IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 3, JUNE 2004957
Comparison of Multistage Gyroamplifiers Operating
in the Frequency-Multiplication Regime With
Gyroamplifiers Operating at a Given
Gregory S. Nusinovich, Fellow, IEEE, Oleksandr V. Sinitsyn, J. Rodgers,
Thomas M. Antonsen, Jr., Senior Member, IEEE, Victor L. Granatstein, Life Fellow, IEEE, and N. C. Luhmann, Jr.
Abstract—The operation of gyrodevices at cyclotron harmonics
is very attractive because of the possibility to reduce the magnetic
field requirement by
number. In recent years, two methods of harmonic operation in
multistage gyroamplifiers have been actively studied: operation at
a given harmonic in all stages and operation in the frequency-mul-
tiplying regime when the input stage operates at a lower harmonic
than the output one. The present paper is aimed at making a com-
parative analysis of these two schemes of operation. To do this, a
simple analytical method is developed, which allows one to qual-
itatively describe saturation effects in both schemes and compare
such performance characteristics as the efficiency, gain, and band-
pros and cons of both schemes.
is the cyclotron harmonic
Index Terms—Cyclotron harmonics, frequency multiplication,
in gyrodevices the magnetic field required for obtaining the cy-
clotron radiation at a given frequency is inversely proportional
to the cyclotron resonance harmonic number, , the gyrotron
operation at harmonics
time. In multistage gyroamplifiers, two concepts of harmonic
operation are known: 1) regime of frequency multiplication 
(for instance, operation of theinput stage at thefundamental cy-
, and the output stage operation at the
) and 2) operation of all stages at the same
. Both concepts have been studied
for a long time. However, so far, no comparative discussion of
their potentials and comparative analysis of these two concepts
have been carried out. Such a comparative analysis is done in
the present paper.
has been studied for a long
Manuscript received September 23, 2003; revised January 23, 2004. This
work was supported by the Multidisciplinary University Research Initiative on
Innovative Vacuum Electronics sponsored by the Air Force Office of Scientific
G. S. Nusinovich, O. V. Sinitsyn, J. Rodgers, T. M. Antonsen, Jr., and V. L.
Granatstein are with the Institute for Research in Electronics and Applied
Physics, University of Maryland, College Park, MD 20742-3511 USA (e-mail:
N. C. Luhmann, Jr. is with the University of California, Davis, CA 95616
Digital Object Identifier 10.1109/TPS.2004.827590
Before starting to describe its content, it makes sense to
discuss possible potential applications of both schemes and
their “pros and cons.” Clearly, the frequency multiplication
is a purely nonlinear process, occurring due to the nonlinear
properties of an electron beam, which being modulated at
a given frequency generates in the electron current density,
in addition to this frequency, also its harmonics. Therefore,
frequency-multiplying gyroamplifiers cannot be used in com-
munication systems that require a high degree of linearity. [For
example, if the carrier is simultaneously modulated by several
channels as in frequency-division multiple-access systems
(FDMA)]. At the same time, however, such devices can be used
in various radar and accelerator applications (see, for instance,
corresponding chapters in ) as well as in single-channel
digital communication systems possibly using code-division
multiple access (CDMA).
An obvious advantage of frequency-multiplying gyroampli-
fiers is operation of the first stage at the fundamental cyclotron
resonance. In comparison with a similar stage operating at
higher harmonic, operation at the fundamental can be realized
at a lower order mode that improves the stability of operation
and simplifies the design and configuration of an input coupler.
Also, the electron beam is more strongly coupled to the
radio-frequency (RF) field at the fundamental resonance than
at harmonics. The difference in the coupling can be roughly
estimated based on the fact that, in accordance with the Schott
formula (see, e.g., [4, eq. (74.8)]), the intensity of cyclotron
radiation, when electron orbital velocities are reasonably small,
is proportional to
normalized to the speed of light.) Therefore, one can use in
frequency-multiplying devices lower power drivers, which
also can operate at lower frequencies. This is a very important
advantage, especially at millimeter (and shorter) wavelengths,
where the availability of drivers of high enough power is a
Since the use of solid-state drivers allows the most compact
solution, it makes sense to briefly discuss the present state-
of-the-art in solid-state sources in the range of the wavelength
of our interest. As shown in , the maximum power of such
sources ranges from 10 to 20 W in the X-band to about 1 W
in the W-band and then falls rapidly at higher frequencies (less
is the electron orbital velocity
0093-3813/04$20.00 © 2004 IEEE
958 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 3, JUNE 2004
which allows one to use lower frequency for driving the device,
is here important. It should be noted that these power levels can
especially in the case of quasi-optical power combining , .
As an illustration of the advances, the work at Sanders can be
circuits (MMICs) arranged in a “tray” architecture resulted in
producing 35 W power at 61 GHz with a 60-dB small-signal
a vacuum tube as a W-band coupled-cavity traveling wave tube
(TWT) (millitron) provides 5-kW peak power with a 10% duty
with the bandwidth in excess of 1% .
After mentioning these more or less obvious arguments, let
us note that such issues as the preference of one of the dis-
cussed above schemes of gyroamplifiers with regard to the ef-
ficiency, gain, bandwidth, and other performance characteris-
tics is not obvious. Just an analysis of these issues is in the
focus of the present paper. The paper is organized as follows.
