained in recent experiments with the relativistic S-band magnetron by Sayapin etal [Appl. Phys. Lett. 95, 074101 (2009)], having a resonance load and without special measures being taken to suppress the microwaves reflected from the load, are presented. The numerical simulations were based on the model which considers a magnetron as a traveling wave resonator coupled with external resonator. In these simulations, experimentally determined parameters of the magnetron and resonator and their coupling coefficient were used. It was found that, under certain conditions, the electromagnetic wave reflected from the resonator leads to an increase in the efficiency of the magnetron operation. Taking into account microwave energy compression in the resonator, one obtains a microwave power comparable with the power of the electron beam in the magnetron. Also, it was shown that the magnetron traveling wave acquires a phase shift due to its interaction with the amplified wave of the resonator. This phase shift can be comparable with the phase of the electron spoke with respect to the maximum of the decelerating phase of the microwave electric field. The latter could be a reason for the quenching of the microwave generation and the fast decay of the microwave power in the resonator found in experiments.

Full text

Numerical simulation of the magnetron operation with resonance load
A. Sayapin and Y. E. Krasika兲
Department of Physics, Technion, Haifa 32000, Israel
共Received 10 January 2010; accepted 10 February 2010; published online 5 April 2010兲
The results of numerical simulations and a comparison with experimental data obtained in recent
experiments with the relativistic S-band magnetron by Sayapin et al. 关Appl. Phys. Lett. 95, 074101
共2009兲兴, having a resonance load and without special measures being taken to suppress the
microwaves reflected from the load, are presented. The numerical simulations were based on the
model which considers a magnetron as a traveling wave resonator coupled with external resonator.
In these simulations, experimentally determined parameters of the magnetron and resonator and
their coupling coefficient were used. It was found that, under certain conditions, the electromagnetic
wave reflected from the resonator leads to an increase in the efficiency of the magnetron operation.
Taking into account microwave energy compression in the resonator, one obtains a microwave
power comparable with the power of the electron beam in the magnetron. Also, it was shown that
the magnetron traveling wave acquires a phase shift due to its interaction with the amplified wave
of the resonator. This phase shift can be comparable with the phase of the electron spoke
with respect to the maximum of the decelerating phase of the microwave electric field. The latter
could be a reason for the quenching of the microwave generation and the fast decay of the
microwave power in the resonator found in experiments. © 2010 American Institute of Physics.
关doi:10.1063/1.3359679兴
I. INTRODUCTION
High-power microwave 共HPM兲pulses with power
108–1010 W can be used in electron acceleration, plasma
heating, communication, radars, and other applications.1,2In
general, there are two main approaches to HPM generation
as follows: the application of various microwave tubes pow-
ered by high-power 共109–1011 W兲, high-voltage
共105–106V兲generators having a pulse duration of 10−7
−10−6 s, and microwave compressors.3,4In the case of short
共a few nanoseconds兲duration HPM, microwave compressors
become preferable because they do not require high-power
generators as primary power supplies. Microwave compres-
sor operation is based on relatively slow storage of the mi-
crowave energy in the resonator, followed by switching of
the stored energy to the load during a time much shorter than
the storage time.
In the case of primary microwave generators having a
power of several megawatts, the microwave energy storage
time in the resonator is in the microsecond time scale. The
latter could cause microwave breakdown inside the
resonator.5,6In order to avoid this problem, one can use a
primary microwave generator that supplies microwave
pulses of 10−7 s duration and a moderate power of tens of
megawatt. For instance, in Ref. 7it was shown that the use
of an S-band relativistic magnetron with a resonator load
allows one to obtain up to six-fold increase in the microwave
power 共Pmax ⬇0.9 GW兲with compression ratio kc=
␶
in /
␶
out
=24. Here
␶
in is the stored time and
␶
out =2l/Vgr is the dura-
tion of the microwave output pulse, where lis the length of
the resonator and Vgr is the electromagnetic 共e/m兲wave
group velocity. In this research, excitation of the resonator by
the relativistic magnetron was achieved in the way similar to
that achieved with the nonrelativistic magnetron. Namely,
adjustment of the coupling between the resonator and mag-
netron, the length of the transition waveguide section, and a
ferrite-based isolator were applied to decrease the influence
of the e/m wave reflected from the resonator on the magne-
tron operation.