Section II briefly describes a general formalism, which we use
in our analysis. Section III contains the results of the analysis
of both schemes of interest for several multistage gyroampli-
fiers: two-cavity gyroklystrons, three-cavity gyroklystrons, and
two-stage gyro-TWTs. Also, some results of simulations for an
ultrahigh-gain gyro-TWT with distributed losses operating ei-
tion IV, we discuss the results obtained and consider a specific
example illustrating the application of our formalism to pos-
sible experimental conditions. Finally, Section V summarizes
II. GENERAL FORMALISM
Our consideration is based on the general formalism devel-
oped for gyrodevices a long time ago . Recently, the theory
of gyroklystrons based on this formalism was analyzed in de-
tail in  and . The derivation of corresponding equations
describing the gyro-TWT and some results of the studies of
gyro-TWTs operating at cyclotron harmonics are given in .
Since the equations have been described in detail in those ref-
erences, here we will only outline the issues in this formalism,
was dictated by the fact that it is very difficult, if not impos-
sible, to draw any conclusion just from comparison of different
designs, while we will attempt to make these conclusions with
the use of the general formalism.
A. Gyroklystron Formalism
The operation of the gyroklystron can be described by the set
of equations that determine the electron motion in the cavities
and drift regions, and equations that describe the field excita-
tion in the cavities. Consideration of electron motion is greatly
simplified in thecase of a so-called point gap model also known
in linear-beam klystrons. (Originally, Yulpatov proposed to use
this model for studying the gyroklystrons ; later, Caplan
 used it for analyzing the small-signal gain of a five-cavity
gyroklystron.) This model represents a gyroklystron circuit as
a set of short cavities separated by long drift sections. When
the cavities are short enough, it is possible to neglect there the
effects associated with electron orbital bunching, and hence to
consider there only electron energy modulation by the RF field.
electron cyclotron frequency due to the energy modulation pro-
ceeds in the long drift regions. In the framework of this model,
the equations for electron motion can be integrated analytically
even in large-signal regimes.
The field excitation in the cavities can be described by the
balance equations (assuming that we consider the stationary op-
eration of a device). For the input cavity, the balance equation
as , 
respect to the RF field in the first cavity. (In the framework
of the point-gap model, it can be calculated analytically.) The
first and second terms in the denominator describe the beam
loading and frequency pulling effects, respectively. In the latter,
is the normalized detuning of the
with respect to the cold-cavity frequency
. In (1), is the normalized intensity of the field excited in
the input cavity by the drive signal. This intensity relates to the
power of the driver,
, as 
is the electron beam susceptibility with
cavity having the loaded
is the beam power associated with the electron gyra-
andare the beam voltage and current, respectively,
is the initial orbital velocity of electrons normalized to the
speed of light, and
energy normalized to the rest energy. (The derivation of (2) is
given in .)
In (1) and (2), we used the normalized beam current param-
eter, which for cylindrical cavities operating at
is the coupling or externalfactor of the input
is the initial electron
is the cyclotron resonance harmonic number,
is the normalized axial coordinate, the function
describes the axial distribution of the cavity field,
is the eigenvalue of the operating mode in a cavity of a
, is the radius of electron guiding centers in a thin
annular electron beam, and
is the Bessel function of the first
The most important for our consideration is the fact that this
normalized beam current parameter is proportional to
This means that, for all other parameters being equal, the nor-
times smaller than for operation at the funda-
mental cyclotron resonance. So, in addition to the issue of the
availability of high-power drivers at different frequencies, we
NUSINOVICH et al.: COMPARISON OF MULTISTAGE GYROAMPLIFIERS OPERATING IN THE FREQUENCY-MULTIPLICATION REGIME959
get here one more factor, which weakens the RF field amplitude
in the case of operation at harmonics.
Then, the integration of equations for electron motion in the
drift sections and in other cavities yields the expression for the
susceptibility as the function of the bunching parameter,
length of the drift section. Note that in frequency-multiplying
devices, in order to have the bunches in all beamlets in the
same phase with respect to the RF field of the output stage,
the azimuthal indices of operating modes and the cyclotron har-
monic numbers should obey the phase synchronization con-
, . In the case of multicavity
gyroklystrons, there are more than one bunching parameters,
which account for the bunching caused by the modulation in
corresponding cavities. Then, the efficiency can be calculated
is the normalized
where the orbital efficiency relates to the imaginary part of the
susceptibility of the beam with respect to the output cavity field
As was mentioned above, in the framework of the point-gap
model this susceptibility can be calculated analytically. Corre-
spondingly, one can easily find that the maximum orbital effi-
ciency of the two-cavity gyroklystron is equal to
(In the case of a larger number of cavities, the expression for
the efficiency is more complicated.) In (6),
of operation at the same harmonic in all stages and larger than
one in frequency-multiplying devices.
The gain, in accordance with  and , can be presented
as a superposition of two terms
depend on the signal frequency variation. For the two-cavity
gyroklystron, it is equal to
is the constant part of the gain, which does not
of the gain, which for the two-cavity gyroklystron is equal to
is the frequency-dependent variable part
sence of bunching effects, which for the point-gap model is
(see ), and
parameters now should be given for the second cavity. Also,
is the bunching parameter independent on the
beam loading and frequency pulling effects, while the previ-
ously introduced bunching parameter
account. As follows from (1), the parameters are related as
is the susceptibility of the second cavity in the ab-
is determined by (3), where all
takes these effects into
Equations (8) and (9) are written for the large-signal regime of
operation. In the small-signal regimes, the bunching parameter
is small, and therefore the Bessel functions describing the sat-
uration effects can be presented in a simple polynomial form,
fication of these equations (see  and ). These equations
become especially simple when the normalized beam current
parameter is small enough, so one can neglect the beam loading
and frequency pulling effects in (1) and below. For our analysis,
it is important that, afterthe polynomial expansionof the Bessel
function, the term in the figure brackets in (9) becomes propor-
, i.e., in the case of small bunching parame-
ters the gain of frequency-multiplying devices becomes much
smaller than that in the case of operation at a given harmonic.
This is, of course, obvious because the frequency multiplication
is a purely nonlinear effect.
, that causes a certain simpli-
B. Gyro-TWTs Formalism
There are also some analytical methods to analyze the op-
eration of multistage gyro-TWTs. The simplest one was pro-
posed in  and then used in  and in later publications.
This model of a gyro-TWT, in which there are input and output
waveguides separated by a sever, is based on the following as-
sumptions. First, we assume that the input waveguide operates
in the small-signal regime, which is valid for practically all
known devices. Then, the orbital electron bunching in the drift
region between the input and output waveguides causes the ap-
pearance of harmonics of the signal frequency in the electron
current density, whichis the nonlinearprocess thatcan be easily
analyzed analytically as in gyroklystrons. Finally, we assume
that the output waveguideis reasonably short, and hence, can be
considered in a specified current approximation. This approx-
imation implies that we consider the excitation of the output
waveguide by a given current, but neglect the effect of the ex-
cited wave on the electrons.
Without going into further description of this formalism, let
us present here the expression for the gain
second waveguide (this parameter is quite similar to that given
above by (3) for a cavity; see  for more details),
normalized amplitude of the wave at the entrance to the input
waveguide, which depends not only on the input power, but
is the normalized beam current parameter for the
960IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 3, JUNE 2004
also on the choice of mode and the resonance harmonic number
. As follows from this dependence,
in which the drive power should also be multiplied by the
coupling impedance of a beam to the wave, which is similar to
the last term in the right-hand side of (3), the operation in the
input waveguide at the fundamental increases the amplitude
of the excited wave at a given level of the drive power. This
increase becomes even larger when in the input waveguide
operating at the fundamental resonance a lower order mode
is used than in the case of the second harmonic operation.
The bunching parameter in (11) is proportional to the product
of the normalized length of the drift section to the function
describing the wave amplification in the input waveguide .
This function describing the wave amplification in the input
waveguide, in the framework of the small-signal theory, is lin-
early proportional to the input wave amplitude,
that, in the case of operation at the fundamental resonance in
the input waveguide, the saturation can be achieved at shorter
lengths of drift sections. Also important is the fact that in the
case of operation in lower order modes at the fundamental
resonance it is easier to provide the stability, since lesser
number of parasitic waves can be excited. (This statement,
of course, does not mean that one cannot provide stability of
the operation at the second cyclotron harmonic; see  for a
convincing example of the successful operation at the second
harmonic at the 200-kW level in the
The gain given by (11) can again be represented as a sum of
the constant and variable parts. In the small-signal regime, cor-
responding expressions can be properly simplified. Of course,
expanding the Bessel function in (11) as a polynomial, we get
thesameconclusion aboutthegaindependence onthebunching
parameter as we made above for gyroklystrons. So, in general,
for all frequency-multiplying devices, the same dependence of
the output power on the input power,
The Bessel function present in (11) also determines the nor-
malized intensity of the outgoing radiation
. This means
, is valid,
This intensity, in accordance with the energy conservation law,
determines the orbital efficiency of the device
which in the framework of the specified current approximation
is small. (As was shown in , this approximation is valid for
the normalized lengths of the output waveguide not exceeding
4 and 2 for the normalized beam current parameters
to 0.01 and 0.1, respectively. In longer waveguides, the effect of
the excited wave on the electron motion becomes significant.)
A. Two-Cavity Gyroklystron
The orbital efficiency of the two-cavity gyroklystron (GKL)
was givenabove by (6). As follows from this equation, the max-
imum efficiency of the frequency-doubling GKL with
the optimal drift section length is shorter in frequency-multiplying GKLs and
vice versa. The solid line corresponds to the case of equal coupling impedances
in input cavities of both schemes; the broken line illustrates the case when
the input cavity operates at the ??
- and ??
Regions of preferable operation in one of two schemes: below the lines
- mode in the 1-2 and 2-2
andis approximately equal to 0.12 and the corre-
sponding optimum value of the bunching parameter is equal to
. Similarly, the maximum orbital efficiency of the
GKL, in which both cavities operate at the second harmonic
is equal to 0.17 and
lowing, we will designate these schemes of GKLs by their har-
and“1-2 scheme,” the case
and so on.
So,atthefirst glance,the2-2 schemeoffersmuchhighereffi-
into account here. First, numerical analysis of such schemes of
GKLs with a finite length of output cavities carried out in 
shows that the maximum efficiencies of both schemes with an
closer: 0.60 and 0.65 for the 1-2 and 2-2 schemes, respectively.