In our recent paper,8results of the operation of the
S-band relativistic magnetron with a resonance load, when
no special measures were taken to suppress the e/m waves
reflected from the load were presented. The superior opera-
tion of an S-band relativistic magnetron powered by a Linear
Induction Accelerator with ⱕ400 kV, ⱕ4 kA, and
⬃150 ns output pulses was revealed. Namely, it was shown
that, under optimal conditions, the efficiency of the micro-
wave generation increases by ⬃40% and the generated mi-
crowave power reaches that of the electron beam. In the
present paper, the results of numerical simulations and their
comparison with experimental data obtained in these experi-
ments are presented.
II. MODEL OF THE MAGNETRON COUPLED WITH
RESONATOR
Let us consider the magnetron as a traveling wave reso-
nator formed by six radial short-circuit rectangular
waveguides placed azimuthally symmetrically. The surfaces
of the cylindrical cathode and anode surfaces opposite to it
can be considered as strip lines which connect these
waveguides. The magnetron is connected with the resonator
of length Lvia a window with coupling coefficient k. Inside
the magnetron will be considered clockwise and counter
wise e/m waves and inside the resonator 共the part between
cross-sections S3and S4shown in Fig. 1兲will be considered
a兲Electronic mail: fnkrasik@physics.technion.ac.il.
JOURNAL OF APPLIED PHYSICS 107, 074501 共2010兲
0021-8979/2010/107共7兲/074501/7/$30.00 © 2010 American Institute of Physics107, 074501-1

forward and backward e/m waves. The amplitude of these
e/m waves will be determined at a set of the cross-sections Sj
共j=1,2,3,and4兲with definitions aj
mn and bj
mn for the for-
ward and reflected e/m waves, respectively 共see Fig. 1兲using
two different discrete time intervals which will be marked as
m=0,1,2,3,... and n=0,1,2,3,... 共see Fig. 2兲. The dura-
tion of interval mis determined by the time tr, which the e/m
wave requires to propagate forth and back inside the resona-
tor
tr=2L/Vg
r,共1兲
where Vgr
ris the group velocity of the e/m wave excited in
the resonator. The duration of interval nis determined by the
time tmag of the e/m wave circulation in the magnetron:
tmag =2
␲
reff/Vg
mag,共2兲
where Vg
mag is the e/m wave group velocity along the trajec-
tory with radius reff. The value of the coupling coefficient
will be determined using “cold” measurements 共see Sec. II兲
as k=冑1−共a3
10 /b3
10兲2when a1
10 =a2
10 =0 which is correct for
t⬍2
␲
reff /Vg
mag.