Second, it is important to bear in mind that optimal values of
bunching parameters can correspond to different lengths of the
drift sections, because the amplitude of the RF field excited in
the input cavity, as follows from (1) through (3), depends on the
resonant cyclotron harmonic number. Clearly, it is preferable to
have a device in which the drift section length is shorter, be-
cause such a device is more compact and its operation is less
sensitive to electron velocity spread and fluctuations in oper-
ating parameters. In our case, the preference depends not only
on the harmonic number
, but also on the choice of the op-
erating mode. The results are shown in Fig. 1. Here, the solid
and dotted lines show the borders between the region of shorter
drift section length for a 1-2 scheme (below the line) and a 2-2
scheme (above the line). The solid line corresponds to the case
when in both schemes the same operating mode is chosen; the
dotted line corresponds to the case when the input cavity oper-
- or -modeinthe1-2and 2-2schemes,
respectively. (It is assumed that
. In the fol-
-factors of input cavities and
NUSINOVICH et al.: COMPARISON OF MULTISTAGE GYROAMPLIFIERS OPERATING IN THE FREQUENCY-MULTIPLICATION REGIME961
the drive power in both cases are the same.) This figure shows
erence depends on such parameters as the beam voltage and the
orbital-to-axial velocity ratio, which determine the electron or-
operates at the lowest order
preferable in a larger region of the parameters. (In accordance
with the above mention phase synchronization condition, when
with the azimuthal index equal to one, the output cavity, which
operates at the second harmonic, should utilize a mode with the
azimuthal index equal to two, e.g.,
Let us now compare the gain and bandwidth of the two
schemes. The constant part of the gain, which is given by (8),
can be the same in both schemes when the output cavities
operate at the same mode, the drift section length is the
same, and the beam loading in the input cavity is negligibly
small, so the difference in the factor
The variable gain given by (9) is, however, different. As we
already mentioned in the previous section, there is no sense
to discuss the small-signal gain in frequency-multiplying
devices, because this gain is negative. Therefore, it is better
to compare the variable gain in two schemes in the cases
when the bunching parameter is optimal for each of them.
The results are shown in Fig. 2. Here, (a) and (b) show the
variable gain for 1-2 and 2-2 schemes, respectively. The gain
is shown as the function of the normalized signal frequency
is the mean frequency of the cold-cavity
frequencies) for several values of the stagger-tuning parameter
-mode, this scheme becomes
there does not matter.
. For the 1-2 scheme,
describes the difference in
equal to 0.12, which is the optimal value of
shows that in the case of zero detuning, when the variable gain
So, the 2-2 scheme exhibits the gain higher by 6 dB in this case.
Results shown in Fig. 2 allow one also to analyze the effect of
stagger-tuning on the bandwidth enlargement at the expense of
the gain degradation. Correspondingresults are shown in Fig. 3,
where solid and dotted lines correspond to the 2-2 and 1-2
schemes, respectively. These results show that in both schemes
the stagger tuning allows one to increase the bandwidth in
practically the same fashion. In comparison with the case of no
stagger tuning, the maximum increase in bandwidth due to the
stagger tuning is 4.83  and 5.3  times in schemes with
and, respectively. However, this bandwidth
enlargement corresponds to the gain degradation by 16.7 dB
and about 17 dB
gain and bandwidth has been analyzed in more detail in 
-factors of cavities, was taken to be
for this scheme
. The tradeoff between the
B. Three-Cavity Gyroklystron
The bunching processes in multicavity gyroklystrons, of
course, are more complicated than those in the two-cavity
device considered above. These processes can be characterized
(a) 1-2 scheme with the optimum bunching parameter ?
(b) 2-2 scheme with ?
Modification of the large-signal variable gain with the stagger tuning:
? ????, and
Fig. 3.Gain degradation and bandwidth enlargement with stagger tuning.
962IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 3, JUNE 2004
by a so-called “bunching” efficiency,
orbital efficiency introduced above as
is the harmonic number in the output cavity. This bunching
efficiency is determined by the intensity of the corresponding
harmonic in the electron current density exciting the output
. Such harmonics in three- and four-cavity GKLs
and their effect on the efficiency, gain, and bandwidth of GKLs
were studied in detail in , and –. Therefore, here
we will only outline the issues that are the most important for
our comparative analysis.
As shown in , the bunching efficiency in 1-2-2 and
2-2-2 schemes can exceed 0.6 (so, the orbital efficiency can
be higher than 0.3 in both cases). The 1-1-2 scheme yields a
lower bunching efficiency exceeding 0.4 only. It is interesting
to note that in the 1-2-2 scheme the efficiency is maximal
when the second cavity is located closer to the input one,
while in the 2-2-2 scheme it should be located closer to the
output cavity. (This conclusion is the same as one done in
 for a three-cavity GKL, in which all cavities operate at
the fundamental resonance.) The optimal values of the first
bunching parameter for 1-2-2 and 2-2-2 schemes are about 2.5
and 1.5, respectively. The ratio of these values is approximately
the same as the ratio of optimal bunching parameters for
two-cavity 1-2 and 2-2 schemes considered above. Therefore,
the results presented above (in particular, Fig. 1) are valid for
the present consideration as well. The optimal values of the
second bunching parameter for different schemes are different
. These values can be realized by adjusting not only the
second drift section length, but also the
cavity. (The latter, however, affects the bandwidth.)
The gain of three-cavity GKLs can be analyzed in the same
manner as in two-cavity devices. Since corresponding expres-
sions for the gain now are more complicated, it makes sense
to restrict our consideration by the case of operation close to
the small-signal one, when the bunching parameter is small,
and hence, the Bessel functions can be represented by poly-
nomials. Corresponding expressions readily show that the op-
gain, is 2/3 and 1/2 of the total drift length for 1-2-2 and 2-2-2
schemes, respectively. For this choice of the second cavity po-
sition and negligibly small beam loading in the input cavity, the
constant part of the gain in the 1-2-2 scheme is smaller than that
in the 2-2-2 scheme by
that the variable gain in frequency-multiplying GKLs contains
also an additional term, which describes the dependence on the
[cf. (9)]. At small
the difference in gains for 1-2-2 and 2-2-2 schemes even larger;
however, at large
’s, this term can increase the gain.