Let us consider the case when time intervals tr=Ntmag,
where N=2,3,4,... 共see Fig. 2兲. In general, boundary con-
ditions require the e/m waves’ amplitudes to be equal at
cross-section S4:
b4
mn =−a4
mn.共3兲
The amplitude of the traveling e/m wave excited in the mag-
netron when it operates with a matched load 共b4
mn ⬅a3
mn =0兲,
will be defined using index “0,” i.e., a10
mn,b20
mn, and the am-
plitude of the e/m wave which the magnetron radiates into
the matched load as b30
mn. In the case of an arbitrary load,
during the first time interval m=0, t=0+trone obtains the
following amplitudes of the e/m waves:
a1
0n=a10
0n,
a2
0n=0,
a3
0n=0,
b1
0n=0,
b2
0n=b20
0n,
b3
0n=ik共a10
0n+a20
0n兲=b30
0n.共4兲
Now one can write that in the following intervals mand n
共m=1,2,3,...,Mand n=0,1,2,3,... ,N兲the amplitudes of
the e/m waves in the magnetron and external resonator can
be determined as a solution of the following equations:
a1
mn =b2
m−1ne
␯
+,n=0,m= 1,2,3. .. M,
a2
mn =b1
m−1ne
␯
−,
a3
mn =b3
m−1ne
␺
,
b1
mn =冑1−k2a2
mn +ik冑2/2a3
mn,
b2
mn =冑1−k2a1
mn +ik冑2/2a3
mn,
b3
mn =冑1−k2a3
mn +ik共a1
mn +a2
mn兲,
a1
mn =b2
mn−1e
␯
+,n= 1,2,3 ... N,m= 1,2,3 ... M,
a2
mn =b1
mn−1e
␯
−,
a3
mn =b3
mn−1e
␺
,
b1
mn =冑1−k2a2
mn +ik冑2/2a3
mn,
b2
mn =冑1−k2a1
mn +ik冑2/2a3
mn,
b3
mn =冑1−k2a3
mn +ik共a1
mn +a2
mn兲.共5兲
Here,
␺
,
␯
+,
␯
−are complex values the imaginary parts of
which determine the change in the phase of the e/m waves
propagating in the external resonator and magnetron, respec-
tively, the real parts of
␺
,
␯
−determine the decay of e/m
waves during propagation in the resonator and magnetron,
respectively, and the real part of
␯
+determines the change in
the amplitude of the e/m wave propagating in the same di-
rection as the electron spoke in the magnetron.
FIG. 1. 共a兲Scheme of the magnetron coupled with external resonator via
coupling window. 共b兲Magnetron as a traveling wave resonator formed by
radial short-circuit rectangular waveguides placed azimuthally symmetri-
cally. S1,S2,S3,andS4are cross-sections where reflections of e/m waves
occur.
FIG. 2. Microwave pulse generated by magnetron at the matched load with
indicated discrete time intervals nand mwhich were used in simulations.
074501-2 A. Sayapin and Y. E. Krasik J. Appl. Phys. 107, 074501 共2010兲

In the case of numerical simulations, we used the values
of k,
␯
−,tmand real part of
␺
, which were determined using
the “cold” measurements with the magnetron, resonator, and
network analyzer. The imaginary part of parameter
␯
+was
estimated using the data obtained with the magnetron oper-
ating with the matched load. The imaginary part of
␺
was
variable parameter in the simulations.
III. PARAMETERS OF RESONATORS
First, the frequency characteristic of the magnetron was
determined using a set-up consisting of a network analyzer
共Agilent Technologies E8257D兲, a directional coupler with
coaxial adapters and a magnetron with B-dot loop 共see Fig.
3兲. The B-dot loop with a diameter of 5 mm was installed
inside the magnetron at its side plane. The signal produced at
the output of the network analyzer was supplied to the mag-
netron via the directional coupler. The signal either from the
directional coupler or from the B-dot 共see Fig. 3兲was regis-
tered at the analyzer input. The results of these measure-
ments 共see Fig. 4兲showed that in the frequency range 2.75–
3.25 GHz one obtains a reflected e/m wave S11 with a
frequency f⬇3.08 GHz and a minimum amplitude 共⫺7dB兲.
This means that at this frequency almost all the network
analyzer microwave power is absorbed by the magnetron. In
the case of frequencies outside the range of the magnetron
resonance frequency, the level of the reflected signal was
around ⫺1 dB. A large value of the reflection coefficient
from the magnetron input and the finite matching value of
the waveguide adaptors 共see Fig. 4兲require additional mea-
surements in order to exclude that, 共a兲the obtained resonance
is not related to the resonance frequency of the waveguide
placed between the waveguide adapter and the input to the
magnetron and 共b兲the obtained resonance is not related to
the resonance frequency of the system waveguide-
magnetron. The measurements showed that in the resonance
frequency range the amplitude of the electric field at the
directional coupler output is smaller than outside the reso-
nance frequency range. Thus, one can conclude that at the
resonance frequency, almost all delivered microwave power
is absorbed by the magnetron. In addition, the measurements
with the B-dot probe confirm that in the range of resonance
frequencies one obtains amplification of the microwave
power within the magnetron.
In order to determine the values of the coupling coeffi-
cient kand time interval tmag 关see Eq. 共2兲兴, microwave pulses
having an approximately rectangular form were delivered to
the input of the magnetron using set-up shown in Fig. 5.In
this set-up, two identical directional couplers FSC16179
共coupling coefficient of ⫺10 dB and directionality of ⫺22
dB兲with coaxial outputs were used. Therefore, it was neces-
sary to use an additional coaxial waveguide adapter to con-
nect the coupler with the magnetron. The microwave pulse
from the Agilent Technologies E8257D generator was sup-
plied to the input of the first directional coupler 共see Fig. 5兲.
A part 共⫺10 dB兲of the power with electric field E1from this
coupler was delivered to the input of the digitizing oscillo-
scope Agilent DSO 80404B 共4 GHz bandwidth兲. The remain-
ing part of the microwave power 共⫺11 dB of the generator
power兲with electric field E2was supplied to the magnetron
input. Thus, the amplitude of the electric field E1measured
by the oscilloscope exceeds the amplitude of the electric field
at the magnetron input on 1 dB. The e/m wave that was
reflected from the magnetron input and with the on ⫺1dB
decreased electric field E2also was detected by the oscillo-
scope 共see Fig. 5兲. In fact, there are several uncertainties
related to the determination of the magnetron reflection co-
efficient using a comparison between the e/m wave reflected
from the magnetron and the e/m wave from the first direc-
tional coupler. Indeed, it is difficult to synchronize the begin-
ning of the reflected and coupler’s microwave pulses. Also,
there is a ⬃40 cm waveguide between the second coupler
and the magnetron. The latter leads to a change in the micro-
wave pulse form because of these waveguide dispersion pa-
FIG. 3. Scheme for measurements of the magnetron frequency characteris-
tic.共1兲Directional coupler, 共2兲coaxial waveguide adaptors, 共3兲magnetron,
共4兲B-dot loop, and 共5兲network analyzer 共Agilent Technologies E8257D兲.
FIG. 4. 共Color online兲Frequency characteristic of the magnetron. S11 is the
amplitude of the e/m wave reflected from the magnetron input, Sc12 is the
amplitude of the e/m wave at the input to the magnetron, and Sb12 is the
B-dot amplitude.
FIG. 5. Scheme for measurements of the transition processes in the magne-
tron. 共1兲Pulsed microwave generator, 共2兲coaxial waveguide transition sec-
tion, 共3兲directional coupler, 共4兲magnetron, 共5兲digitizing oscilloscope, and
共6兲waveguide.
074501-3 A. Sayapin and Y. E. Krasik J. Appl. Phys. 107, 074501 共2010兲

rameters. Therefore, the e/m wave reflected from the magne-
tron input was compared with the e/m wave reflected from
the magnetron input when a conductive disk having the mag-
netron anode radius was placed at the magnetron input. Dur-
ing these measurements, the microwave signal from the first
directional coupler was used to control the stability of the
microwave pulses. Here let us note that the magnetron input
was connected with the second coupler via a coaxial adapter.
Therefore, a part of the microwave power which was re-
flected from this adapter gave some input to the amplitude of
e/m wave E2. In order to determine the phase and amplitude
of this reflected e/m wave,a2mlength waveguide 共6 in Fig.
5兲was installed between the adapter and magnetron input.