The bandwidth of these two schemes can be analyzed using
the expressions for the variable part of the gain, which are quite
similar, but more complicated than (9). Some results, neverthe-
ysis, for instance, was done for a 1-2-2 scheme with two last
cavities having equal
-factors, equal values of the normalized
beam current parameter
cold-cavity frequencies symmetrically situated with respect to
cific case the maximum bandwidth can be realized when the pa-
, which relates to the
-factor of the second
’s, this term makes
. It was shown that in this spe-
the stagger tuning parameter
is equal to 7.9. In the case of such a maximal stagger tuning the
bandwidth expressed in terms of the detuning
32-dB gain degradation.
Theseresults canbe comparedwith similarresultsfor a2-2-2
scheme, which in terms of normalized parameters is identical
to the 1-1-1 scheme studied in . In this case, the optimal
is 0.25 and the maximum bandwidth corresponds
to the stagger-tuning parameter equal to 6.066. This stagger
tuning increases the normalized bandwidth from 1.2 (when
) to 17.0, i.e., by more than 14 times, at the expense of
the gain degradation by about 36 dB. So, in this sense both
schemes look quite similar. Of course, the gain-bandwidth
product, which is often treated as a figure-of-merit of various
amplifiers, corresponds to smaller stagger-tunings than those
that yield the largest bandwidth. (This issue was discussed
elsewhere , .)
is equal to 0.0504 and
C. Two-Stage Gyro-TWT
Comparison of two concepts of gyro-TWTs leads to the con-
clusions that are quite similar to those derived for two-cavity
GKLs above. The constant part of the gain in the both fre-
quency-multiplying gyro-TWT and two-stage gyro-TWT oper-
ating at the second harmonic in both stages is the same. The
ized by the ratio of the Bessel function of the order
to the input wave amplitude. Therefore, the gain degradation
caused by the variable gain is smaller in the case of operation at
a given harmonic in all stages. Also, at small bunching param-
eters, the operation at a given harmonic
gain; however, the saturation can be realized faster in the case
of frequency-multiplying devices.
In addition to the comparison based on the use of a very
simplified approach described in Section II-B, we have also
carried out some numerical studies with the use of nonlinear
theory. It was considered a two-stage gyro-TWT experimen-
tally studied at the University of Maryland. We assumed that
the beam parameters such as the beam voltage and current, and
the orbital-to-axial velocity ratio are fixed: 60 kV, 5 A, and 1.0,
respectively. We also assumed that the lengths of all sections
and their transverse dimensions are the same as in the experi-
ment. Then, we did simulations for three cases: 1) operation in
both stages at the fundamental cyclotron resonance (operating
in both stages); 2) 1-2 scheme with the op-
respectively; and 3) 2-2 scheme where the operating mode is
in both stages.
Some results of the simulations are shown in Figs. 4–6.
Fig. 4 shows the case of operation at the fundamental in both
stages. In Fig. 4(a), where vertical lines separate the regions
of the first waveguide, drift section and output waveguide, the
axial dependencies of the output power are shown for several
values of the cyclotron resonance detuning in the output stage
yields a higher
in the first and second stage,
NUSINOVICH et al.: COMPARISON OF MULTISTAGE GYROAMPLIFIERS OPERATING IN THE FREQUENCY-MULTIPLICATION REGIME963
operating at the fundamental harmonic in both stages for several values of the
cyclotron resonance detuning in the output stage. (b) Dependence of the output
power on the normalized cyclotron resonance detuning for 50- and 100-W input
(a) Axial dependence of the output power in a two-stage gyro-TWT
; this detuning in the
input stage was taken equal to zero because this corresponds
to the maximum growth rate of the wave. (The variation in the
detuning can be realized by a weak tapering of the external
magnetic field.) As one can see, in this case the length of the
output section is long enough to reach saturation when the input
power equals 100 W. Resulting dependencies of the output
power on the detuning in the output stage are shown in Fig. 4(b)
for 50- and 100-W input power. For 50-W input power, the
output waveguide is not long enough to reach the maximum
power and efficiency. Therefore, the maximum power shown in
Fig. 4(b) for this case is lower than for 100 W. Results shown
in Fig. 4(b) indicate that this device can operate with the gain
exceeding 30 dB.
Fig. 5 shows some results of simulations for the frequency-
doubling gyro-TWT. Here figures (a) and (b) correspond to 50-
gyro-TWT for several values of the cyclotron resonance detuning in the output
stage for (a) 50 W and (b) 100 W levels of the input power.
Axial dependence of the output power in a frequency-doubling
and 100-W input power, respectively. In this case, as follows
of 100-W input power it is necessary to increase the length of
the output stage by about 60%. This will result in about 40-kW
output power, while in the1-1 scheme themaximum output was
higher than 100 kW for the same 100-W input power, as shown
in Fig. 4(b).
Finally, Fig. 6 shows some results of simulations for the 2-2
scheme. In thiscase, as follows from Fig.6(a), to reach the peak
a device to variation of the external magnetic field. As one can
see, the efficient operation is possible in the range of magnetic
1% or less of the nominal value. As follows from
Fig. 6(b), the bandwidth corresponding to this nominal value of
the magnetic field,
, is about 1.7%.
964IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 3, JUNE 2004
power on the operating frequency in this tube for several values of the external
magnetic field (lengths of the input waveguide, the drift section, and the output
waveguide are equal to 15.2, 9.0, and 14.6 cm, respectively).
(a) Axial dependence of the output power in a two-stage gyro-TWT
D. Ultrahigh-Gain Gyro-TWT
Regardless of the chosen operating harmonic, the problem
of obtaining high-frequency drive sources capable of saturating
the gyroamplifier in question can be relaxed somewhat if the
gain is high enough. Recently, single-stage gyro-TWTs with
distributed loss have been investigated as a means of achieving
gains as high as 70 dB while maintaining operation free of
oscillations from reflective and absolute instabilities .