This additional waveguide allows one to determine the am-
plitude and phase of the e/m wave reflected from the adapter
prior to the appearance of the e/m wave reflected from the
magnetron input. The waveforms of the e/m waves reflected
from the magnetron input with and without the conductive
plate are shown in Fig. 6. Now, the amplitudes of the e/m
waves E2reflected from the magnetron input, with and with-
out the metal plate at its input, were determined taking into
account the e/m wave reflected from the adapter with correct
phase and amplitude. The time-dependent ratio ⌫between
the electric field amplitudes of the reflected e/m waves with
and without the metal plate at the magnetron input and the
amplitudes of the reflected e/m wave at the resonance and
out of resonance frequencies are shown in Fig. 7. One can
see that within the first 4–6 ns, the reflection coefficient is
⌫=0.81⫾0.05. In the case of a uniform resonator one can
expect a constant value of ⌫for the first time interval, tmag.
Thus, the instant when ⌫starts to change its value can be
used to determine the time interval tmag. In the case of a
magnetron resonator with nonuniformities along the path of
the e/m wave propagation within the first time interval tmag,
one obtains some uncertainties 共⫾6%兲in the determination
of the tmag value. Nevertheless, the comparison of the electric
field waveforms obtained with different frequencies 关see Fig.
7共b兲兴allows one to determine accurately the value of tmag
⬇4 ns. The ratio ⌫between the reflected and forward e/m
wave amplitudes at the beginning of the magnetron excita-
tion is related to the coupling coefficient kas, ⌫=冑1−k2.
Thus, using the data shown in Fig. 7, one determines the
value of k⬇−4.7 dB. Now, using the values of kand S11
共see Fig. 3兲the real part
␯
−can be determined.
In the case of a magnetron excitation by an external
generator 共a3=a0兲and assuming that all the transition pro-
cesses are finished, one can write the following equation for
b3electric field amplitude of e/m wave 共definitions see in
Fig. 1兲:9
b3=a0
冉
冑1−k2−k22冑2
1−e
␯
−冑1−k2
冊
.共6兲
At the resonance frequency, the imaginary part of the com-
plex value
␯
−is equal to l
␲
, where l=0,1,2,3,.... Thus, the
real part of
␯
−which is responsible for the decay of the e/m
wave in the resonator can be determined as
Re
␯
−=ln 1
冑1−k2
冉
1− k22冑2
冑1−k2−S11
冊
,共7兲
where S11 =b3/a0was determined experimentally 共see Fig.
3兲. For the complex value of
␺
, the imaginary part of which
is responsible for the phase shift between the e/m waves a3
and b3, this part was determined by the position of the mov-
ing plunger 共see Fig. 1兲. The real part of
␺
which is respon-
sible for the decay of e/m waves a3and b3was determined
using the datasheet of the copper waveguide RG 48/U 共cross
sectional area 72⫻34 mm2兲.
IV. NUMERICAL SIMULATION OF ELECTRIC FIELDS
IN THE MAGNETRON AND EXTERNAL
RESONATOR
In Sec. III, all the parameters necessary for the calcula-
tion of the electric field in the magnetron and external gen-
erator were determined except for the complex value
␯
+. Let
us note that the imaginary parts of complex values of
␯
+,
␯
−
determine the phase shift of e/m waves a1and b2and a2and
FIG. 6. 共Color online兲The waveforms of the e/m waves reflected from the
magnetron input without 共1兲and with 共2兲conductive plate at the magnetron
input. 3 is the e/m wave reflected from the coaxial adapter 共see Fig. 4, item
2兲. Measurements were carried out at the resonance frequency of 3.08 GHz.
FIG. 7. 共Color online兲共a兲The time dependence of the ratio ⌫between the
amplitudes of the electric field of the reflected e/m waves without and with
metal plate at the magnetron input and 共b兲the amplitudes of the reflected
e/m wave at the 共1兲resonance fr=3080 MHz and 共2兲out of resonance fr
=3180 MHz frequencies.
074501-4 A. Sayapin and Y. E. Krasik J. Appl. Phys. 107, 074501 共2010兲

b1, respectively 共see Fig. 1兲. The real part of the value
␯
−
determines the decay of the e/m wave b1during one magne-
tron’s circle. The real part of the
␯
+determines the amplifi-
cation of the e/m wave b2due to interaction with the electron
spoke, the rotation of which is synchronized with the b2e/m
wave. Here let us note that the phase shifts of e/m waves a1
and b2and a2and b1共the imaginary parts of complex values
of
␯
+,
␯
−兲are determined by not only the magnetron design
but also the parameters of the expanding cathode and anode
plasmas. Therefore, in the simulations the imaginary parts of
␯
+,
␯
−were assumed to be equal to each other and were
considered as variable parameters.
Now, let us determine the value of the real part of
␯
+
averaged for the time interval ⌬t=tmag for the case of the
magnetron operating with matched load 共a3=0兲. First, the
experimentally obtained value b30
0nwas used to calculate e/m
waves’ electric field amplitude using Eq. 共5兲for the first time
interval m=0. Further, the obtained electric field amplitudes
were used to calculate the amplitude of the e/m waves for the
next time interval 共m=1, n=1兲for the value of
␯
+=0, i.e., for
the case when the e/m wave a1amplification was absent. The
calculated electric field of the e/m wave b3
11 was compared
with the experimentally obtained electric field of the same
e/m wave b30
11. The latter allows one to determine the ampli-
fication of the e/m wave b3
11 in the magnetron, i.e., the real
part of the value of
␯
+
␯
11
+共m=1,n=1兲=ln
兺
l=1
L
兩b03
11共l兲兩
兺
l=1
L
兩b3
11共l兲兩
.共8兲
Here L=672 is the number of the experimental sampling data
points of the e/m wave electric field amplitude within the
time interval n=1. Now, using the value of
␯
11
+, the electric
field amplitudes of the e/m waves were determined. Further,
these amplitudes were used for the calculation of the electric
field amplitude of the e/m waves at the next time intervals n
and mwith the same algorithm as for the time interval n
=1 and m= 1. The simulations continued up to M= 7, which
corresponds the microwave pulse duration of ⬃120 ns mea-
sured in the experiments described in Ref. 8.
In the case of the resonance load 共see Eq. 共5兲兲, when the
phase shift between the e/m waves a3and b2is not signifi-
cant, one can assume that the amplification of the e/m wave
b2共e/m wave b2is the superposition of e/m waves in the
magnetron a1and resonator a3兲occurs with an increment of
␯
+共m,n兲. Indeed, in this case the maximum space-charge
density of the electron spoke remains at the same retarding
phase with respect to the e/m wave b2. In the case of the
finite value of the phase shift between the e/m waves a3and
b2, this assumption also remains correct if the time tst, which
the spoke electron space-charge requires to recover its phase
with respect to the initial 共undisturb兲retarding phase of the
e/m wave b2,istst ⬍tmag. Thus, in these cases the variable
parameters which determine the phase shifts are only imagi-
nary parts of
␺
,
␯
values. The calculated electric field ampli-
tudes for some values of
␺
,
␯
are shown in Fig. 8. Here, the
maximal calculated values of electric fields are normalized
on the electric field amplitudes that were obtained in
experiments8when the magnetron operates with the matched
load. One can see that the maximal amplification 共up to fac-
tor 8兲of the e/m wave in the resonator coincides with that in
the magnetron for different phase differences between the
e/m waves b3and a1. In order to achieve the maximal am-
plitude of the e/m wave in the resonator, one has to adjust in
a proper way phase difference
␺
either by changing the mag-
netron design 共this would change the value of
␯
兲or the ex-
ternal resonator 共this would change the value of
␺
兲or both of
them.
Now, analyzing the experimental data presented in Ref.