Conceivably, systems with such high gain gyroamplifiers could
achieve output powers as high as several hundred kilowatts in
W-band using low-noise solid-state drivers. In view of the ob-
vious advantages of such configurations, not the least of which
is saving the cost of an intermediate vacuum tube amplifier,
there are other issues that require attention. Depending on the
application, signal integrity issues such as linearity, noise, and
harmonic gyro-TWT configurations with distributed loss applied to the first
15 cm of the interaction circuit followed by 7 cm of high conductivity wall.
The arrows indicate the location of the beam guiding center radius.
Schematic of MAGY model for (a) fundamental and (b) second
cost reduction. It seemed prudent to consider whether similarly
high gains could be achieved in distributed loss harmonic
gyro-TWTs as well.
A numerical study using the Maryland Gyrotron code
(MAGY)  was conducted to compare the performance of
gyro-TWTs with distributed loss operating at the fundamental
and second harmonic of the cyclotron frequency. Fig. 7 shows
a schematic of the two models we considered. The basic
parameters governing synchronism between the fast cyclotron
and EM waves were fixed by comparing a
circuit with radius of 5.5 mm with a
circuit with radius 10.07 mm, both with a length of 22 cm.
Thus, the cutoff frequencies and dispersion characteristics for
the operating modes in both cases are the same, and maximum
gainwas expectedarounda frequencyof35 GHz.Themagnetic
field strength for grazing intersection
could be determined
NUSINOVICH et al.: COMPARISON OF MULTISTAGE GYROAMPLIFIERS OPERATING IN THE FREQUENCY-MULTIPLICATION REGIME965
MAGY PARAMETERS FOR COMPARISON OF DISTRIBUTED LOSS GYRO-TWTS
ativistic mass factors and
waveguide. Equation (14) shows that this field depends on the
appliedvoltageandtheelectronorbital toaxial velocityratio.In
our simulations, we usedthe electronbeam voltageand electron
velocity pitch fixed at 80 kV and 1.0, respectively. The beam
current and distributed loss rate were left as free parameters and
optimized for each case for maximum gain. Table I summarizes
the model parameters, and Fig. 8 gives the results.
The results show that, while impressive fundamental gain
is realized, the compression slope has fairly dramatic conse-
quences on output phase. In applications where linearity is
important, the output power should be “backed-off” below
the peak value to minimize AM-PM conversion. We note
that neither model was comprehensively optimized for all
performance figures of merit; rather, the importance was direct
comparison. In the second harmonic case, 45 dBm of drive
power was needed for saturation while in the fundamental
harmonic case the device saturates at less than 20 dBm. This
value (45 dBm) is above the capability of solid-state sources
in the millimeter-wave region. The results should not be
interpreted to mean that a distributed-loss, second harmonic
gyro-TWT could not achieve higher gain and meet this crite-
rion. Higher perpendicular velocity (with bandwidth reduction)
and increased gain length may mitigate this shortcoming and
should be investigated further.
is the ratio of the unperturbed total to axial rel-
is the cutoff frequency of the
Let us illustrate our theoretical analysis by one more ex-
ample. Assume (as it was done in  for 1-1 and 1-2 schemes
of GKLs) that we have an electron beam, which was used in the
experiments with a W-band gyroklystron at NRL : 65-kV
beam voltage, 6-A beam current, orbital-to-axial velocity ratio
equal to 1.4. Let us consider the performance characteristics
of two-cavity GKLs driven by such a beam for the following
three cases: 1) 1-1 scheme with both cavities operating at the
-mode; 2) 1-2 scheme with the first cavity operating
at the fundamental at the same
cavity operating at the second cyclotron harmonic at the
-mode; and 3) 2-2 scheme with both cavities operating
-mode. In all cases, for the sake of simplicity, we
-mode and the second
versus drive power for (a) fundamental, and (b) second harmonic gyro-TWTs
with distributed loss placed in the interaction circuit to achieve high gain and
MAGY simulation results showing the output power, gain, and phase
will neglect the beam loading and frequency pulling effects.
This allows us to simplify the relation given by (1) between
the normalized field amplitude in the first cavity and the
amplitude of the field excited by a driver there. Then, in order
to estimate the relation between the latter amplitude and the
drive power given by (2), assume that the axial profile of the
input cavity is sinusoidal and the cavity length is equal to one
wavelength; also assume that a thin annular beam is positioned
in the peak of the coupling impedance to the operating mode,
-factor is equal to 100, and the cavity is critically
. For this set of parameters, (2)
section length can be found under assumption that for a given
drive power and the case of exact resonance between the
signal frequency and the cold-cavity frequency the bunching
parameterhas an optimal value. For instance, if we want tohave
an optimal bunching parameter,
the normalized length of the drift section should be equal to
that corresponds to
similar fashion, we can also evaluate the normalized parameters
. Then, the optimal drift
. In a
966 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 3, JUNE 2004
different schemes in the framework of the point-gap model.
Dependence of the output power and gain on the input power for
for two other schemes. In the case of a 1-2 scheme, it was
assumed that the beam position remains the same as for the
1-1 scheme; it was also assumed that, to provide the maximum
bandwidth in the case of the stagger-tuning, in accordance with
, we should have the
-factor of the second cavity about
2.9 times higher than that for the first one, i.e., it should be
. This value of the
cavities in the third case (2-2 scheme). Of course, the bunching
parameters were chosen to be optimal for each case.
on the drive power in all three schemes are shown in Fig. 9.