7, one can conclude that the rather sophisticated form of the
microwave pulse 共for instance, a steplike increase in the rise
of the microwave power兲can be explained by the redistribu-
tion of the generated microwave energy between the e/m
waves a1and a2in the magnetron and the reflected e/m wave
a3propagating in the resonator. The calculations showed also
that the experimentally obtained amplification of the e/m
wave in the resonator can be explained by the more efficient
microwave generation inside the magnetron.
Finally, the case of the commonly used ferrite isolators
共the absence of the e/m waves from the resonator toward the
magnetron兲when a magnetron with moderate power is used
for the microwave resonator pumping was considered 共see
Fig. 8, curve 3兲. In these simulations in the expressions 关see
Eq. 共5兲兴for the amplitudes of e/m waves b1and b2terms
proportional to the amplitude a3were omitted. One can see
共Fig. 8兲that in this case the maximal value of e/m wave
FIG. 8. Dependencies of the e/m waves electric field amplitude a1共1兲,b3共2兲,
and b3
ⴱ共3兲normalized on the electric field amplitude which one obtains with
the magnetron operating with the matched load on value
␺
共phase shift
between the e/m waves a3and b3兲. Here b3
ⴱis the amplitude of the e/m wave
electric field in the resonator when the magnetron is completely decoupled
from the resonator. 共a兲Im共
␯
兲=0 and 共b兲Im共
␯
兲=
␲
/4.
074501-5 A. Sayapin and Y. E. Krasik J. Appl. Phys. 107, 074501 共2010兲

amplitude b3
ⴱin the resonator does not exceed 3, i.e., it is
almost three times less than that which one can achieve in
the case of a proper coupling between the magnetron and
resonator and which was demonstrated in the experiments
described in Ref. 8. Thus, one can conclude that the increase
in the amplitude of the e/m wave a1in the magnetron due to
the interaction with e/m wave a3of the resonator leads to a
more efficient transference of the potential energy from the
electron space-charge to the e/m wave a1.
Now let us consider the case when the phase difference
between the e/m wave b2and a3cannot be neglected. In the
linear approximation, the power Pof the amplified e/m wave
b2can be determined as:10
dP
dz =J1b2cos
␸
e,共9兲
where z-coordinate is in the direction of the propagation of
the e/m wave b2,J1is the amplitude of the first harmonic of
the alternative current formed by the circulating electron
space-charge in the magnetron, and
␸
eis the phase shift be-
tween the J1and the electric field b2of the e/m wave that is
realized when one obtains amplification of the e/m wave in
the magnetron. Let us assume that
␸
e⬎0. At the cross-
section S2共see Fig. 1兲the value of phase
␸
echanges on ⌬
␺
e
which can be determined as
⌬
␺
e共m,n兲= arctan ⫾ka
ˆ3共m−1,n−1兲cos共
␺
ˆ兲
冑共1−k2兲a
ˆ1
2共m−1,n−1兲+k2a
ˆ3
2共m−1,n−1兲.共10兲
Here a
ˆi共m,n兲is the average value of the superposition of e/m
waves ka3and a1, and
␺
ˆis the phase difference between
these e/m waves. Here let us note that the value
␸
˜
e=
␸
e
+⌬
␺
ecan change the sign as a result of the e/m waves’
superposition. In this case, the electron space-charge will be
displaced with respect to the e/m wave a1to the phase which
would cause electron repulsion. This will cause the electron
spoke to disappear and, respectively, to termination of the
microwave generation in the magnetron. The results of the
simulations for Im
␯
=0 and different values of
␺
resulting in
different
␸
˜
eare shown in Fig. 8.
In Ref. 11, the results of the numerical simulations of the
phase difference between the rotating electron spoke space-
charge and e/m wave in the magnetron versus the value of
the electron space-charge and radial component of the e/m
wave electric field are presented. It was shown that the mi-
crowave generation is realized when this phase difference is
in the range 9–32°. Our simulations showed 共see Fig. 9兲that
the value of the phase difference ⌬
␺
ecan exceed this range.