Since the normalized length of the drift section is rather long,
saturation effects in the schemes without frequency multiplica-
tion appear at input power level of about a few watts. Therefore,
the gain curves for 1-1 and 2-2 schemes shown in Fig. 9(b) that
W have almost linear slops demonstrating
the saturation. Of course, the 1-1 scheme has a superior per-
formance; it also offers a larger bandwidth, since in this case
-factors are lower. So, the main obstacles for the use of
-factor was also used for both
this simple scheme could be either the necessity to create strong
magnetic fields with the use of large solenoids increasing the
mass and volume of a microwave system, or the tendency to
operate at high power levels, for which it is difficult, if not im-
possible, to realize optimal operating conditions in the case of
low-order modes operating at the fundamental resonance. Com-
parison of two latter schemes shows that the 2-2 scheme offers
higher gain and output power than the 1-2 scheme. To the great
extent, this is the result of describing the saturation effects by
the Bessel functions corresponding to the account of the bal-
listic bunching in the drift sections of the point-gap model only.
To consider a more realistic case, we replaced in the formula
for the orbital efficiency given by (6) these functions by the
corresponding polynomials of the bunching parameter
and determined the coefficients
the conditions that approximations given by (15) yield for the
optimal values of bunching parameters (
) the orbital efficiencies equal to their maximum
valuesfound in  (
powerare shownas functionof theratioof thedrivepowertoits
optimum value (100 W). Now the performance characteristics
of the 1-2 and 2-2 schemes are closer, but still the 2-2 scheme
exhibits a higher gain; for instance, for the optimal drive power
the difference in the gain is a little smaller than 5 dB in favor
of the 2-2 scheme. Recall that our analysis done in Section III
showed that the tradeoff between the gain and the bandwidth
due to the stagger tuning in both schemes yields approximately
the same results.
So far, we focused our consideration on the analysis of sev-
width) only. For many applications, various signal distortion ef-
fects are also important. Some of them can be caused by the
effect of the input signal amplitude modulation on the ampli-
tude and phase of the output radiation. Although the effect of
the input signal modulation on the output phase (a so-called
AM/PM conversion, see, e.g., ) cannot be studied in the
framework of the point-gap model , one can easily use this
model for estimating the effect of the input signal modulation
on the amplitude of the output radiation (AM/AM conversion).
In accordance with  and the formalism developed above,
this conversion for a point-gap model can be characterized by
As follows from this simple formula, at small values of the
bunching parameter (small-signal regime) the 2-2 scheme oper-
ates with 1 dB/dB conversion as any linear amplifier, while the
1-2 scheme yields 2 dB/dB conversion, which is caused by the
cific cases of AM signals. Of course, the maximum efficiency
operation corresponds to
noise ratio. As known, the noise of any amplifier is predom-
inantly determined by the noise generated in the input stage
andfor each scheme from
and ). Corre-
in any scheme.
NUSINOVICH et al.: COMPARISON OF MULTISTAGE GYROAMPLIFIERS OPERATING IN THE FREQUENCY-MULTIPLICATION REGIME967
different schemes in the case of corrected polynomial approximation for the
Dependence of the output power and gain on the input power for
by a driver and by an electron beam. (This noise in the fun-
damental harmonic GKLs was studied theoretically and exper-
imentally in  and , respectively.) Then, this noise, to-
gether with a signal, is amplified in other stages of the device.
In the case of operation at a given harmonic, this process has a
linear nature, but it is a nonlinear process in frequency-multi-
plying devices. This nonlinearity weakens the role of noise with
respect to the signal. Therefore, the signal-to-noise ratio in fre-
quency-multiplying devices can be higher than in the devices
operating without multiplication. This issue was recently ana-
lyzed in  in detail.
and the results obtained are applicable for large-orbit gyrode-
vices (LOGs) . As was pointed out in , the theory of
these devices can be based on the formalism developed for con-
ventional, small-orbit gyrodevices. The peculiarity of LOGs is
the necessity to consider the coupling impedance present in the
definition of the normalized beam current parameter given by
immediately yields the rule of mode selection in LOGs ,
, which is the only case of nonzero coupling impedance
for electrons with
. Note that, in order to enhance radia-
tion at high cyclotron harmonics, instead of smooth-wall cylin-
drical cavities/waveguides, various magnetron-type microwave
circuits are often used in LOGs , . In such cases, not
only the numerator, but also denominator in (3) should be prop-
A simple theory is developed that allows one to make
a comparative analysis of various schemes of multistage
gyroamplifiers. The results of the analysis done for the fre-
quency-multiplying devices and devices, which operate at a
characteristics considered are better for the devices operating
at a given harmonic (gain and efficiency), while others (such
as the bandwidth) can be better in frequency-multiplying
gyroklystrons where the operation of the input stage at the
fundamental resonance allows one to use the low-
Without going into details of the analysis performed, it can be
stated that the preference of one of two schemes considered
stronger depends on the practical issues (such as a specific
application, availability of drivers at different frequencies,
ability to provide a stable operation at higher order mode in the
input stage, etc.) than on the details of the saturation effects
and possibilities to increase the bandwidth and gain-bandwidth
product by using the stagger tuning (the latter is essentially
the same in both schemes). Thus, it makes sense to continue
analysis of both schemes.