Here let us note that in simulations the maximal values of
⌬
␺
ewere obtained at 40 and 65 ns with respect to the be-
ginning of the microwave generation.
Thus, the value of ⌬
␺
e, which depends on the resonator
phase length
␺
, can lead to the change in the sign of the
phase difference between the rotating electron spoke space-
charge and the e/m wave in the magnetron. This leads to an
unstable mode of microwave generation and, respectively,
one can obtain quenching of the microwave generation.
V. SUMMARY
Simplified numerical simulations of the operation of a
magnetron coupled with an external resonator without spe-
cial measures for decoupling the magnetron and resonator
allows one to make the following conclusions:
共1兲An increase in the amplitude of the e/m wave stored in
the resonator is accompanied by an amplification in the
amplitude of the e/m wave in the magnetron. This means
an increase in the efficiency of the magnetron operation,
i.e., an increase in the electron beam energy transfer to
the microwaves, which agrees with the experimental re-
sults described in Ref. 8.
共2兲It was shown that the superposition of the e/m waves of
the magnetron and resonator may lead to a significant
change in the phase difference between the magnetron
e/m wave and electron spoke space-charge. This could
be a reason of the disappearance of the electron spoke
and, respectively, in the termination of the microwave
generation, which was obtained in recent experiments
described in Ref. 8.
ACKNOWLEDGMENTS
The authors are deeply indebted to Dr. A. Shlapakovsky
and Dr. Yu. Bliokh for fruitful discussions and comments.
The authors also thank Mr. Y. Hadas and T. Kweller for help
in simulations and comparison with experimental data. This
FIG. 9. Maximal values of the phase difference ⌬
␺
evs phase ⌿at time
delays with respect to the beginning of the microwave generation, 共1兲40 ns
and 共2兲65 ns.
074501-6 A. Sayapin and Y. E. Krasik J. Appl. Phys. 107, 074501 共2010兲

research was supported by the Center for Absorption in Sci-
ence, Ministry of Immigrant Absorption, State of Israel and
Technion under Grant No. 2010209.
1High Power Microwave Sources, edited by V. L. Granatstein and I. A.
Alexeff 共Artech House, Norwood, MA, 1987兲.
2R. J. Barker and E. Schamiloglu, High-Power Microwave Sources and
Technologies 共IEEE, New York, 2001兲.
3J. Benford, J. A. Swegle, and E. Schamiloglu, High Power Microwaves,
2nd ed. 共Taylor & Francis, New York, 2007兲.
4S. H. Gold and G. S. Nusinovich, Rev. Sci. Instrum. 68, 3945 共1997兲.
5J. T. Mayhan, R. L. Fante, R. O’Keefe, R. Elkin, J. Klugerman, and J. Yos,
J. Appl. Phys. 42, 5362 共1971兲.
6A. N. Didenko and Y. G. Yushkov, Powerful Microwave Nanosecond
Pulses 共Energoatimizdat, Moscow, 1984兲.
7A. N. Didenko, I. I. Vintizenko, A. I. Mashchenko, A. I. Ryabchikov, G. P.
Fomenko, P. Y. Chumerin, and Y. G. Yushkov, Dokl. Akad. Nauk 44,344
共1999兲.
8A. Sayapin, Y. Hadas, and Y. E. Krasik, Appl. Phys. Lett. 95, 074101
共2009兲.
9L. J. Milosevic and R. Vutey, IEEE Trans. Microwave Theory Tech. 6,136
共1958兲.
10Wang Yuan-ling, IEEE Trans. Nucl. Sci. 30, 3024 共1983兲.
11K. M. Basravi, T. I. Frolova, and G. I. Churiumov, Visnyk of Sumy State
University Ser. Physics, Mathematics and Mechanics 1,103共2007兲.
074501-7 A. Sayapin and Y. E. Krasik J. Appl. Phys. 107, 074501 共2010兲