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Gregory S. Nusinovich (SM’92–F’00) received the
B.Sc., M.Sc., and Ph.D. degrees from Gorky State
In 1968, he joined the Gorky Radiophysical
Research Institute, Gorky. From 1977 to 1990,
he was a Senior Research Scientist and Head of
the Research Group at the Institute of Applied
Physics, the Academy of Sciences of the U.S.S.R.,
Gorky. From 1968 to 1990, his scientific interests
included developing high-power millimeter- and
submillimeter-wave gyrotrons. He was a Member of the Scientific Council on
Physical Electronics of the Academy of Sciences of the U.S.S.R. In 1991, he
immigrated to the U.S. and joined the Research Staff at the Institute for Plasma
Research (presently, the Institute for Research in Electronics and Applied
Physics), University of Maryland, College Park. Since 1991, he has also served
as a Consultant to the Science Applications International Corporation, McLean,
VA, the Physical Sciences Corporation, Alexandria, VA, and Omega-P, Inc.,
New Haven, CT. He is the author of Introduction to the Physics of Gyrotrons
(Baltimore, MD: The Johns Hopkins University Press, 2004). He has authored
and coauthored more than 150 papers published in refereed journals. His
current research interests include the study of high-power electromagnetic
radiation from various types of microwave sources.
Dr. Nusinovich is a Member of the Executive Committee of the Plasma Sci-
ence and Application Committee of the IEEE Nuclear and Plasma Sciences
Society. He is a Fellow of the American Physical Society (APS). In 1996 and
1999, he was a Guest Editor of the Special Issues of the IEEE TRANSACTIONS
ON PLASMA SCIENCE on High-Power Microwave Generation and on Cyclotron
Resonance Masers and Gyrotrons, respectively. Presently, he is an Associate
Editor of the IEEE TRANSACTIONS ON PLASMA SCIENCE.
Oleksandr V. Sinitsyn was born in Kharkiv,
Ukraine, in 1978. He received the B.S. degree in
radio physics from V. Karazin Kharkiv National
University, Kharkiv, Ukraine, in 2000 and the M.S.
degree in electrical engineering from the University
of Maryland, College Park, in 2002. He is currently
working toward the Ph.D. degree in electrical and
computer engineering at the University of Maryland.
His research interests include theory and design of
high-power microwave sources.
J. Rodgers, photograph and biography not available at the time of publication.
NUSINOVICH et al.: COMPARISON OF MULTISTAGE GYROAMPLIFIERS OPERATING IN THE FREQUENCY-MULTIPLICATION REGIME969 Download full-text
Thomas M. Antonsen, Jr. (M’82–SM’96) was born
in Hackensack, NJ, in 1950. He received the B.S. de-
gree in electrical engineering in 1973, and his M.S.
and Ph.D. degrees in 1976 and 1977, all from Cor-
nell University, Ithaca, NY.
He was a National Research Council Post Doc-
toral Fellow at the Naval Research Laboratory, from
1976 to 1977, and a Research Scientist in the Re-
search Laboratory of Electronics at MIT, Cambridge,
from 1977 to 1980. In 1980, he moved to the Univer-
sity of Maryland, College Park, where he joined the
faculty of the Departments of Electrical Engineering and Physics in 1984. He
is currently a Professor of physics and electrical and computer engineering. He
has held visiting appointments at the Institute for Theoretical Physics (UCSB),
Physique Theorique, Ecole Polytechnique, Palaiseau, France. He served as the
Acting Director of the Institute for Plasma Research, University of Maryland
from 1998 to 2000. His research interests include the theory of magnetically
confined plasmas, the theory and design of high-power sources of coherent ra-
laser pulses and plasmas. He is the author and coauthor of over 200 journal ar-
ticles and co-author of the book Principles of Free-Electron Lasers.
Dr. Antonsen has served on the editorial board of Physical Review Letters,
The Physics of Fluids, and Comments on Plasma Physics. He was selected as
a Fellow of the Division of Plasma Physics of the American Physical Society
in 1986. In 1999, he was a co-recipient of the Robert L. Woods award for Ex-
cellence in Vacuum Electronics Technology, and in 2003 he received the IEEE
Plasma Science and Applications Award.
Victor L. Granatstein (S’59–M’64–SM’86–F’02–
LF’02) received the Ph.D. degree in electrical
engineering from Columbia University, New York,
After a year of postdoctoral work at Columbia, he
wasa ResearchScientist at BellTelephoneLaborato-
ries from 1964 to 1972, where he studied microwave
scattering from turbulent plasma. In 1972, he joined
the Naval Research Laboratory (NRL), Washington,
DC, as a Research Physicist, and from 1978 to 1983,
heservedasHead ofNRL’sHighPower Electromag-
netic Radiation Branch. In August 1983, he became a Professor in the Electrical
Engineering Department, University of Maryland, College Park. From 1988 to
1998, he was Director of the Institute for Plasma Research at the University of
Maryland. He spent semesters in both 1994 and 2003 as a Visiting Professor
at the University of Tel Aviv, Tel Aviv, Israel. He is currently leading studies
of high pulsed power gyroklystron amplifiers for driving high-gradient electron
accelerators, efficiency enhancement in high average power gyrotron oscilla-
tors for heating and stabilizing magnetic fusion plasmas, advanced concepts in
high-power, wideband millimeter-wave amplifiers for radar applications, and
the effect of high-power microwave pulses on electronic components and sys-
tems. He has co-authored more than 220 research papers in scientific journals
and has co-edited three books. He holds a number of patents on active and pas-
sive microwave devices.
Dr.GranatsteinisaFellowoftheAmericanPhysical Society.Hehas received
a number of major research awards including the E.O. Hulbert Annual Science
Dexter Conrad Award for Scientific Achievement (awarded by the Secretary of
the Navy in 1981), the IEEE Plasma Science and Applications Award in 1991,
and the Robert L. Woods Award for Excellence in Electronics Technology in
N. C. Luhmann, Jr., photograph and biography not available at the time of